[
    {
        "formal": "Set.Finite.biUnion ** \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Sort w \u03b3 : Type x \u03b9 : Type u_1 s : Set \u03b9 hs : Set.Finite s t : \u03b9 \u2192 Set \u03b1 ht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i) \u22a2 Set.Finite (\u22c3 i \u2208 s, t i) ** classical\n  cases hs\n  haveI := fintypeBiUnion s t fun i hi => (ht i hi).fintype\n  apply toFinite ** \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Sort w \u03b3 : Type x \u03b9 : Type u_1 s : Set \u03b9 hs : Set.Finite s t : \u03b9 \u2192 Set \u03b1 ht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i) \u22a2 Set.Finite (\u22c3 i \u2208 s, t i) ** cases hs ** case intro \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Sort w \u03b3 : Type x \u03b9 : Type u_1 s : Set \u03b9 t : \u03b9 \u2192 Set \u03b1 ht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i) a\u271d : Fintype \u2191s \u22a2 Set.Finite (\u22c3 i \u2208 s, t i) ** haveI := fintypeBiUnion s t fun i hi => (ht i hi).fintype ** case intro \u03b1 : Type u \u03b2 : Type v \u03b9\u271d : Sort w \u03b3 : Type x \u03b9 : Type u_1 s : Set \u03b9 t : \u03b9 \u2192 Set \u03b1 ht : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Finite (t i) a\u271d : Fintype \u2191s this : Fintype \u2191(\u22c3 x \u2208 s, t x) \u22a2 Set.Finite (\u22c3 i \u2208 s, t i) ** apply toFinite ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mul_upcrossingsBefore_le ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b \u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 k in Finset.range N, upcrossingStrat a b f N k \u03c9 * (f (k + 1) - f k) \u03c9 ** by_cases hN : N = 0 ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 \u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 k in Finset.range N, upcrossingStrat a b f N k \u03c9 * (f (k + 1) - f k) \u03c9 ** simp_rw [upcrossingStrat, Finset.sum_mul, \u2190\n  Set.indicator_mul_left _ _ (fun x \u21a6 (f (x + 1) - f x) \u03c9), Pi.one_apply, Pi.sub_apply, one_mul] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 \u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264     \u2211 x in Finset.range N,       \u2211 x_1 in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N x_1 \u03c9) (upperCrossingTime a b f N (x_1 + 1) \u03c9))           (fun a => f (a + 1) \u03c9 - f a \u03c9) x ** rw [Finset.sum_comm] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 \u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264     \u2211 y in Finset.range N,       \u2211 x in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N y \u03c9) (upperCrossingTime a b f N (y + 1) \u03c9))           (fun a => f (a + 1) \u03c9 - f a \u03c9) x ** simp_rw [h\u2081] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 h\u2082 :   \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a) \u2264     \u2211 k in Finset.range N,       (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9) \u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264     \u2211 x in Finset.range N,       (stoppedValue f (upperCrossingTime a b f N (x + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N x) \u03c9) ** refine' le_trans _ h\u2082 ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 h\u2082 :   \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a) \u2264     \u2211 k in Finset.range N,       (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9) \u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a) ** rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : N = 0 \u22a2 (b - a) * \u2191(upcrossingsBefore a b f N \u03c9) \u2264 \u2211 k in Finset.range N, upcrossingStrat a b f N k \u03c9 * (f (k + 1) - f k) \u03c9 ** simp [hN] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 \u22a2 \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 ** intro k ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 k : \u2115 \u22a2 \u2211 n in Finset.range N,       Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))         (fun m => f (m + 1) \u03c9 - f m \u03c9) n =     stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 ** rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i \u2208 Set.Ico\n  (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9)) (Finset.range N) =\n  Finset.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9)),\n  Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ,\n  Finset.sum_range_sub fun n => f n \u03c9, Finset.sum_range_sub fun n => f n \u03c9, neg_sub,\n  sub_add_sub_cancel] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 k : \u2115 \u22a2 f (upperCrossingTime a b f N (k + 1) \u03c9) \u03c9 - f (lowerCrossingTime a b f N k \u03c9) \u03c9 =     stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 ** rfl ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 k : \u2115 \u22a2 Finset.filter (fun i => i \u2208 Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))       (Finset.range N) =     Finset.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9) ** ext i ** case a \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 k i : \u2115 \u22a2 i \u2208       Finset.filter (fun i => i \u2208 Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))         (Finset.range N) \u2194     i \u2208 Finset.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9) ** simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico,\n  and_iff_right_iff_imp, and_imp] ** case a \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 k i : \u2115 \u22a2 lowerCrossingTime a b f N k \u03c9 \u2264 i \u2192 i < upperCrossingTime a b f N (k + 1) \u03c9 \u2192 i < N ** exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 \u22a2 \u2211 _k in Finset.range (upcrossingsBefore a b f N \u03c9), (b - a) \u2264     \u2211 k in Finset.range (upcrossingsBefore a b f N \u03c9),       (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9) ** refine' Finset.sum_le_sum fun i hi =>\n  le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab _ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 i : \u2115 hi : i \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) \u22a2 i < upcrossingsBefore a b f N \u03c9 ** rwa [Finset.mem_range] at hi ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 \u22a2 \u2211 k in Finset.range (upcrossingsBefore a b f N \u03c9),       (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9) \u2264     \u2211 k in Finset.range N,       (stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9) ** refine' Finset.sum_le_sum_of_subset_of_nonneg\n  (Finset.range_subset.2 (upcrossingsBefore_le f \u03c9 hab)) fun i _ hi => _ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 i : \u2115 x\u271d : i \u2208 Finset.range N hi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) \u22a2 0 \u2264 stoppedValue f (upperCrossingTime a b f N (i + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N i) \u03c9 ** by_cases hi' : i = upcrossingsBefore a b f N \u03c9 ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 i : \u2115 x\u271d : i \u2208 Finset.range N hi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) hi' : i = upcrossingsBefore a b f N \u03c9 \u22a2 0 \u2264 stoppedValue f (upperCrossingTime a b f N (i + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N i) \u03c9 ** subst hi' ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 x\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N hi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) \u22a2 0 \u2264     stoppedValue f (upperCrossingTime a b f N (upcrossingsBefore a b f N \u03c9 + 1)) \u03c9 -       stoppedValue f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9)) \u03c9 ** simp only [stoppedValue] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 x\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N hi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) \u22a2 0 \u2264     f (upperCrossingTime a b f N (upcrossingsBefore a b f N \u03c9 + 1) \u03c9) \u03c9 -       f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9 ** rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 x\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N hi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) \u22a2 0 \u2264 f N \u03c9 - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9 ** by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 x\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N hi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N \u22a2 0 \u2264 f N \u03c9 - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9 ** rw [heq, sub_self] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 x\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N hi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) heq : \u00aclowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N \u22a2 0 \u2264 f N \u03c9 - f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9 ** rw [sub_nonneg] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 x\u271d : upcrossingsBefore a b f N \u03c9 \u2208 Finset.range N hi : \u00acupcrossingsBefore a b f N \u03c9 \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) heq : \u00aclowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9 = N \u22a2 f (lowerCrossingTime a b f N (upcrossingsBefore a b f N \u03c9) \u03c9) \u03c9 \u2264 f N \u03c9 ** exact le_trans (stoppedValue_lowerCrossingTime heq) hf ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 i : \u2115 x\u271d : i \u2208 Finset.range N hi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) hi' : \u00aci = upcrossingsBefore a b f N \u03c9 \u22a2 0 \u2264 stoppedValue f (upperCrossingTime a b f N (i + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N i) \u03c9 ** rw [sub_eq_zero_of_upcrossingsBefore_lt hab] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 i : \u2115 x\u271d : i \u2208 Finset.range N hi : \u00aci \u2208 Finset.range (upcrossingsBefore a b f N \u03c9) hi' : \u00aci = upcrossingsBefore a b f N \u03c9 \u22a2 upcrossingsBefore a b f N \u03c9 < i ** rw [Finset.mem_range, not_lt] at hi ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : a \u2264 f N \u03c9 hab : a < b hN : \u00acN = 0 h\u2081 :   \u2200 (k : \u2115),     \u2211 n in Finset.range N,         Set.indicator (Set.Ico (lowerCrossingTime a b f N k \u03c9) (upperCrossingTime a b f N (k + 1) \u03c9))           (fun m => f (m + 1) \u03c9 - f m \u03c9) n =       stoppedValue f (upperCrossingTime a b f N (k + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N k) \u03c9 i : \u2115 x\u271d : i \u2208 Finset.range N hi : upcrossingsBefore a b f N \u03c9 \u2264 i hi' : \u00aci = upcrossingsBefore a b f N \u03c9 \u22a2 upcrossingsBefore a b f N \u03c9 < i ** exact lt_of_le_of_ne hi (Ne.symm hi') ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.mkMetric'.eq_iSup_nat ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r : \u211d\u22650\u221e \u03bc : OuterMeasure X s : Set X m : Set X \u2192 \u211d\u22650\u221e \u22a2 mkMetric' m = \u2a06 n, pre m (\u2191n)\u207b\u00b9 ** ext1 s ** case h \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r : \u211d\u22650\u221e \u03bc : OuterMeasure X s\u271d : Set X m : Set X \u2192 \u211d\u22650\u221e s : Set X \u22a2 \u2191(mkMetric' m) s = \u2191(\u2a06 n, pre m (\u2191n)\u207b\u00b9) s ** rw [iSup_apply] ** case h \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r : \u211d\u22650\u221e \u03bc : OuterMeasure X s\u271d : Set X m : Set X \u2192 \u211d\u22650\u221e s : Set X \u22a2 \u2191(mkMetric' m) s = \u2a06 i, \u2191(pre m (\u2191i)\u207b\u00b9) s ** refine' tendsto_nhds_unique (mkMetric'.tendsto_pre_nat m s)\n  (tendsto_atTop_iSup fun k l hkl => mkMetric'.mono_pre_nat m hkl s) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.stoppedProcess_eq' ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n : \u2115 this : Set.indicator {a | n \u2264 \u03c4 a} (u n) = Set.indicator {a | n + 1 \u2264 \u03c4 a} (u n) + Set.indicator {a | \u03c4 a = n} (u n) \u22a2 stoppedProcess u \u03c4 n =     Set.indicator {a | n + 1 \u2264 \u03c4 a} (u n) + \u2211 i in Finset.range (n + 1), Set.indicator {a | \u03c4 a = i} (u i) ** rw [stoppedProcess_eq, this, Finset.sum_range_succ_comm, \u2190 add_assoc] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n : \u2115 \u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) = Set.indicator {a | n + 1 \u2264 \u03c4 a} (u n) + Set.indicator {a | \u03c4 a = n} (u n) ** ext x ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n : \u2115 x : \u03a9 \u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) x = (Set.indicator {a | n + 1 \u2264 \u03c4 a} (u n) + Set.indicator {a | \u03c4 a = n} (u n)) x ** rw [add_comm, Pi.add_apply, \u2190 Set.indicator_union_of_not_mem_inter] ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n : \u2115 x : \u03a9 \u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) x = Set.indicator ({a | \u03c4 a = n} \u222a {a | n + 1 \u2264 \u03c4 a}) (u n) x ** simp_rw [@eq_comm _ _ n, @le_iff_eq_or_lt _ _ n, Nat.succ_le_iff] ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n : \u2115 x : \u03a9 \u22a2 Set.indicator {a | n = \u03c4 a \u2228 n < \u03c4 a} (u n) x = Set.indicator ({a | n = \u03c4 a} \u222a {a | n < \u03c4 a}) (u n) x ** rfl ** case h.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n : \u2115 x : \u03a9 \u22a2 \u00acx \u2208 {a | \u03c4 a = n} \u2229 {a | n + 1 \u2264 \u03c4 a} ** rintro \u27e8h\u2081, h\u2082\u27e9 ** case h.h.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n : \u2115 x : \u03a9 h\u2081 : x \u2208 {a | \u03c4 a = n} h\u2082 : x \u2208 {a | n + 1 \u2264 \u03c4 a} \u22a2 False ** exact (Nat.succ_le_iff.1 h\u2082).ne h\u2081.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.option_casesOn ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 o : \u03b1 \u2192 Option \u03b2 f : \u03b1 \u2192 \u03c3 g : \u03b1 \u2192 \u03b2 \u2192 \u03c3 ho : Primrec o hf : Primrec f hg : Primrec\u2082 g a : \u03b1 \u22a2 (Nat.casesOn (encode (o a)) (encode (f a)) fun b =>       Nat.pred (encode (Option.bind (decode (encode (a, b).1)) fun a => Option.map (g a) (decode b)))) =     encode (Option.casesOn (o a) (f a) (g a)) ** cases' o a with b <;> simp [encodek] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.funext_fin ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n : \u2115 p : MvPolynomial (Fin n) R h : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0 \u22a2 p = 0 ** induction' n with n ih ** case zero R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n : \u2115 p\u271d : MvPolynomial (Fin n) R h\u271d : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p\u271d = 0 p : MvPolynomial (Fin Nat.zero) R h : \u2200 (x : Fin Nat.zero \u2192 R), \u2191(eval x) p = 0 \u22a2 p = 0 ** apply (MvPolynomial.isEmptyRingEquiv R (Fin 0)).injective ** case zero.a R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n : \u2115 p\u271d : MvPolynomial (Fin n) R h\u271d : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p\u271d = 0 p : MvPolynomial (Fin Nat.zero) R h : \u2200 (x : Fin Nat.zero \u2192 R), \u2191(eval x) p = 0 \u22a2 \u2191(isEmptyRingEquiv R (Fin 0)) p = \u2191(isEmptyRingEquiv R (Fin 0)) 0 ** rw [RingEquiv.map_zero] ** case zero.a R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n : \u2115 p\u271d : MvPolynomial (Fin n) R h\u271d : \u2200 (x : Fin n \u2192 R), \u2191(eval x) p\u271d = 0 p : MvPolynomial (Fin Nat.zero) R h : \u2200 (x : Fin Nat.zero \u2192 R), \u2191(eval x) p = 0 \u22a2 \u2191(isEmptyRingEquiv R (Fin 0)) p = 0 ** convert h finZeroElim ** case succ R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n\u271d : \u2115 p\u271d : MvPolynomial (Fin n\u271d) R h\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0 n : \u2115 ih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 p : MvPolynomial (Fin (Nat.succ n)) R h : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0 \u22a2 p = 0 ** apply (finSuccEquiv R n).injective ** case succ.a R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n\u271d : \u2115 p\u271d : MvPolynomial (Fin n\u271d) R h\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0 n : \u2115 ih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 p : MvPolynomial (Fin (Nat.succ n)) R h : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0 \u22a2 \u2191(finSuccEquiv R n) p = \u2191(finSuccEquiv R n) 0 ** simp only [AlgEquiv.map_zero] ** case succ.a R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n\u271d : \u2115 p\u271d : MvPolynomial (Fin n\u271d) R h\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0 n : \u2115 ih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 p : MvPolynomial (Fin (Nat.succ n)) R h : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0 \u22a2 \u2191(finSuccEquiv R n) p = 0 ** refine Polynomial.funext fun q => ?_ ** case succ.a R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n\u271d : \u2115 p\u271d : MvPolynomial (Fin n\u271d) R h\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0 n : \u2115 ih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 p : MvPolynomial (Fin (Nat.succ n)) R h : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0 q : MvPolynomial (Fin n) R \u22a2 Polynomial.eval q (\u2191(finSuccEquiv R n) p) = Polynomial.eval q 0 ** rw [Polynomial.eval_zero] ** case succ.a R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n\u271d : \u2115 p\u271d : MvPolynomial (Fin n\u271d) R h\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0 n : \u2115 ih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 p : MvPolynomial (Fin (Nat.succ n)) R h : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0 q : MvPolynomial (Fin n) R \u22a2 Polynomial.eval q (\u2191(finSuccEquiv R n) p) = 0 ** apply ih fun x => ?_ ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : Infinite R n\u271d : \u2115 p\u271d : MvPolynomial (Fin n\u271d) R h\u271d : \u2200 (x : Fin n\u271d \u2192 R), \u2191(eval x) p\u271d = 0 n : \u2115 ih : \u2200 {p : MvPolynomial (Fin n) R}, (\u2200 (x : Fin n \u2192 R), \u2191(eval x) p = 0) \u2192 p = 0 p : MvPolynomial (Fin (Nat.succ n)) R h : \u2200 (x : Fin (Nat.succ n) \u2192 R), \u2191(eval x) p = 0 q : MvPolynomial (Fin n) R x : Fin n \u2192 R \u22a2 \u2191(eval x) (Polynomial.eval q (\u2191(finSuccEquiv R n) p)) = 0 ** calc _ = _ := eval_polynomial_eval_finSuccEquiv p _\n     _ = 0 := h _ ** Qed",
        "informal": ""
    },
    {
        "formal": "measure_le_lintegral_thickenedIndicatorAux ** \u03b1 : Type u_1 \u03b2 : Type u_2 E\u271d : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E\u271d inst\u271d : PseudoEMetricSpace \u03b1 \u03bc : Measure \u03b1 E : Set \u03b1 E_mble : MeasurableSet E \u03b4 : \u211d \u22a2 \u2191\u2191\u03bc E \u2264 \u222b\u207b (a : \u03b1), thickenedIndicatorAux \u03b4 E a \u2202\u03bc ** convert_to lintegral \u03bc (E.indicator fun _ => (1 : \u211d\u22650\u221e)) \u2264 lintegral \u03bc (thickenedIndicatorAux \u03b4 E) ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 E\u271d : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E\u271d inst\u271d : PseudoEMetricSpace \u03b1 \u03bc : Measure \u03b1 E : Set \u03b1 E_mble : MeasurableSet E \u03b4 : \u211d \u22a2 \u2191\u2191\u03bc E = lintegral \u03bc (indicator E fun x => 1) ** rw [lintegral_indicator _ E_mble] ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 E\u271d : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E\u271d inst\u271d : PseudoEMetricSpace \u03b1 \u03bc : Measure \u03b1 E : Set \u03b1 E_mble : MeasurableSet E \u03b4 : \u211d \u22a2 \u2191\u2191\u03bc E = \u222b\u207b (a : \u03b1) in E, 1 \u2202\u03bc ** simp only [lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E\u271d : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E\u271d inst\u271d : PseudoEMetricSpace \u03b1 \u03bc : Measure \u03b1 E : Set \u03b1 E_mble : MeasurableSet E \u03b4 : \u211d \u22a2 lintegral \u03bc (indicator E fun x => 1) \u2264 lintegral \u03bc (thickenedIndicatorAux \u03b4 E) ** apply lintegral_mono ** case hfg \u03b1 : Type u_1 \u03b2 : Type u_2 E\u271d : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E\u271d inst\u271d : PseudoEMetricSpace \u03b1 \u03bc : Measure \u03b1 E : Set \u03b1 E_mble : MeasurableSet E \u03b4 : \u211d \u22a2 (fun a => indicator E (fun x => 1) a) \u2264 fun a => thickenedIndicatorAux \u03b4 E a ** apply indicator_le_thickenedIndicatorAux ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.extract ** l m r ml mm mr : List Char b e : Pos \u22a2 \u2203 l' r',     ValidFor l' mm r'       (Substring.extract         { str := { data := l ++ (ml ++ mm ++ mr) ++ r }, startPos := { byteIdx := utf8Len l },           stopPos := { byteIdx := utf8Len l + utf8Len (ml ++ mm ++ mr) } }         { byteIdx := utf8Len ml } { byteIdx := utf8Len ml + utf8Len mm }) ** simp [Substring.extract] ** l m r ml mm mr : List Char b e : Pos \u22a2 \u2203 l' r',     ValidFor l' mm r'       (if utf8Len ml + utf8Len mm \u2264 utf8Len ml then { str := \"\", startPos := 0, stopPos := 0 }       else         { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },           startPos :=             Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }               ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),           stopPos :=             Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }               ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) }) ** split ** case inl l m r ml mm mr : List Char b e : Pos h\u271d : utf8Len ml + utf8Len mm \u2264 utf8Len ml \u22a2 \u2203 l' r', ValidFor l' mm r' { str := \"\", startPos := 0, stopPos := 0 } ** next h =>\nrw [utf8Len_eq_zero.1 <| Nat.le_zero.1 <| (Nat.add_le_add_iff_left _ _ 0).1 h]\nexact \u27e8[], [], \u27e8\u27e9\u27e9 ** l m r ml mm mr : List Char b e : Pos h : utf8Len ml + utf8Len mm \u2264 utf8Len ml \u22a2 \u2203 l' r', ValidFor l' mm r' { str := \"\", startPos := 0, stopPos := 0 } ** rw [utf8Len_eq_zero.1 <| Nat.le_zero.1 <| (Nat.add_le_add_iff_left _ _ 0).1 h] ** l m r ml mm mr : List Char b e : Pos h : utf8Len ml + utf8Len mm \u2264 utf8Len ml \u22a2 \u2203 l' r', ValidFor l' [] r' { str := \"\", startPos := 0, stopPos := 0 } ** exact \u27e8[], [], \u27e8\u27e9\u27e9 ** case inr l m r ml mm mr : List Char b e : Pos h\u271d : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml \u22a2 \u2203 l' r',     ValidFor l' mm r'       { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },         startPos :=           Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }             ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),         stopPos :=           Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }             ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) } ** next h =>\nrefine \u27e8l ++ ml, mr ++ r, .of_eq _ (by simp) ?_ ?_\u27e9 <;>\n  simp [Nat.min_eq_min] <;> rw [Nat.min_eq_right] <;>\n  try simp [Nat.add_le_add_iff_left, Nat.le_add_right]\nrw [Nat.add_assoc] ** l m r ml mm mr : List Char b e : Pos h : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml \u22a2 \u2203 l' r',     ValidFor l' mm r'       { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },         startPos :=           Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }             ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),         stopPos :=           Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }             ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) } ** refine \u27e8l ++ ml, mr ++ r, .of_eq _ (by simp) ?_ ?_\u27e9 <;>\n  simp [Nat.min_eq_min] <;> rw [Nat.min_eq_right] <;>\n  try simp [Nat.add_le_add_iff_left, Nat.le_add_right] ** case refine_2 l m r ml mm mr : List Char b e : Pos h : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml \u22a2 utf8Len l + (utf8Len ml + utf8Len mm) = utf8Len l + utf8Len ml + utf8Len mm ** rw [Nat.add_assoc] ** l m r ml mm mr : List Char b e : Pos h : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml \u22a2 { str := { data := l ++ (ml ++ (mm ++ (mr ++ r))) },           startPos :=             Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }               ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml }),           stopPos :=             Pos.min { byteIdx := utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) }               ({ byteIdx := utf8Len l } + { byteIdx := utf8Len ml + utf8Len mm }) }.str.data =     l ++ ml ++ mm ++ (mr ++ r) ** simp ** case refine_2 l m r ml mm mr : List Char b e : Pos h : \u00acutf8Len ml + utf8Len mm \u2264 utf8Len ml \u22a2 utf8Len l + (utf8Len ml + utf8Len mm) \u2264 utf8Len l + (utf8Len ml + (utf8Len mm + utf8Len mr)) ** simp [Nat.add_le_add_iff_left, Nat.le_add_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_iUnionLift ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d : Countable \u03b9 t : \u03b9 \u2192 Set \u03b1 f : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 htf :   \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),     f i { val := x, property := hxi } = f j { val := x, property := hxj } T : Set \u03b1 hT : T \u2286 \u22c3 i, t i htm : \u2200 (i : \u03b9), MeasurableSet (t i) hfm : \u2200 (i : \u03b9), Measurable (f i) s : Set \u03b2 hs : MeasurableSet s \u22a2 MeasurableSet (iUnionLift t f htf T hT \u207b\u00b9' s) ** rw [preimage_iUnionLift] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t\u271d u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d : Countable \u03b9 t : \u03b9 \u2192 Set \u03b1 f : (i : \u03b9) \u2192 \u2191(t i) \u2192 \u03b2 htf :   \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 t i) (hxj : x \u2208 t j),     f i { val := x, property := hxi } = f j { val := x, property := hxj } T : Set \u03b1 hT : T \u2286 \u22c3 i, t i htm : \u2200 (i : \u03b9), MeasurableSet (t i) hfm : \u2200 (i : \u03b9), Measurable (f i) s : Set \u03b2 hs : MeasurableSet s \u22a2 MeasurableSet (inclusion hT \u207b\u00b9' \u22c3 i, inclusion (_ : t i \u2286 \u22c3 i, t i) '' (f i \u207b\u00b9' s)) ** exact .preimage (.iUnion fun i => .image_inclusion _ (htm _) (hfm i hs)) (measurable_inclusion _) ** Qed",
        "informal": ""
    },
    {
        "formal": "List.join_eq_joinTR ** \u22a2 @join = @joinTR ** funext \u03b1 l ** case h.h \u03b1 : Type u_1 l : List (List \u03b1) \u22a2 join l = joinTR l ** rw [\u2190 List.bind_id, List.bind_eq_bindTR] ** case h.h \u03b1 : Type u_1 l : List (List \u03b1) \u22a2 bindTR l id = joinTR l ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.finSuccEquiv_support ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R \u22a2 Polynomial.support (\u2191(finSuccEquiv R n) f) = Finset.image (fun m => \u2191m 0) (support f) ** ext i ** case a R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R i : \u2115 \u22a2 i \u2208 Polynomial.support (\u2191(finSuccEquiv R n) f) \u2194 i \u2208 Finset.image (fun m => \u2191m 0) (support f) ** rw [Polynomial.mem_support_iff, Finset.mem_image, nonzero_iff_exists] ** case a R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R i : \u2115 \u22a2 (\u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) a \u2260 0) \u2194 \u2203 a, a \u2208 support f \u2227 \u2191a 0 = i ** constructor ** case a.mp R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R i : \u2115 \u22a2 (\u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) a \u2260 0) \u2192 \u2203 a, a \u2208 support f \u2227 \u2191a 0 = i ** rintro \u27e8m, hm\u27e9 ** case a.mp.intro R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R i : \u2115 m : Fin n \u2192\u2080 \u2115 hm : \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) m \u2260 0 \u22a2 \u2203 a, a \u2208 support f \u2227 \u2191a 0 = i ** refine' \u27e8cons i m, _, cons_zero _ _\u27e9 ** case a.mp.intro R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R i : \u2115 m : Fin n \u2192\u2080 \u2115 hm : \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) m \u2260 0 \u22a2 cons i m \u2208 support f ** rw [\u2190 support_coeff_finSuccEquiv] ** case a.mp.intro R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R i : \u2115 m : Fin n \u2192\u2080 \u2115 hm : \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) m \u2260 0 \u22a2 m \u2208 support (Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) ** simpa using hm ** case a.mpr R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R i : \u2115 \u22a2 (\u2203 a, a \u2208 support f \u2227 \u2191a 0 = i) \u2192 \u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) i) a \u2260 0 ** rintro \u27e8m, h, rfl\u27e9 ** case a.mpr.intro.intro R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R m : Fin (n + 1) \u2192\u2080 \u2115 h : m \u2208 support f \u22a2 \u2203 a, \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) (\u2191m 0)) a \u2260 0 ** refine' \u27e8tail m, _\u27e9 ** case a.mpr.intro.intro R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R m : Fin (n + 1) \u2192\u2080 \u2115 h : m \u2208 support f \u22a2 \u2191(Polynomial.coeff (\u2191(finSuccEquiv R n) f) (\u2191m 0)) (tail m) \u2260 0 ** rwa [\u2190 coeff, \u2190 mem_support_iff, support_coeff_finSuccEquiv, cons_tail] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.max_erase_ne_self ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 \u22a2 Finset.max (erase s x) \u2260 \u2191x ** by_cases s0 : (s.erase x).Nonempty ** case pos F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 s0 : Finset.Nonempty (erase s x) \u22a2 Finset.max (erase s x) \u2260 \u2191x ** refine' ne_of_eq_of_ne (coe_max' s0).symm _ ** case pos F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 s0 : Finset.Nonempty (erase s x) \u22a2 \u2191(max' (erase s x) s0) \u2260 \u2191x ** exact WithBot.coe_eq_coe.not.mpr (max'_erase_ne_self _) ** case neg F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 s0 : \u00acFinset.Nonempty (erase s x) \u22a2 Finset.max (erase s x) \u2260 \u2191x ** rw [not_nonempty_iff_eq_empty.mp s0, max_empty] ** case neg F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 s0 : \u00acFinset.Nonempty (erase s x) \u22a2 \u22a5 \u2260 \u2191x ** exact WithBot.bot_ne_coe ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.withDensity\u1d65_rnDeriv_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : s \u226a\u1d65 toENNRealVectorMeasure \u03bc \u22a2 withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = s ** rw [absolutelyContinuous_ennreal_iff, (_ : \u03bc.toENNRealVectorMeasure.ennrealToMeasure = \u03bc),\n  totalVariation_absolutelyContinuous_iff] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc \u22a2 withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = s ** ext1 i hi ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(withDensity\u1d65 \u03bc (rnDeriv s \u03bc)) i = \u2191s i ** rw [withDensity\u1d65_apply (integrable_rnDeriv _ _) hi, rnDeriv, integral_sub,\n  withDensity_rnDeriv_toReal_eq h.1 hi, withDensity_rnDeriv_toReal_eq h.2 hi] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) = \u2191s i ** conv_rhs => rw [\u2190 s.toSignedMeasure_toJordanDecomposition] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) =     \u2191(JordanDecomposition.toSignedMeasure (toJordanDecomposition s)) i ** erw [VectorMeasure.sub_apply] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) =     \u2191(toSignedMeasure (toJordanDecomposition s).posPart) i - \u2191(toSignedMeasure (toJordanDecomposition s).negPart) i ** rw [toSignedMeasure_apply_measurable hi, toSignedMeasure_apply_measurable hi] ** case h.hg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 Integrable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x) ** rw [\u2190 integrableOn_univ] ** case h.hg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 IntegrableOn (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x)) Set.univ ** refine' \u27e8_, hasFiniteIntegral_toReal_of_lintegral_ne_top _\u27e9 ** case h.hg.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x))     (Measure.restrict \u03bc Set.univ) ** apply Measurable.aestronglyMeasurable ** case h.hg.refine'_1.hf \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 Measurable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x) ** measurability ** case h.hg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 \u222b\u207b (x : \u03b1) in Set.univ, Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4 ** rw [set_lintegral_univ] ** case h.hg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : (toJordanDecomposition s).posPart \u226a \u03bc \u2227 (toJordanDecomposition s).negPart \u226a \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 \u222b\u207b (x : \u03b1), Measure.rnDeriv (toJordanDecomposition s).negPart \u03bc x \u2202\u03bc \u2260 \u22a4 ** exact (lintegral_rnDeriv_lt_top _ _).ne ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc h : totalVariation s \u226a ennrealToMeasure (toENNRealVectorMeasure \u03bc) \u22a2 ennrealToMeasure (toENNRealVectorMeasure \u03bc) = \u03bc ** exact equivMeasure.right_inv \u03bc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_lintegral_max ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g s : Set \u03b1 \u22a2 \u222b\u207b (x : \u03b1) in s, max (f x) (g x) \u2202\u03bc =     \u222b\u207b (x : \u03b1) in s \u2229 {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in s \u2229 {x | g x < f x}, f x \u2202\u03bc ** rw [lintegral_max hf hg, restrict_restrict, restrict_restrict, inter_comm s, inter_comm s] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g s : Set \u03b1 \u22a2 MeasurableSet {x | g x < f x}  \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g s : Set \u03b1 \u22a2 MeasurableSet {x | f x \u2264 g x} ** exacts [measurableSet_lt hg hf, measurableSet_le hf hg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_le_setAverage_pos ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s \u22a2 0 < \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} ** refine' pos_iff_ne_zero.2 fun H => _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 \u22a2 False ** replace H : (\u03bc.restrict s) {x | f x \u2264 \u2a0d a in s, f a \u2202\u03bc} = 0 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 \u22a2 False ** haveI := Fact.mk h\u03bc\u2081.lt_top ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 this : Fact (\u2191\u2191\u03bc s < \u22a4) \u22a2 False ** refine' (integral_sub_average (\u03bc.restrict s) f).not_gt _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 this : Fact (\u2191\u2191\u03bc s < \u22a4) \u22a2 0 < \u222b (x : \u03b1) in s, f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc \u2202\u03bc ** refine' (set_integral_pos_iff_support_of_nonneg_ae _ _).2 _ ** case H \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 \u22a2 \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 ** rwa [restrict_apply\u2080, inter_comm] ** case H \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 \u22a2 NullMeasurableSet {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} ** exact AEStronglyMeasurable.nullMeasurableSet_le hf.1 aestronglyMeasurable_const ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 this : Fact (\u2191\u2191\u03bc s < \u22a4) \u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc ** refine' eq_bot_mono (measure_mono fun x hx => _) H ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 this : Fact (\u2191\u2191\u03bc s < \u22a4) x : \u03b1 hx : x \u2208 {x | (fun x => OfNat.ofNat 0 x \u2264 (fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) x) x}\u1d9c \u22a2 x \u2208 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} ** simp only [Pi.zero_apply, sub_nonneg, mem_compl_iff, mem_setOf_eq, not_le] at hx ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 this : Fact (\u2191\u2191\u03bc s < \u22a4) x : \u03b1 hx : f x < \u2a0d (a : \u03b1) in s, f a \u2202\u03bc \u22a2 x \u2208 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} ** exact hx.le ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 this : Fact (\u2191\u2191\u03bc s < \u22a4) \u22a2 IntegrableOn (fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) s ** exact hf.sub (integrableOn_const.2 <| Or.inr <| lt_top_iff_ne_top.2 h\u03bc\u2081) ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 this : Fact (\u2191\u2191\u03bc s < \u22a4) \u22a2 0 < \u2191\u2191\u03bc ((support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc) \u2229 s) ** rwa [pos_iff_ne_zero, inter_comm, \u2190 diff_compl, \u2190 diff_inter_self_eq_diff, measure_diff_null] ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 this : Fact (\u2191\u2191\u03bc s < \u22a4) \u22a2 \u2191\u2191\u03bc ((support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc)\u1d9c \u2229 s) = 0 ** refine' eq_bot_mono (measure_mono _) (measure_inter_eq_zero_of_restrict H) ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d h\u03bc : \u2191\u2191\u03bc s \u2260 0 h\u03bc\u2081 : \u2191\u2191\u03bc s \u2260 \u22a4 hf : IntegrableOn f s H : \u2191\u2191(Measure.restrict \u03bc s) {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} = 0 this : Fact (\u2191\u2191\u03bc s < \u22a4) \u22a2 (support fun x => f x - \u2a0d (a : \u03b1) in s, f a \u2202\u03bc)\u1d9c \u2229 s \u2286 {x | f x \u2264 \u2a0d (a : \u03b1) in s, f a \u2202\u03bc} \u2229 s ** exact inter_subset_inter_left _ fun a ha => (sub_eq_zero.1 <| of_not_not ha).le ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.set_set ** \u03b1 : Type ?u.17292 a : Array \u03b1 i : Fin (size a) v v' : \u03b1 \u22a2 i.val < size (set a i v) ** simp [i.2] ** \u03b1 : Type u_1 a : Array \u03b1 i : Fin (size a) v v' : \u03b1 \u22a2 set (set a i v) { val := i.val, isLt := (_ : i.val < size (set a i v)) } v' = set a i v' ** simp [set, List.set_set] ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.take ** l m r : List Char s : Substring h : ValidFor l m r s n : Nat \u22a2 ValidFor l (List.take n m) (List.drop n m ++ r) (Substring.take s n) ** have : Substring.nextn {..} .. = _ := h.nextn (m\u2081 := []) n ** l m r : List Char s : Substring h : ValidFor l m r s n : Nat this :   Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n { byteIdx := utf8Len [] } =     { byteIdx := utf8Len [] + utf8Len (List.take n m) } \u22a2 ValidFor l (List.take n m) (List.drop n m ++ r) (Substring.take s n) ** simp at this ** l m r : List Char s : Substring h : ValidFor l m r s n : Nat this :   Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =     { byteIdx := utf8Len (List.take n m) } \u22a2 ValidFor l (List.take n m) (List.drop n m ++ r) (Substring.take s n) ** simp [Substring.take, this] ** l m r : List Char s : Substring h : ValidFor l m r s n : Nat this :   Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =     { byteIdx := utf8Len (List.take n m) } \u22a2 ValidFor l (List.take n m) (List.drop n m ++ r)     { str := s.str, startPos := s.startPos, stopPos := s.startPos + { byteIdx := utf8Len (List.take n m) } } ** simp [h.str, h.startPos, h.stopPos] ** l m r : List Char s : Substring h : ValidFor l m r s n : Nat this :   Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =     { byteIdx := utf8Len (List.take n m) } \u22a2 ValidFor l (List.take n m) (List.drop n m ++ r)     { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },       stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } } ** rw [\u2190 List.take_append_drop n m] at h ** l m r : List Char s : Substring n : Nat h : ValidFor l (List.take n m ++ List.drop n m) r s this :   Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =     { byteIdx := utf8Len (List.take n m) } \u22a2 ValidFor l (List.take n m) (List.drop n m ++ r)     { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },       stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } } ** refine .of_eq _ ?_ (by simp) (by simp) ** l m r : List Char s : Substring n : Nat h : ValidFor l (List.take n m ++ List.drop n m) r s this :   Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =     { byteIdx := utf8Len (List.take n m) } \u22a2 { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },           stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }.str.data =     l ++ List.take n m ++ (List.drop n m ++ r) ** conv => lhs; rw [\u2190 List.take_append_drop n m] ** l m r : List Char s : Substring n : Nat h : ValidFor l (List.take n m ++ List.drop n m) r s this :   Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =     { byteIdx := utf8Len (List.take n m) } \u22a2 { str := { data := l ++ (List.take n m ++ List.drop n m ++ r) }, startPos := { byteIdx := utf8Len l },           stopPos :=             { byteIdx := utf8Len l } +               { byteIdx := utf8Len (List.take n (List.take n m ++ List.drop n m)) } }.str.data =     l ++ List.take n m ++ (List.drop n m ++ r) ** simp [-List.take_append_drop, Nat.add_assoc] ** l m r : List Char s : Substring n : Nat h : ValidFor l (List.take n m ++ List.drop n m) r s this :   Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =     { byteIdx := utf8Len (List.take n m) } \u22a2 { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },           stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }.startPos.byteIdx =     utf8Len l ** simp ** l m r : List Char s : Substring n : Nat h : ValidFor l (List.take n m ++ List.drop n m) r s this :   Substring.nextn { str := s.str, startPos := s.startPos, stopPos := s.stopPos } n 0 =     { byteIdx := utf8Len (List.take n m) } \u22a2 { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },           stopPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) } }.stopPos.byteIdx =     utf8Len l + utf8Len (List.take n m) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Path.zoom_fill' ** \u03b1 : Type u_1 cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 path : Path \u03b1 \u22a2 fill' (zoom cut t path) = fill path t ** induction t generalizing path with\n| nil => rfl\n| node _ _ _ _ iha ihb => unfold zoom; split <;> [apply iha; apply ihb; rfl] ** case nil \u03b1 : Type u_1 cut : \u03b1 \u2192 Ordering path : Path \u03b1 \u22a2 fill' (zoom cut nil path) = fill path nil ** rfl ** case node \u03b1 : Type u_1 cut : \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 iha : \u2200 (path : Path \u03b1), fill' (zoom cut l\u271d path) = fill path l\u271d ihb : \u2200 (path : Path \u03b1), fill' (zoom cut r\u271d path) = fill path r\u271d path : Path \u03b1 \u22a2 fill' (zoom cut (node c\u271d l\u271d v\u271d r\u271d) path) = fill path (node c\u271d l\u271d v\u271d r\u271d) ** unfold zoom ** case node \u03b1 : Type u_1 cut : \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 iha : \u2200 (path : Path \u03b1), fill' (zoom cut l\u271d path) = fill path l\u271d ihb : \u2200 (path : Path \u03b1), fill' (zoom cut r\u271d path) = fill path r\u271d path : Path \u03b1 \u22a2 fill'       (match cut v\u271d with       | Ordering.lt => zoom cut l\u271d (left c\u271d path v\u271d r\u271d)       | Ordering.gt => zoom cut r\u271d (right c\u271d l\u271d v\u271d path)       | Ordering.eq => (node c\u271d l\u271d v\u271d r\u271d, path)) =     fill path (node c\u271d l\u271d v\u271d r\u271d) ** split <;> [apply iha; apply ihb; rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "FinEnum.mem_pi ** \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a \u22a2 f \u2208 pi xs fun x => toList (\u03b2 x) ** induction' xs with xs_hd xs_tl xs_ih <;> simp [pi, -List.map_eq_map, monad_norm, functor_norm] ** case nil \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a f : (a : \u03b1) \u2192 a \u2208 [] \u2192 \u03b2 a \u22a2 f = fun x h => (_ : False).elim ** ext a \u27e8\u27e9 ** case cons \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a \u22a2 \u2203 a, (\u2203 a_1, a = Pi.cons xs_hd xs_tl a_1) \u2227 \u2203 a_1, (a_1 \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = a a_1 ** exists Pi.cons xs_hd xs_tl (f _ (List.mem_cons_self _ _)) ** case cons \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a \u22a2 (\u2203 a, Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) = Pi.cons xs_hd xs_tl a) \u2227     \u2203 a, (a \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) a ** constructor ** case cons.left \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a \u22a2 \u2203 a, Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) = Pi.cons xs_hd xs_tl a  case cons.right \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a \u22a2 \u2203 a, (a \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) a ** exact \u27e8_, rfl\u27e9 ** case cons.right \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a \u22a2 \u2203 a, (a \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) a ** exists Pi.tail f ** case cons.right \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a \u22a2 (Pi.tail f \u2208 pi xs_tl fun x => toList (\u03b2 x)) \u2227     f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) (Pi.tail f) ** constructor ** case cons.right.left \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a \u22a2 Pi.tail f \u2208 pi xs_tl fun x => toList (\u03b2 x) ** apply xs_ih ** case cons.right.right \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a \u22a2 f = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) (Pi.tail f) ** ext x h ** case cons.right.right.h.h \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a x : \u03b1 h : x \u2208 xs_hd :: xs_tl \u22a2 f x h = Pi.cons xs_hd xs_tl (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) (Pi.tail f) x h ** simp only [Pi.cons] ** case cons.right.right.h.h \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a x : \u03b1 h : x \u2208 xs_hd :: xs_tl \u22a2 f x h =     if h' : x = xs_hd then cast (_ : \u03b2 xs_hd = \u03b2 x) (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl))     else Pi.tail f x (_ : x \u2208 xs_tl) ** split_ifs ** case pos \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a x : \u03b1 h : x \u2208 xs_hd :: xs_tl h\u271d : x = xs_hd \u22a2 f x h = cast (_ : \u03b2 xs_hd = \u03b2 x) (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) ** subst x ** case pos \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a h : xs_hd \u2208 xs_hd :: xs_tl \u22a2 f xs_hd h = cast (_ : \u03b2 xs_hd = \u03b2 xs_hd) (f xs_hd (_ : xs_hd \u2208 xs_hd :: xs_tl)) ** rfl ** case neg \u03b1 : Type u \u03b2\u271d : \u03b1 \u2192 Type v \u03b2 : \u03b1 \u2192 Type (max u u_1) inst\u271d\u00b9 : FinEnum \u03b1 inst\u271d : (a : \u03b1) \u2192 FinEnum (\u03b2 a) xs : List \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 xs \u2192 \u03b2 a xs_hd : \u03b1 xs_tl : List \u03b1 xs_ih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 xs_tl \u2192 \u03b2 a), f \u2208 pi xs_tl fun x => toList (\u03b2 x) f : (a : \u03b1) \u2192 a \u2208 xs_hd :: xs_tl \u2192 \u03b2 a x : \u03b1 h : x \u2208 xs_hd :: xs_tl h\u271d : \u00acx = xs_hd \u22a2 f x h = Pi.tail f x (_ : x \u2208 xs_tl) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_stronglyMeasurable_limit_of_tendsto_ae ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc h_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) \u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) ** borelize \u03b2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc h_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) this\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d : BorelSpace \u03b2 \u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) ** obtain \u27e8g, _, hg\u27e9 :\n  \u2203 (g : \u03b1 \u2192 \u03b2) (_ : Measurable g), \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) :=\n  measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).aemeasurable) h_ae_tendsto ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc h_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) this\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d : BorelSpace \u03b2 g : \u03b1 \u2192 \u03b2 w\u271d : Measurable g hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) ** have Hg : AEStronglyMeasurable g \u03bc := aestronglyMeasurable_of_tendsto_ae _ hf hg ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc h_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) this\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d : BorelSpace \u03b2 g : \u03b1 \u2192 \u03b2 w\u271d : Measurable g hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) Hg : AEStronglyMeasurable g \u03bc \u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) ** refine' \u27e8Hg.mk g, Hg.stronglyMeasurable_mk, _\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc h_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) this\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d : BorelSpace \u03b2 g : \u03b1 \u2192 \u03b2 w\u271d : Measurable g hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) Hg : AEStronglyMeasurable g \u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x)) ** filter_upwards [hg, Hg.ae_eq_mk] with x hx h'x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc h_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) this\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d : BorelSpace \u03b2 g : \u03b1 \u2192 \u03b2 w\u271d : Measurable g hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) Hg : AEStronglyMeasurable g \u03bc x : \u03b1 hx : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) h'x : g x = AEStronglyMeasurable.mk g Hg x \u22a2 Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x)) ** rwa [h'x] at hx ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r : \u211d\u22650\u221e \u03bc : OuterMeasure X s\u271d : Set X m : Set X \u2192 \u211d\u22650\u221e s : Set X \u22a2 Tendsto (fun r => \u2191(pre m r) s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191(mkMetric' m) s)) ** rw [\u2190 map_coe_Ioi_atBot, tendsto_map'_iff] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r : \u211d\u22650\u221e \u03bc : OuterMeasure X s\u271d : Set X m : Set X \u2192 \u211d\u22650\u221e s : Set X \u22a2 Tendsto ((fun r => \u2191(pre m r) s) \u2218 Subtype.val) atBot (\ud835\udcdd (\u2191(mkMetric' m) s)) ** simp only [mkMetric', OuterMeasure.iSup_apply, iSup_subtype'] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m\u271d : Set X \u2192 \u211d\u22650\u221e r : \u211d\u22650\u221e \u03bc : OuterMeasure X s\u271d : Set X m : Set X \u2192 \u211d\u22650\u221e s : Set X \u22a2 Tendsto ((fun r => \u2191(pre m r) s) \u2218 Subtype.val) atBot (\ud835\udcdd (\u2a06 i, \u2191(pre m \u2191i) s)) ** exact tendsto_atBot_iSup fun r r' hr => mono_pre _ hr _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_multiset_prod ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : F \u22a2 \u2200 (a : List (Set \u03b1)),     \u2191f '' Multiset.prod (Quotient.mk (List.isSetoid (Set \u03b1)) a) =       Multiset.prod (Multiset.map (fun s => \u2191f '' s) (Quotient.mk (List.isSetoid (Set \u03b1)) a)) ** simpa only [Multiset.quot_mk_to_coe, Multiset.coe_prod, Multiset.coe_map] using\n  image_list_prod f ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.coeFn_abs ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2078 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b2\u271d inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : TopologicalSpace \u03b4 \u03b2 : Type u_5 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : Lattice \u03b2 inst\u271d\u00b2 : TopologicalLattice \u03b2 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : TopologicalAddGroup \u03b2 f : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 \u2191|f| =\u1d50[\u03bc] fun x => |\u2191f x| ** simp_rw [abs_eq_sup_neg] ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2078 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b2\u271d inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : TopologicalSpace \u03b4 \u03b2 : Type u_5 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : Lattice \u03b2 inst\u271d\u00b2 : TopologicalLattice \u03b2 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : TopologicalAddGroup \u03b2 f : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 \u2191(f \u2294 -f) =\u1d50[\u03bc] fun x => \u2191f x \u2294 -\u2191f x ** filter_upwards [AEEqFun.coeFn_sup f (-f), AEEqFun.coeFn_neg f] with x hx_sup hx_neg ** case h \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2078 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b2\u271d inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : TopologicalSpace \u03b4 \u03b2 : Type u_5 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : Lattice \u03b2 inst\u271d\u00b2 : TopologicalLattice \u03b2 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : TopologicalAddGroup \u03b2 f : \u03b1 \u2192\u2098[\u03bc] \u03b2 x : \u03b1 hx_sup : \u2191(f \u2294 -f) x = \u2191f x \u2294 \u2191(-f) x hx_neg : \u2191(-f) x = (-\u2191f) x \u22a2 \u2191(f \u2294 -f) x = \u2191f x \u2294 -\u2191f x ** rw [hx_sup, hx_neg, Pi.neg_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.analyticSet_range_of_polishSpace ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03b2 : Type u_3 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b2 \u2192 \u03b1 f_cont : Continuous f \u22a2 AnalyticSet (range f) ** cases isEmpty_or_nonempty \u03b2 ** case inl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03b2 : Type u_3 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b2 \u2192 \u03b1 f_cont : Continuous f h\u271d : IsEmpty \u03b2 \u22a2 AnalyticSet (range f) ** rw [range_eq_empty] ** case inl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03b2 : Type u_3 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b2 \u2192 \u03b1 f_cont : Continuous f h\u271d : IsEmpty \u03b2 \u22a2 AnalyticSet \u2205 ** exact analyticSet_empty ** case inr \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03b2 : Type u_3 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b2 \u2192 \u03b1 f_cont : Continuous f h\u271d : Nonempty \u03b2 \u22a2 AnalyticSet (range f) ** rw [AnalyticSet] ** case inr \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03b2 : Type u_3 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b2 \u2192 \u03b1 f_cont : Continuous f h\u271d : Nonempty \u03b2 \u22a2 range f = \u2205 \u2228 \u2203 f_1, Continuous f_1 \u2227 range f_1 = range f ** obtain \u27e8g, g_cont, hg\u27e9 : \u2203 g : (\u2115 \u2192 \u2115) \u2192 \u03b2, Continuous g \u2227 Surjective g :=\n  exists_nat_nat_continuous_surjective \u03b2 ** case inr.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03b2 : Type u_3 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b2 \u2192 \u03b1 f_cont : Continuous f h\u271d : Nonempty \u03b2 g : (\u2115 \u2192 \u2115) \u2192 \u03b2 g_cont : Continuous g hg : Surjective g \u22a2 range f = \u2205 \u2228 \u2203 f_1, Continuous f_1 \u2227 range f_1 = range f ** refine' Or.inr \u27e8f \u2218 g, f_cont.comp g_cont, _\u27e9 ** case inr.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03b2 : Type u_3 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PolishSpace \u03b2 f : \u03b2 \u2192 \u03b1 f_cont : Continuous f h\u271d : Nonempty \u03b2 g : (\u2115 \u2192 \u2115) \u2192 \u03b2 g_cont : Continuous g hg : Surjective g \u22a2 range (f \u2218 g) = range f ** rw [hg.range_comp] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.all_iff ** s : String p : Char \u2192 Bool \u22a2 all s p = true \u2194 \u2200 (c : Char), c \u2208 s.data \u2192 p c = true ** simp [all_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm'_add_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hq1 : 1 \u2264 q \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264     (\u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc) ^ (1 / q) ** refine' ENNReal.rpow_le_rpow _ (by simp [le_trans zero_le_one hq1] : 0 \u2264 1 / q) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hq1 : 1 \u2264 q \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc ** refine' lintegral_mono fun a => ENNReal.rpow_le_rpow _ (le_trans zero_le_one hq1) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hq1 : 1 \u2264 q a : \u03b1 \u22a2 \u2191\u2016(f + g) a\u2016\u208a \u2264 ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ** simp only [Pi.add_apply, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hq1 : 1 \u2264 q \u22a2 0 \u2264 1 / q ** simp [le_trans zero_le_one hq1] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.le_index_mul ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) \u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V ** obtain \u27e8s, h1s, h2s\u27e9 := index_elim K.isCompact K\u2080.interior_nonempty ** case intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 \u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V ** obtain \u27e8t, h1t, h2t\u27e9 := index_elim K\u2080.isCompact hV ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V \u22a2 index (\u2191K) V \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) V ** rw [\u2190 h2s, \u2190 h2t, mul_comm] ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V \u22a2 index (\u2191K) V \u2264 Finset.card t * Finset.card s ** refine' le_trans _ Finset.card_mul_le ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V \u22a2 index (\u2191K) V \u2264 Finset.card (t * s) ** apply Nat.sInf_le ** case intro.intro.intro.intro.hm G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V \u22a2 Finset.card (t * s) \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} ** refine' \u27e8_, _, rfl\u27e9 ** case intro.intro.intro.intro.hm G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V \u22a2 t * s \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} ** rw [mem_setOf_eq] ** case intro.intro.intro.intro.hm G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V \u22a2 \u2191K \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V ** refine' Subset.trans h1s _ ** case intro.intro.intro.intro.hm G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V \u22a2 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V ** apply iUnion\u2082_subset ** case intro.intro.intro.intro.hm.h G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V \u22a2 \u2200 (i : G), i \u2208 s \u2192 (fun h => i * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V ** intro g\u2081 hg\u2081 ** case intro.intro.intro.intro.hm.h G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V g\u2081 : G hg\u2081 : g\u2081 \u2208 s \u22a2 (fun h => g\u2081 * h) \u207b\u00b9' \u2191K\u2080 \u2286 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V ** rw [preimage_subset_iff] ** case intro.intro.intro.intro.hm.h G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V g\u2081 : G hg\u2081 : g\u2081 \u2208 s \u22a2 \u2200 (a : G), g\u2081 * a \u2208 \u2191K\u2080 \u2192 a \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V ** intro g\u2082 hg\u2082 ** case intro.intro.intro.intro.hm.h G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V g\u2081 : G hg\u2081 : g\u2081 \u2208 s g\u2082 : G hg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080 \u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V ** have := h1t hg\u2082 ** case intro.intro.intro.intro.hm.h G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V g\u2081 : G hg\u2081 : g\u2081 \u2208 s g\u2082 : G hg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080 this : g\u2081 * g\u2082 \u2208 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V \u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V ** rcases this with \u27e8_, \u27e8g\u2083, rfl\u27e9, A, \u27e8hg\u2083, rfl\u27e9, h2V\u27e9 ** case intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V g\u2081 : G hg\u2081 : g\u2081 \u2208 s g\u2082 : G hg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080 g\u2083 : G hg\u2083 : g\u2083 \u2208 t h2V : g\u2081 * g\u2082 \u2208 (fun h => (fun h => g\u2083 * h) \u207b\u00b9' V) hg\u2083 \u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V ** rw [mem_preimage, \u2190 mul_assoc] at h2V ** case intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G V : Set G hV : Set.Nonempty (interior V) s : Finset G h1s : \u2191K \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191K\u2080 h2s : Finset.card s = index \u2191K \u2191K\u2080 t : Finset G h1t : \u2191K\u2080 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V h2t : Finset.card t = index (\u2191K\u2080) V g\u2081 : G hg\u2081 : g\u2081 \u2208 s g\u2082 : G hg\u2082 : g\u2081 * g\u2082 \u2208 \u2191K\u2080 g\u2083 : G hg\u2083 : g\u2083 \u2208 t h2V : g\u2083 * g\u2081 * g\u2082 \u2208 V \u22a2 g\u2082 \u2208 \u22c3 g \u2208 t * s, (fun h => g * h) \u207b\u00b9' V ** exact mem_biUnion (Finset.mul_mem_mul hg\u2083 hg\u2081) h2V ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.pdf.IsUniform.mul_pdf_integrable ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 \u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x) ** by_cases hsupp : volume s = \u221e ** case neg \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u00ac\u2191\u2191volume s = \u22a4 \u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x) ** constructor ** case neg.right \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u00ac\u2191\u2191volume s = \u22a4 \u22a2 HasFiniteIntegral fun x => x * ENNReal.toReal (pdf X \u2119 x) ** refine' hasFiniteIntegral_mul huX _ ** case neg.right \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u00ac\u2191\u2191volume s = \u22a4 \u22a2 \u222b\u207b (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) x \u2260 \u22a4 ** set ind := (volume s)\u207b\u00b9 \u2022 (1 : \u211d \u2192 \u211d\u22650\u221e) ** case neg.right \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u00ac\u2191\u2191volume s = \u22a4 ind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1 \u22a2 \u222b\u207b (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x \u2260 \u22a4 ** have : \u2200 x, \u2191\u2016x\u2016\u208a * s.indicator ind x = s.indicator (fun x => \u2016x\u2016\u208a * ind x) x := fun x =>\n  (s.indicator_mul_right (fun x => \u2191\u2016x\u2016\u208a) ind).symm ** case neg.right \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u00ac\u2191\u2191volume s = \u22a4 ind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1 this : \u2200 (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x = Set.indicator s (fun x => \u2191\u2016x\u2016\u208a * ind x) x \u22a2 \u222b\u207b (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x \u2260 \u22a4 ** simp only [this, lintegral_indicator _ hms, mul_one, Algebra.id.smul_eq_mul, Pi.one_apply,\n  Pi.smul_apply] ** case neg.right \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u00ac\u2191\u2191volume s = \u22a4 ind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1 this : \u2200 (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x = Set.indicator s (fun x => \u2191\u2016x\u2016\u208a * ind x) x \u22a2 \u222b\u207b (x : \u211d) in s, \u2191\u2016x\u2016\u208a * (\u2191\u2191volume s)\u207b\u00b9 \u2260 \u22a4 ** rw [lintegral_mul_const _ measurable_nnnorm.coe_nnreal_ennreal] ** case neg.right \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u00ac\u2191\u2191volume s = \u22a4 ind : \u211d \u2192 \u211d\u22650\u221e := (\u2191\u2191volume s)\u207b\u00b9 \u2022 1 this : \u2200 (x : \u211d), \u2191\u2016x\u2016\u208a * Set.indicator s ind x = Set.indicator s (fun x => \u2191\u2016x\u2016\u208a * ind x) x \u22a2 (\u222b\u207b (a : \u211d) in s, \u2191\u2016a\u2016\u208a) * (\u2191\u2191volume s)\u207b\u00b9 \u2260 \u22a4 ** refine' (ENNReal.mul_lt_top (set_lintegral_lt_top_of_isCompact hsupp hcs continuous_nnnorm).ne\n  (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hns)).ne).ne ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u2191\u2191volume s = \u22a4 \u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x) ** have : pdf X \u2119 =\u1d50[volume] 0 := by\n  refine' ae_eq_trans huX _\n  simp [hsupp, ae_eq_refl] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u2191\u2191volume s = \u22a4 this : pdf X \u2119 =\u1da0[ae volume] 0 \u22a2 Integrable fun x => x * ENNReal.toReal (pdf X \u2119 x) ** refine' Integrable.congr (integrable_zero _ _ _) _ ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u2191\u2191volume s = \u22a4 this : pdf X \u2119 =\u1da0[ae volume] 0 \u22a2 (fun x => 0) =\u1da0[ae volume] fun x => x * ENNReal.toReal (pdf X \u2119 x) ** rw [(by simp : (fun x => 0 : \u211d \u2192 \u211d) = fun x => x * (0 : \u211d\u22650\u221e).toReal)] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u2191\u2191volume s = \u22a4 this : pdf X \u2119 =\u1da0[ae volume] 0 \u22a2 (fun x => x * ENNReal.toReal 0) =\u1da0[ae volume] fun x => x * ENNReal.toReal (pdf X \u2119 x) ** refine'\n  Filter.EventuallyEq.mul (ae_eq_refl _) (Filter.EventuallyEq.fun_comp this.symm ENNReal.toReal) ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u2191\u2191volume s = \u22a4 \u22a2 pdf X \u2119 =\u1da0[ae volume] 0 ** refine' ae_eq_trans huX _ ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u2191\u2191volume s = \u22a4 \u22a2 Set.indicator s ((\u2191\u2191volume s)\u207b\u00b9 \u2022 1) =\u1da0[ae volume] 0 ** simp [hsupp, ae_eq_refl] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u2191\u2191volume s = \u22a4 this : pdf X \u2119 =\u1da0[ae volume] 0 \u22a2 (fun x => 0) = fun x => x * ENNReal.toReal 0 ** simp ** case neg.left \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E X : \u03a9 \u2192 \u211d s : Set \u211d hms : MeasurableSet s hns : \u2191\u2191volume s \u2260 0 inst\u271d : IsFiniteMeasure \u2119 hcs : IsCompact s huX : IsUniform X s \u2119 hsupp : \u00ac\u2191\u2191volume s = \u22a4 \u22a2 AEStronglyMeasurable (fun x => x * ENNReal.toReal (pdf X \u2119 x)) volume ** exact aestronglyMeasurable_id.mul\n  (measurable_pdf X \u2119).aemeasurable.ennreal_toReal.aestronglyMeasurable ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i \u22a2 \u2191m (\u22c3 n, s n) = \u2a06 n, \u2191m (s n) ** refine' m.iUnion_of_tendsto_zero atTop _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i \u22a2 Tendsto (fun k => \u2191m ((\u22c3 n, s n) \\ s k)) atTop (\ud835\udcdd 0) ** refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 \u22a2 \u2191m ((\u22c3 n, s n) \\ s n) \u2264 \u2211' (k : \u2115), \u2191m (s (k + n + 1) \\ s (k + n)) ** refine' (m.mono _).trans (m.iUnion _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 \u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n) ** have h' : Monotone s := @monotone_nat_of_le_succ (Set \u03b1) _ _ h_mono ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s \u22a2 (\u22c3 n, s n) \\ s n \u2286 \u22c3 i, s (i + n + 1) \\ s (i + n) ** simp only [diff_subset_iff, iUnion_subset_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s \u22a2 \u2200 (i : \u2115), s i \u2286 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n) ** intro i x hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s i : \u2115 x : \u03b1 hx : x \u2208 s i \u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n) ** have : \u2203i, x \u2208 s i := by exists i ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s i : \u2115 x : \u03b1 hx : x \u2208 s i this : \u2203 i, x \u2208 s i \u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n) ** rcases Nat.findX this with \u27e8j, hj, hlt\u27e9 ** case mk.intro \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s i : \u2115 x : \u03b1 hx : x \u2208 s i this : \u2203 i, x \u2208 s i j : \u2115 hj : x \u2208 s j hlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m \u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n) ** clear hx i ** case mk.intro \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s x : \u03b1 this : \u2203 i, x \u2208 s i j : \u2115 hj : x \u2208 s j hlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m \u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n) ** cases' le_or_lt j n with hjn hnj ** case mk.intro.inr \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s x : \u03b1 this : \u2203 i, x \u2208 s i j : \u2115 hj : x \u2208 s j hlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m hnj : n < j \u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n) ** have : j - (n + 1) + n + 1 = j := by rw [add_assoc, tsub_add_cancel_of_le hnj.nat_succ_le] ** case mk.intro.inr \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s x : \u03b1 this\u271d : \u2203 i, x \u2208 s i j : \u2115 hj : x \u2208 s j hlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m hnj : n < j this : j - (n + 1) + n + 1 = j \u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n) ** refine' Or.inr (mem_iUnion.2 \u27e8j - (n + 1), _, hlt _ _\u27e9) ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s i : \u2115 x : \u03b1 hx : x \u2208 s i \u22a2 \u2203 i, x \u2208 s i ** exists i ** case mk.intro.inl \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s x : \u03b1 this : \u2203 i, x \u2208 s i j : \u2115 hj : x \u2208 s j hlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m hjn : j \u2264 n \u22a2 x \u2208 s n \u222a \u22c3 i, s (i + n + 1) \\ s (i + n) ** exact Or.inl (h' hjn hj) ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s x : \u03b1 this : \u2203 i, x \u2208 s i j : \u2115 hj : x \u2208 s j hlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m hnj : n < j \u22a2 j - (n + 1) + n + 1 = j ** rw [add_assoc, tsub_add_cancel_of_le hnj.nat_succ_le] ** case mk.intro.inr.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s x : \u03b1 this\u271d : \u2203 i, x \u2208 s i j : \u2115 hj : x \u2208 s j hlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m hnj : n < j this : j - (n + 1) + n + 1 = j \u22a2 x \u2208 s (j - (n + 1) + n + 1) ** rwa [this] ** case mk.intro.inr.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m\u271d m : OuterMeasure \u03b1 s : \u2115 \u2192 Set \u03b1 h_mono : \u2200 (n : \u2115), s n \u2286 s (n + 1) h0 : \u2211' (k : \u2115), \u2191m (s (k + 1) \\ s k) \u2260 \u22a4 inst\u271d : (i : \u2115) \u2192 DecidablePred fun x => x \u2208 s i n : \u2115 h' : Monotone s x : \u03b1 this\u271d : \u2203 i, x \u2208 s i j : \u2115 hj : x \u2208 s j hlt : \u2200 (m : \u2115), m < j \u2192 \u00acx \u2208 s m hnj : n < j this : j - (n + 1) + n + 1 = j \u22a2 j - (n + 1) + n < j ** rw [\u2190 Nat.succ_le_iff, Nat.succ_eq_add_one, this] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.upcrossingsBefore_le ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 hab : a < b \u22a2 upcrossingsBefore a b f N \u03c9 \u2264 N ** by_cases hN : N = 0 ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 hab : a < b hN : N = 0 \u22a2 upcrossingsBefore a b f N \u03c9 \u2264 N ** subst hN ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 hab : a < b \u22a2 upcrossingsBefore a b f 0 \u03c9 \u2264 0 ** rw [upcrossingsBefore_zero] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 hab : a < b hN : \u00acN = 0 \u22a2 upcrossingsBefore a b f N \u03c9 \u2264 N ** refine' csSup_le \u27e80, zero_lt_iff.2 hN\u27e9 fun n (hn : _ < N) => _ ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d N n\u271d m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 hab : a < b hN : \u00acN = 0 n : \u2115 hn : upperCrossingTime a b f N n \u03c9 < N \u22a2 n \u2264 N ** by_contra hnN ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d N n\u271d m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 hab : a < b hN : \u00acN = 0 n : \u2115 hn : upperCrossingTime a b f N n \u03c9 < N hnN : \u00acn \u2264 N \u22a2 False ** exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x ** obtain \u27e8f, hf\u27e9 := exists_measurableEmbedding_real \u03a9 ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x ** set \u03c1' : Measure (\u03b1 \u00d7 \u211d) := \u03c1.map (Prod.map id f) with h\u03c1'def ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x ** have h\u03c1' : \u03c1'.fst = \u03c1.fst ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x ** have h\u03c1'' : \u2200\u1d50 x \u2202\u03c1'.fst, kernel.map \u03ba f hf.measurable x = \u03c1'.condKernel x ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1', \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x ** rw [h\u03c1'] at h\u03c1'' ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x ** suffices : \u2200\u1d50 x \u2202\u03c1.fst, \u2200 s, MeasurableSet s \u2192\n  ((\u03c1.map (Prod.map id f)).condKernel x) s = (\u03c1.condKernel x) (f \u207b\u00b9' s) ** case this \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2200 (s : Set \u211d),       MeasurableSet s \u2192         \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) ** have hprod : (\u03c1.map (Prod.map id f)).fst = \u03c1.fst ** case this \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2200 (s : Set \u211d),       MeasurableSet s \u2192         \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) ** suffices : \u03c1.map (Prod.map id f) =\n  (kernel.const Unit (\u03c1.map (Prod.map id f)).fst \u2297\u2096\n    kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f hf.measurable)) () ** case this \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 \u22a2 Measure.map (Prod.map id f) \u03c1 =     \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096           kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))       () ** ext s hs ** case this.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.map (Prod.map id f) \u03c1) s =     \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096                 kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))             ())       s ** have hinteq : \u2200 x, (\u03c1.condKernel x).map f {c | (x, c) \u2208 s} =\n    \u03c1.condKernel x {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} ** case this.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s hinteq :   \u2200 (x : \u03b1),     \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =       \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} \u22a2 \u2191\u2191(Measure.map (Prod.map id f) \u03c1) s =     \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096                 kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))             ())       s ** simp only [hprod, kernel.compProd_apply _ _ _ hs, kernel.prodMkLeft_apply,\n  kernel.map_apply _ hf.measurable, hinteq, Set.mem_preimage, Prod_map, id_eq,\n  kernel.lintegral_const] ** case this.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s hinteq :   \u2200 (x : \u03b1),     \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =       \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} \u22a2 \u2191\u2191(Measure.map (Prod.map id f) \u03c1) s = \u222b\u207b (x : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, f c) \u2208 s} \u2202Measure.fst \u03c1 ** rw [Measure.map_apply (measurable_id.prod_map hf.measurable) hs, \u2190 lintegral_condKernel_mem] ** case h\u03c1' \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 \u22a2 Measure.fst \u03c1' = Measure.fst \u03c1 ** ext s hs ** case h\u03c1'.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.fst \u03c1') s = \u2191\u2191(Measure.fst \u03c1) s ** rw [h\u03c1'def, Measure.fst_apply, Measure.fst_apply, Measure.map_apply] ** case h\u03c1'.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.fst \u207b\u00b9' s)) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s)  case h\u03c1'.h.hf \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 Measurable (Prod.map id f)  case h\u03c1'.h.hs \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 MeasurableSet (Prod.fst \u207b\u00b9' s)  case h\u03c1'.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 MeasurableSet s  case h\u03c1'.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 MeasurableSet s ** exacts [rfl, Measurable.prod measurable_fst <| hf.measurable.comp measurable_snd,\n  measurable_fst hs, hs, hs] ** case h\u03c1'' \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1', \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x ** refine' eq_condKernel_of_measure_eq_compProd_real \u03c1' (kernel.map \u03ba f hf.measurable) _ ** case h\u03c1'' \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u03c1' = \u2191(kernel.const Unit (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) () ** ext s hs ** case h\u03c1''.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2191\u2191\u03c1' s = \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ()) s ** simp only [Measure.map_apply (measurable_id.prod_map hf.measurable) hs] ** case h\u03c1''.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' s) =     \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096                 kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f)))             ())       s ** conv_lhs => congr; rw [h\u03ba] ** case h\u03c1''.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) ()) (Prod.map id f \u207b\u00b9' s) =     \u2191\u2191(\u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096                 kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f)))             ())       s ** rw [kernel.compProd_apply _ _ _ hs, kernel.compProd_apply _ _ _\n  (measurable_id.prod_map hf.measurable hs), (_ : (\u03c1.map (Prod.map id f)).fst = \u03c1.fst)] ** case h\u03c1''.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u222b\u207b (b : \u03b1),       \u2191\u2191(\u2191(kernel.prodMkLeft Unit \u03ba) ((), b))         {c | (b, c) \u2208 Prod.map id f \u207b\u00b9' s} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) () =     \u222b\u207b (b : \u03b1),       \u2191\u2191(\u2191(kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ((), b))         {c | (b, c) \u2208 s} \u2202\u2191(kernel.const Unit (Measure.fst \u03c1)) () ** congr ** case h\u03c1''.h.e_f \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 (fun b => \u2191\u2191(\u2191(kernel.prodMkLeft Unit \u03ba) ((), b)) {c | (b, c) \u2208 Prod.map id f \u207b\u00b9' s}) = fun b =>     \u2191\u2191(\u2191(kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ((), b)) {c | (b, c) \u2208 s} ** ext x ** case h\u03c1''.h.e_f.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s x : \u03b1 \u22a2 \u2191\u2191(\u2191(kernel.prodMkLeft Unit \u03ba) ((), x)) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} =     \u2191\u2191(\u2191(kernel.prodMkLeft Unit (kernel.map \u03ba f (_ : Measurable f))) ((), x)) {c | (x, c) \u2208 s} ** simp only [Set.mem_preimage, Prod_map, id_eq, kernel.prodMkLeft_apply, kernel.map_apply] ** case h\u03c1''.h.e_f.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s x : \u03b1 \u22a2 \u2191\u2191(\u2191\u03ba x) {c | (x, f c) \u2208 s} = \u2191\u2191(Measure.map f (\u2191\u03ba x)) {c | (x, c) \u2208 s} ** rw [Measure.map_apply hf.measurable] ** case h\u03c1''.h.e_f.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s x : \u03b1 \u22a2 \u2191\u2191(\u2191\u03ba x) {c | (x, f c) \u2208 s} = \u2191\u2191(\u2191\u03ba x) (f \u207b\u00b9' {c | (x, c) \u2208 s}) ** rfl ** case h\u03c1''.h.e_f.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s x : \u03b1 \u22a2 MeasurableSet {c | (x, c) \u2208 s} ** exact measurable_prod_mk_left hs ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 ** rw [Measure.fst_map_prod_mk] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 Measure.map (fun a => (Prod.map id f a).1) \u03c1 = Measure.fst \u03c1 ** simp only [Prod_map, id_eq] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 Measure.map (fun a => a.1) \u03c1 = Measure.fst \u03c1 ** rfl ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 Measurable fun a => (Prod.map id f a).2 ** exact (hf.measurable.comp measurable_snd) ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x this :   \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2200 (s : Set \u211d),       MeasurableSet s \u2192         \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x ** filter_upwards [h\u03c1'', this] with x hx h ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x this :   \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2200 (s : Set \u211d),       MeasurableSet s \u2192         \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) x : \u03b1 hx : \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x h :   \u2200 (s : Set \u211d),     MeasurableSet s \u2192       \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) \u22a2 \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x ** rw [kernel.map_apply] at hx ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x this :   \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2200 (s : Set \u211d),       MeasurableSet s \u2192         \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) x : \u03b1 hx : Measure.map f (\u2191\u03ba x) = \u2191(Measure.condKernel \u03c1') x h :   \u2200 (s : Set \u211d),     MeasurableSet s \u2192       \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) \u22a2 \u2191\u03ba x = \u2191(Measure.condKernel \u03c1) x ** ext s hs ** case h.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x this :   \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2200 (s : Set \u211d),       MeasurableSet s \u2192         \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) x : \u03b1 hx : Measure.map f (\u2191\u03ba x) = \u2191(Measure.condKernel \u03c1') x h :   \u2200 (s : Set \u211d),     MeasurableSet s \u2192       \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) s : Set \u03a9 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191\u03ba x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) s ** rw [\u2190 Set.preimage_image_eq s hf.injective,\n  \u2190 Measure.map_apply hf.measurable <| hf.measurableSet_image.2 hs, hx,\n  h _ <| hf.measurableSet_image.2 hs] ** case hprod \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x \u22a2 Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 ** ext s hs ** case hprod.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.fst (Measure.map (Prod.map id f) \u03c1)) s = \u2191\u2191(Measure.fst \u03c1) s ** rw [Measure.fst_apply hs,\n  Measure.map_apply (measurable_id.prod_map hf.measurable) (measurable_fst hs),\n  \u2190 Set.preimage_comp, Measure.fst_apply hs] ** case hprod.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191\u03c1 (Prod.fst \u2218 Prod.map id f \u207b\u00b9' s) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' s) ** rfl ** case this \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 this :   Measure.map (Prod.map id f) \u03c1 =     \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096           kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))       () \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2200 (s : Set \u211d),       MeasurableSet s \u2192         \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) ** have heq := eq_condKernel_of_measure_eq_compProd_real _ _ this ** case this \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 this :   Measure.map (Prod.map id f) \u03c1 =     \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096           kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))       () heq :   \u2200\u1d50 (x : \u03b1) \u2202Measure.fst (Measure.map (Prod.map id f) \u03c1),     \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x =       \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2200 (s : Set \u211d),       MeasurableSet s \u2192         \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) ** rw [hprod] at heq ** case this \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 this :   Measure.map (Prod.map id f) \u03c1 =     \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096           kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))       () heq :   \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x =       \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2200 (s : Set \u211d),       MeasurableSet s \u2192         \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) ** filter_upwards [heq] with x hx s hs ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 this :   Measure.map (Prod.map id f) \u03c1 =     \u2191(kernel.const Unit (Measure.fst (Measure.map (Prod.map id f) \u03c1)) \u2297\u2096           kernel.prodMkLeft Unit (kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)))       () heq :   \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1,     \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x =       \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x x : \u03b1 hx :   \u2191(kernel.map (Measure.condKernel \u03c1) f (_ : Measurable f)) x = \u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x s : Set \u211d hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(Measure.condKernel (Measure.map (Prod.map id f) \u03c1)) x) s = \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' s) ** rw [\u2190 hx, kernel.map_apply, Measure.map_apply hf.measurable hs] ** case hinteq \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2200 (x : \u03b1),     \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =       \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} ** intro x ** case hinteq \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s x : \u03b1 \u22a2 \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =     \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} ** rw [Measure.map_apply hf.measurable] ** case hinteq \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s x : \u03b1 \u22a2 \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) (f \u207b\u00b9' {c | (x, c) \u2208 s}) =     \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} ** rfl ** case hinteq \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s x : \u03b1 \u22a2 MeasurableSet {c | (x, c) \u2208 s} ** exact measurable_prod_mk_left hs ** case this.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s hinteq :   \u2200 (x : \u03b1),     \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =       \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) a) {x | (a, x) \u2208 Prod.map id f \u207b\u00b9' s} \u2202Measure.fst \u03c1 =     \u222b\u207b (x : \u03b1), \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, f c) \u2208 s} \u2202Measure.fst \u03c1 ** rfl ** case this.h.hs \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } inst\u271d : IsFiniteKernel \u03ba h\u03ba : \u03c1 = \u2191(kernel.const Unit (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft Unit \u03ba) () f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 h\u03c1'def : \u03c1' = Measure.map (Prod.map id f) \u03c1 h\u03c1' : Measure.fst \u03c1' = Measure.fst \u03c1 h\u03c1'' : \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, \u2191(kernel.map \u03ba f (_ : Measurable f)) x = \u2191(Measure.condKernel \u03c1') x hprod : Measure.fst (Measure.map (Prod.map id f) \u03c1) = Measure.fst \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s hinteq :   \u2200 (x : \u03b1),     \u2191\u2191(Measure.map f (\u2191(Measure.condKernel \u03c1) x)) {c | (x, c) \u2208 s} =       \u2191\u2191(\u2191(Measure.condKernel \u03c1) x) {c | (x, c) \u2208 Prod.map id f \u207b\u00b9' s} \u22a2 MeasurableSet (Prod.map id f \u207b\u00b9' s) ** exact measurable_id.prod_map hf.measurable hs ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec\u2082.curry ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 f : \u03b1 \u00d7 \u03b2 \u2192 \u03c3 \u22a2 Primrec\u2082 (Function.curry f) \u2194 Primrec f ** rw [\u2190 uncurry, Function.uncurry_curry] ** Qed",
        "informal": ""
    },
    {
        "formal": "ack_strict_mono_left' ** m : \u2115 _h : 0 < m + 1 \u22a2 ack 0 0 < ack (m + 1) 0 ** simpa using one_lt_ack_succ_right m 0 ** m n : \u2115 h : 0 < m + 1 \u22a2 ack 0 (n + 1) < ack (m + 1) (n + 1) ** rw [ack_zero, ack_succ_succ] ** m n : \u2115 h : 0 < m + 1 \u22a2 n + 1 + 1 < ack m (ack (m + 1) n) ** apply lt_of_le_of_lt (le_trans _ <| add_le_add_left (add_add_one_le_ack _ _) m) (add_lt_ack _ _) ** m n : \u2115 h : 0 < m + 1 \u22a2 n + 1 + 1 \u2264 m + (m + 1 + n + 1) ** linarith ** m\u2081 m\u2082 : \u2115 h : m\u2081 + 1 < m\u2082 + 1 \u22a2 ack (m\u2081 + 1) 0 < ack (m\u2082 + 1) 0 ** simpa using ack_strict_mono_left' 1 ((add_lt_add_iff_right 1).1 h) ** m\u2081 m\u2082 n : \u2115 h : m\u2081 + 1 < m\u2082 + 1 \u22a2 ack (m\u2081 + 1) (n + 1) < ack (m\u2082 + 1) (n + 1) ** rw [ack_succ_succ, ack_succ_succ] ** m\u2081 m\u2082 n : \u2115 h : m\u2081 + 1 < m\u2082 + 1 \u22a2 ack m\u2081 (ack (m\u2081 + 1) n) < ack m\u2082 (ack (m\u2082 + 1) n) ** exact\n  (ack_strict_mono_left' _ <| (add_lt_add_iff_right 1).1 h).trans\n    (ack_strictMono_right _ <| ack_strict_mono_left' n h) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.toJordanDecomposition_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 j : JordanDecomposition \u03b1 h : s = toSignedMeasure j \u22a2 toJordanDecomposition s = j ** rw [h, toJordanDecomposition_toSignedMeasure] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpIndL1_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G \u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x ** by_cases hs : MeasurableSet s ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G hs : MeasurableSet s \u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G hs : \u00acMeasurableSet s \u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G hs : MeasurableSet s \u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x ** by_cases h\u03bcs : \u03bc s = \u221e ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G hs : \u00acMeasurableSet s \u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x ** simp_rw [condexpIndL1_of_not_measurableSet hs] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G hs : \u00acMeasurableSet s \u22a2 0 = c \u2022 0 ** rw [smul_zero] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = \u22a4 \u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x ** simp_rw [condexpIndL1_of_measure_eq_top h\u03bcs] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = \u22a4 \u22a2 0 = c \u2022 0 ** rw [smul_zero] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G hs : MeasurableSet s h\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4 \u22a2 condexpIndL1 hm \u03bc s (c \u2022 x) = c \u2022 condexpIndL1 hm \u03bc s x ** simp_rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs h\u03bcs] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) c : \u211d x : G hs : MeasurableSet s h\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4 \u22a2 condexpIndL1Fin hm hs h\u03bcs (c \u2022 x) = c \u2022 condexpIndL1Fin hm hs h\u03bcs x ** exact condexpIndL1Fin_smul hs h\u03bcs c x ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_smul_left' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f : \u03b1 \u2192 E \u22a2 setToFun \u03bc T' hT' f = c \u2022 setToFun \u03bc T hT f ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f : \u03b1 \u2192 E hf : Integrable f \u22a2 setToFun \u03bc T' hT' f = c \u2022 setToFun \u03bc T hT f ** simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left' hT hT' c h_smul] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 setToFun \u03bc T' hT' f = c \u2022 setToFun \u03bc T hT f ** simp_rw [setToFun_undef _ hf, smul_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "parallelepiped_single ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a : \u03b9 \u2192 \u211d \u22a2 (parallelepiped fun i => Pi.single i (a i)) = uIcc 0 a ** ext x ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d \u22a2 (x \u2208 parallelepiped fun i => Pi.single i (a i)) \u2194 x \u2208 uIcc 0 a ** simp_rw [Set.uIcc, mem_parallelepiped_iff, Set.mem_Icc, Pi.le_def, \u2190 forall_and, Pi.inf_apply,\n  Pi.sup_apply, \u2190 Pi.single_smul', Pi.one_apply, Pi.zero_apply, \u2190 Pi.smul_apply',\n  Finset.univ_sum_single (_ : \u03b9 \u2192 \u211d)] ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d \u22a2 (\u2203 t h, x = fun i => (t \u2022 a) i) \u2194 \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1 ** constructor ** case h.mp \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d \u22a2 (\u2203 t h, x = fun i => (t \u2022 a) i) \u2192 \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1 ** rintro \u27e8t, ht, rfl\u27e9 i ** case h.mp.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a t : \u03b9 \u2192 \u211d ht : \u2200 (x : \u03b9), 0 \u2264 t x \u2227 t x \u2264 1 i : \u03b9 \u22a2 0 \u2293 a i \u2264 (fun i => (t \u2022 a) i) i \u2227 (fun i => (t \u2022 a) i) i \u2264 0 \u2294 a i ** specialize ht i ** case h.mp.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a t : \u03b9 \u2192 \u211d i : \u03b9 ht : 0 \u2264 t i \u2227 t i \u2264 1 \u22a2 0 \u2293 a i \u2264 (fun i => (t \u2022 a) i) i \u2227 (fun i => (t \u2022 a) i) i \u2264 0 \u2294 a i ** simp_rw [smul_eq_mul, Pi.mul_apply] ** case h.mp.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a t : \u03b9 \u2192 \u211d i : \u03b9 ht : 0 \u2264 t i \u2227 t i \u2264 1 \u22a2 0 \u2293 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 \u2294 a i ** cases' le_total (a i) 0 with hai hai ** case h.mp.intro.intro.inl \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a t : \u03b9 \u2192 \u211d i : \u03b9 ht : 0 \u2264 t i \u2227 t i \u2264 1 hai : a i \u2264 0 \u22a2 0 \u2293 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 \u2294 a i ** rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] ** case h.mp.intro.intro.inl \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a t : \u03b9 \u2192 \u211d i : \u03b9 ht : 0 \u2264 t i \u2227 t i \u2264 1 hai : a i \u2264 0 \u22a2 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 ** exact \u27e8le_mul_of_le_one_left hai ht.2, mul_nonpos_of_nonneg_of_nonpos ht.1 hai\u27e9 ** case h.mp.intro.intro.inr \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a t : \u03b9 \u2192 \u211d i : \u03b9 ht : 0 \u2264 t i \u2227 t i \u2264 1 hai : 0 \u2264 a i \u22a2 0 \u2293 a i \u2264 t i * a i \u2227 t i * a i \u2264 0 \u2294 a i ** rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai] ** case h.mp.intro.intro.inr \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a t : \u03b9 \u2192 \u211d i : \u03b9 ht : 0 \u2264 t i \u2227 t i \u2264 1 hai : 0 \u2264 a i \u22a2 0 \u2264 t i * a i \u2227 t i * a i \u2264 a i ** exact \u27e8mul_nonneg ht.1 hai, mul_le_of_le_one_left hai ht.2\u27e9 ** case h.mpr \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d \u22a2 (\u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1) \u2192 \u2203 t h, x = fun i => (t \u2022 a) i ** intro h ** case h.mpr \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d h : \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1 \u22a2 \u2203 t h, x = fun i => (t \u2022 a) i ** refine' \u27e8fun i => x i / a i, fun i => _, funext fun i => _\u27e9 ** case h.mpr.refine'_1 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d h : \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1 i : \u03b9 \u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1 ** specialize h i ** case h.mpr.refine'_1 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i \u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1 ** cases' le_total (a i) 0 with hai hai ** case h.mpr.refine'_1.inl \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i hai : a i \u2264 0 \u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1 ** rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h ** case h.mpr.refine'_1.inl \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : a i \u2264 x i \u2227 x i \u2264 0 hai : a i \u2264 0 \u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1 ** exact \u27e8div_nonneg_of_nonpos h.2 hai, div_le_one_of_ge h.1 hai\u27e9 ** case h.mpr.refine'_1.inr \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i hai : 0 \u2264 a i \u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1 ** rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai] at h ** case h.mpr.refine'_1.inr \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : 0 \u2264 x i \u2227 x i \u2264 a i hai : 0 \u2264 a i \u22a2 0 \u2264 (fun i => x i / a i) i \u2227 (fun i => x i / a i) i \u2264 1 ** exact \u27e8div_nonneg h.1 hai, div_le_one_of_le h.2 hai\u27e9 ** case h.mpr.refine'_2 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d h : \u2200 (x_1 : \u03b9), 0 \u2293 a x_1 \u2264 x x_1 \u2227 x x_1 \u2264 0 \u2294 a x_1 i : \u03b9 \u22a2 x i = ((fun i => x i / a i) \u2022 a) i ** specialize h i ** case h.mpr.refine'_2 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i \u22a2 x i = ((fun i => x i / a i) \u2022 a) i ** simp only [smul_eq_mul, Pi.mul_apply] ** case h.mpr.refine'_2 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i \u22a2 x i = x i / a i * a i ** cases' eq_or_ne (a i) 0 with hai hai ** case h.mpr.refine'_2.inl \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i hai : a i = 0 \u22a2 x i = x i / a i * a i ** rw [hai, inf_idem, sup_idem, \u2190 le_antisymm_iff] at h ** case h.mpr.refine'_2.inl \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : 0 = x i hai : a i = 0 \u22a2 x i = x i / a i * a i ** rw [hai, \u2190 h, zero_div, zero_mul] ** case h.mpr.refine'_2.inr \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2076 : Fintype \u03b9 inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \u211d E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \u211d F inst\u271d : DecidableEq \u03b9 a x : \u03b9 \u2192 \u211d i : \u03b9 h : 0 \u2293 a i \u2264 x i \u2227 x i \u2264 0 \u2294 a i hai : a i \u2260 0 \u22a2 x i = x i / a i * a i ** rw [div_mul_cancel _ hai] ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_signed_sum' ** \u03b1\u271d \u03b1 : Type u_1 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 n : \u2115 h : \u2211 i in s, Int.natAbs (f i) \u2264 n \u22a2 \u2203 \u03b2 x sgn g,     (\u2200 (b : \u03b2), \u00acg b \u2208 s \u2192 sgn b = 0) \u2227       Fintype.card \u03b2 = n \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i : \u03b2, if g i = a then \u2191(sgn i) else 0) = f a ** obtain \u27e8\u03b2, _, sgn, g, hg, h\u03b2, hf\u27e9 := exists_signed_sum s f ** case intro.intro.intro.intro.intro.intro \u03b1\u271d \u03b1 : Type u_1 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 n : \u2115 h : \u2211 i in s, Int.natAbs (f i) \u2264 n \u03b2 : Type u_1 w\u271d : Fintype \u03b2 sgn : \u03b2 \u2192 SignType g : \u03b2 \u2192 \u03b1 hg : \u2200 (b : \u03b2), g b \u2208 s h\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a) hf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a \u22a2 \u2203 \u03b2 x sgn g,     (\u2200 (b : \u03b2), \u00acg b \u2208 s \u2192 sgn b = 0) \u2227       Fintype.card \u03b2 = n \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i : \u03b2, if g i = a then \u2191(sgn i) else 0) = f a ** refine'\n  \u27e8Sum \u03b2 (Fin (n - \u2211 i in s, (f i).natAbs)), inferInstance, Sum.elim sgn 0,\n    Sum.elim g (Classical.arbitrary (Fin (n - Finset.sum s fun i => Int.natAbs (f i)) \u2192 \u03b1)),\n      _, by simp [h\u03b2, h], fun a ha => by simp [hf _ ha]\u27e9 ** case intro.intro.intro.intro.intro.intro \u03b1\u271d \u03b1 : Type u_1 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 n : \u2115 h : \u2211 i in s, Int.natAbs (f i) \u2264 n \u03b2 : Type u_1 w\u271d : Fintype \u03b2 sgn : \u03b2 \u2192 SignType g : \u03b2 \u2192 \u03b1 hg : \u2200 (b : \u03b2), g b \u2208 s h\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a) hf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a \u22a2 \u2200 (b : \u03b2 \u2295 Fin (n - \u2211 i in s, Int.natAbs (f i))),     \u00acSum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) b \u2208 s \u2192 Sum.elim sgn 0 b = 0 ** rintro (b | b) hb ** \u03b1\u271d \u03b1 : Type u_1 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 n : \u2115 h : \u2211 i in s, Int.natAbs (f i) \u2264 n \u03b2 : Type u_1 w\u271d : Fintype \u03b2 sgn : \u03b2 \u2192 SignType g : \u03b2 \u2192 \u03b1 hg : \u2200 (b : \u03b2), g b \u2208 s h\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a) hf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a \u22a2 Fintype.card (\u03b2 \u2295 Fin (n - \u2211 i in s, Int.natAbs (f i))) = n ** simp [h\u03b2, h] ** \u03b1\u271d \u03b1 : Type u_1 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 n : \u2115 h : \u2211 i in s, Int.natAbs (f i) \u2264 n \u03b2 : Type u_1 w\u271d : Fintype \u03b2 sgn : \u03b2 \u2192 SignType g : \u03b2 \u2192 \u03b1 hg : \u2200 (b : \u03b2), g b \u2208 s h\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a) hf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a a : \u03b1 ha : a \u2208 s \u22a2 (\u2211 i : \u03b2 \u2295 Fin (n - \u2211 i in s, Int.natAbs (f i)),       if Sum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) i = a then \u2191(Sum.elim sgn 0 i)       else 0) =     f a ** simp [hf _ ha] ** case intro.intro.intro.intro.intro.intro.inl \u03b1\u271d \u03b1 : Type u_1 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 n : \u2115 h : \u2211 i in s, Int.natAbs (f i) \u2264 n \u03b2 : Type u_1 w\u271d : Fintype \u03b2 sgn : \u03b2 \u2192 SignType g : \u03b2 \u2192 \u03b1 hg : \u2200 (b : \u03b2), g b \u2208 s h\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a) hf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a b : \u03b2 hb : \u00acSum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) (Sum.inl b) \u2208 s \u22a2 Sum.elim sgn 0 (Sum.inl b) = 0 ** cases hb (hg _) ** case intro.intro.intro.intro.intro.intro.inr \u03b1\u271d \u03b1 : Type u_1 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 n : \u2115 h : \u2211 i in s, Int.natAbs (f i) \u2264 n \u03b2 : Type u_1 w\u271d : Fintype \u03b2 sgn : \u03b2 \u2192 SignType g : \u03b2 \u2192 \u03b1 hg : \u2200 (b : \u03b2), g b \u2208 s h\u03b2 : Fintype.card \u03b2 = \u2211 a in s, Int.natAbs (f a) hf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b : \u03b2, if g b = a then \u2191(sgn b) else 0) = f a b : Fin (n - \u2211 i in s, Int.natAbs (f i)) hb : \u00acSum.elim g (Classical.arbitrary (Fin (n - \u2211 i in s, Int.natAbs (f i)) \u2192 \u03b1)) (Sum.inr b) \u2208 s \u22a2 Sum.elim sgn 0 (Sum.inr b) = 0 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.ext_iff ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : { x // x \u2208 Lp E p } h : f = g \u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191g ** rw [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.coeFn_comp\u2082Measurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9\u2074 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : PseudoMetrizableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b3 inst\u271d\u2076 : PseudoMetrizableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b4 inst\u271d\u00b9 : OpensMeasurableSpace \u03b4 inst\u271d : SecondCountableTopology \u03b4 g : \u03b2 \u2192 \u03b3 \u2192 \u03b4 hg : Measurable (uncurry g) f\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2 f\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3 \u22a2 \u2191(comp\u2082Measurable g hg f\u2081 f\u2082) =\u1d50[\u03bc] fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a) ** rw [comp\u2082Measurable_eq_mk] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9\u2074 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b2 inst\u271d\u2079 : PseudoMetrizableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b3 inst\u271d\u2076 : PseudoMetrizableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b4 inst\u271d\u00b9 : OpensMeasurableSpace \u03b4 inst\u271d : SecondCountableTopology \u03b4 g : \u03b2 \u2192 \u03b3 \u2192 \u03b4 hg : Measurable (uncurry g) f\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2 f\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3 \u22a2 \u2191(mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)) =\u1d50[\u03bc] fun a =>     g (\u2191f\u2081 a) (\u2191f\u2082 a) ** apply coeFn_mk ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_liminf_le' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e h_meas : \u2200 (n : \u2115), AEMeasurable (f n) \u22a2 \u222b\u207b (a : \u03b1), liminf (fun n => f n a) atTop \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a06 n, \u2a05 i, \u2a05 (_ : i \u2265 n), f i a \u2202\u03bc ** simp only [liminf_eq_iSup_iInf_of_nat] ** Qed",
        "informal": ""
    },
    {
        "formal": "RegularExpression.rmatch_iff_matches' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P : RegularExpression \u03b1 \u22a2 \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ** intro x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P : RegularExpression \u03b1 x : List \u03b1 \u22a2 rmatch P x = true \u2194 x \u2208 matches' P ** induction P generalizing x ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch zero x = true \u2194 x \u2208 matches' zero  case epsilon \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch epsilon x = true \u2194 x \u2208 matches' epsilon  case char \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b a\u271d : \u03b1 x : List \u03b1 \u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)  case plus \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (plus a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (plus a\u271d\u00b9 a\u271d)  case comp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (comp a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (comp a\u271d\u00b9 a\u271d)  case star \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d) ** all_goals\n  try rw [zero_def]\n  try rw [one_def]\n  try rw [plus_def]\n  try rw [comp_def] ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch 0 x = true \u2194 x \u2208 matches' 0  case epsilon \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1  case char \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b a\u271d : \u03b1 x : List \u03b1 \u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)  case plus \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)  case comp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)  case star \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d) ** case zero =>\n  rw [zero_rmatch]\n  tauto ** case epsilon \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1  case char \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b a\u271d : \u03b1 x : List \u03b1 \u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)  case plus \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)  case comp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)  case star \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d) ** case epsilon =>\n  rw [one_rmatch_iff]\n  rfl ** case char \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b a\u271d : \u03b1 x : List \u03b1 \u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d)  case plus \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)  case comp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)  case star \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d) ** case char =>\n  rw [char_rmatch_iff]\n  rfl ** case plus \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d)  case comp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (a\u271d\u00b9 * a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 * a\u271d)  case star \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d) ** case plus _ _ ih\u2081 ih\u2082 =>\n  rw [add_rmatch_iff, ih\u2081, ih\u2082]\n  rfl ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch zero x = true \u2194 x \u2208 matches' zero ** try rw [zero_def] ** case epsilon \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch epsilon x = true \u2194 x \u2208 matches' epsilon ** try rw [one_def] ** case plus \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (plus a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (plus a\u271d\u00b9 a\u271d) ** try rw [plus_def] ** case comp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (comp a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (comp a\u271d\u00b9 a\u271d) ** try rw [comp_def] ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch zero x = true \u2194 x \u2208 matches' zero ** rw [zero_def] ** case epsilon \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch epsilon x = true \u2194 x \u2208 matches' epsilon ** rw [one_def] ** case plus \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (plus a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (plus a\u271d\u00b9 a\u271d) ** rw [plus_def] ** case comp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 a_ih\u271d\u00b9 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 a_ih\u271d : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (comp a\u271d\u00b9 a\u271d) x = true \u2194 x \u2208 matches' (comp a\u271d\u00b9 a\u271d) ** rw [comp_def] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch 0 x = true \u2194 x \u2208 matches' 0 ** rw [zero_rmatch] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 false = true \u2194 x \u2208 matches' 0 ** tauto ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 rmatch 1 x = true \u2194 x \u2208 matches' 1 ** rw [one_rmatch_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 x : List \u03b1 \u22a2 x = [] \u2194 x \u2208 matches' 1 ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b a\u271d : \u03b1 x : List \u03b1 \u22a2 rmatch (char a\u271d) x = true \u2194 x \u2208 matches' (char a\u271d) ** rw [char_rmatch_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b a\u271d : \u03b1 x : List \u03b1 \u22a2 x = [a\u271d] \u2194 x \u2208 matches' (char a\u271d) ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 ih\u2082 : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (a\u271d\u00b9 + a\u271d) x = true \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d) ** rw [add_rmatch_iff, ih\u2081, ih\u2082] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d\u00b9 a\u271d : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch a\u271d\u00b9 x = true \u2194 x \u2208 matches' a\u271d\u00b9 ih\u2082 : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 x \u2208 matches' a\u271d\u00b9 \u2228 x \u2208 matches' a\u271d \u2194 x \u2208 matches' (a\u271d\u00b9 + a\u271d) ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x : List \u03b1 \u22a2 rmatch (P * Q) x = true \u2194 x \u2208 matches' (P * Q) ** simp only [mul_rmatch_iff, comp_def, Language.mul_def, exists_and_left, Set.mem_image2,\n  Set.image_prod] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x : List \u03b1 \u22a2 (\u2203 t u, x = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true) \u2194 x \u2208 matches' (P * Q) ** constructor ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x : List \u03b1 \u22a2 (\u2203 t u, x = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true) \u2192 x \u2208 matches' (P * Q) ** rintro \u27e8x, y, hsum, hmatch\u2081, hmatch\u2082\u27e9 ** case mp.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x\u271d x y : List \u03b1 hsum : x\u271d = x ++ y hmatch\u2081 : rmatch P x = true hmatch\u2082 : rmatch Q y = true \u22a2 x\u271d \u2208 matches' (P * Q) ** rw [ih\u2081] at hmatch\u2081 ** case mp.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x\u271d x y : List \u03b1 hsum : x\u271d = x ++ y hmatch\u2081 : x \u2208 matches' P hmatch\u2082 : rmatch Q y = true \u22a2 x\u271d \u2208 matches' (P * Q) ** rw [ih\u2082] at hmatch\u2082 ** case mp.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x\u271d x y : List \u03b1 hsum : x\u271d = x ++ y hmatch\u2081 : x \u2208 matches' P hmatch\u2082 : y \u2208 matches' Q \u22a2 x\u271d \u2208 matches' (P * Q) ** exact \u27e8x, y, hmatch\u2081, hmatch\u2082, hsum.symm\u27e9 ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x : List \u03b1 \u22a2 x \u2208 matches' (P * Q) \u2192 \u2203 t u, x = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true ** rintro \u27e8x, y, hmatch\u2081, hmatch\u2082, hsum\u27e9 ** case mpr.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x\u271d x y : List \u03b1 hmatch\u2081 : x \u2208 matches' P hmatch\u2082 : y \u2208 matches' Q hsum : (fun x x_1 => x ++ x_1) x y = x\u271d \u22a2 \u2203 t u, x\u271d = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true ** rw [\u2190 ih\u2081] at hmatch\u2081 ** case mpr.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x\u271d x y : List \u03b1 hmatch\u2081 : rmatch P x = true hmatch\u2082 : y \u2208 matches' Q hsum : (fun x x_1 => x ++ x_1) x y = x\u271d \u22a2 \u2203 t u, x\u271d = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true ** rw [\u2190 ih\u2082] at hmatch\u2082 ** case mpr.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 ih\u2081 : \u2200 (x : List \u03b1), rmatch P x = true \u2194 x \u2208 matches' P ih\u2082 : \u2200 (x : List \u03b1), rmatch Q x = true \u2194 x \u2208 matches' Q x\u271d x y : List \u03b1 hmatch\u2081 : rmatch P x = true hmatch\u2082 : rmatch Q y = true hsum : (fun x x_1 => x ++ x_1) x y = x\u271d \u22a2 \u2203 t u, x\u271d = t ++ u \u2227 rmatch P t = true \u2227 rmatch Q u = true ** exact \u27e8x, y, hsum.symm, hmatch\u2081, hmatch\u2082\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 rmatch (star a\u271d) x = true \u2194 x \u2208 matches' (star a\u271d) ** rw [star_rmatch_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 (\u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 t \u2260 [] \u2227 rmatch a\u271d t = true) \u2194 x \u2208 matches' (star a\u271d) ** simp only [ne_eq, matches', Language.kstar_def_nonempty, mem_setOf_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 (\u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true) \u2194     x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []} ** constructor ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 (\u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true) \u2192     x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []}  case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []} \u2192     \u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true ** all_goals\n  rintro \u27e8S, hx, hS\u27e9\n  refine' \u27e8S, hx, _\u27e9\n  intro y\n  specialize hS y ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 \u22a2 x \u2208 {x | \u2203 S, x = join S \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = []} \u2192     \u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true ** rintro \u27e8S, hx, hS\u27e9 ** case mpr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 S : List (List \u03b1) hx : x = join S hS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = [] \u22a2 \u2203 S, x = join S \u2227 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true ** refine' \u27e8S, hx, _\u27e9 ** case mpr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 S : List (List \u03b1) hx : x = join S hS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = [] \u22a2 \u2200 (t : List \u03b1), t \u2208 S \u2192 \u00act = [] \u2227 rmatch a\u271d t = true ** intro y ** case mpr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 S : List (List \u03b1) hx : x = join S hS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = [] y : List \u03b1 \u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true ** specialize hS y ** case mp.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 S : List (List \u03b1) hx : x = join S y : List \u03b1 hS : y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true \u22a2 y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = [] ** rw [\u2190 ih y] ** case mp.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 S : List (List \u03b1) hx : x = join S y : List \u03b1 hS : y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true \u22a2 y \u2208 S \u2192 rmatch a\u271d y = true \u2227 \u00acy = [] ** tauto ** case mpr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 S : List (List \u03b1) hx : x = join S y : List \u03b1 hS : y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = [] \u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 rmatch a\u271d y = true ** rw [ih y] ** case mpr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 a\u271d : RegularExpression \u03b1 ih : \u2200 (x : List \u03b1), rmatch a\u271d x = true \u2194 x \u2208 matches' a\u271d x : List \u03b1 S : List (List \u03b1) hx : x = join S y : List \u03b1 hS : y \u2208 S \u2192 y \u2208 matches' a\u271d \u2227 \u00acy = [] \u22a2 y \u2208 S \u2192 \u00acy = [] \u2227 y \u2208 matches' a\u271d ** tauto ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image_add_left_Ico ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1 inst\u271d\u00b2 : ExistsAddOfLE \u03b1 inst\u271d\u00b9 : LocallyFiniteOrder \u03b1 inst\u271d : DecidableEq \u03b1 a b c : \u03b1 \u22a2 image ((fun x x_1 => x + x_1) c) (Ico a b) = Ico (c + a) (c + b) ** rw [\u2190 map_add_left_Ico, map_eq_image, addLeftEmbedding, Embedding.coeFn_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc N : \u2115 \u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc ** by_cases hab : a < b ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc N : \u2115 hab : a < b \u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc ** exact mul_integral_upcrossingsBefore_le_integral_pos_part_aux hf hab ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc N : \u2115 hab : \u00aca < b \u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc ** rw [not_lt, \u2190 sub_nonpos] at hab ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc N : \u2115 hab\u271d : b \u2264 a hab : b - a \u2264 0 \u22a2 (b - a) * \u222b (x : \u03a9), \u2191(upcrossingsBefore a b f N x) \u2202\u03bc \u2264 \u222b (x : \u03a9), (fun \u03c9 => (f N \u03c9 - a)\u207a) x \u2202\u03bc ** exact le_trans (mul_nonpos_of_nonpos_of_nonneg hab (integral_nonneg fun \u03c9 => Nat.cast_nonneg _))\n  (integral_nonneg fun \u03c9 => LatticeOrderedGroup.pos_nonneg _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_add_const_uIcc ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 (fun x => x + a) \u207b\u00b9' [[b, c]] = [[b - a, c - a]] ** simpa only [add_comm] using preimage_const_add_uIcc a b c ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_le_snorm_top_mul_snorm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc ** by_cases hp_top : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc ** by_cases hp_zero : p = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc ** simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top, snorm_exponent_top, snormEssSup] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : p = \u22a4 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc ** simp_rw [hp_top, snorm_exponent_top] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : p = \u22a4 \u22a2 snormEssSup (fun x => b (f x) (g x)) \u03bc \u2264 snormEssSup f \u03bc * snormEssSup g \u03bc ** refine' le_trans (essSup_mono_ae <| h.mono fun a ha => _) (ENNReal.essSup_mul_le _ _) ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : p = \u22a4 a : \u03b1 ha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a \u22a2 (fun x => \u2191\u2016(fun x => b (f x) (g x)) x\u2016\u208a) a \u2264 ((fun x => \u2191\u2016f x\u2016\u208a) * fun x => \u2191\u2016g x\u2016\u208a) a ** simp_rw [Pi.mul_apply, \u2190 ENNReal.coe_mul, ENNReal.coe_le_coe] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : p = \u22a4 a : \u03b1 ha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a \u22a2 \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a ** exact ha ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : p = 0 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc ** simp only [hp_zero, snorm_exponent_zero, mul_zero, le_zero_iff] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016b (f x) (g x)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264     (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) ** refine' ENNReal.rpow_le_rpow _ (one_div_nonneg.mpr ENNReal.toReal_nonneg) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016b (f x) (g x)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264     \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc ** refine' lintegral_mono_ae (h.mono fun a ha => _) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 a : \u03b1 ha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a \u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a ^ ENNReal.toReal p \u2264 \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p ** rw [\u2190 ENNReal.mul_rpow_of_nonneg _ _ ENNReal.toReal_nonneg] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 a : \u03b1 ha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a \u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a ^ ENNReal.toReal p \u2264 (\u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a) ^ ENNReal.toReal p ** refine' ENNReal.rpow_le_rpow _ ENNReal.toReal_nonneg ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 a : \u03b1 ha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a \u22a2 \u2191\u2016b (f a) (g a)\u2016\u208a \u2264 \u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a ** rw [\u2190 ENNReal.coe_mul, ENNReal.coe_le_coe] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 a : \u03b1 ha : \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a \u22a2 \u2016b (f a) (g a)\u2016\u208a \u2264 \u2016f a\u2016\u208a * \u2016g a\u2016\u208a ** exact ha ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) \u2264     (\u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^       (1 / ENNReal.toReal p) ** refine' ENNReal.rpow_le_rpow _ _ ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264     \u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc  case refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 0 \u2264 1 / ENNReal.toReal p ** swap ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2264     \u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc ** refine' lintegral_mono_ae _ ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc,     \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2264       essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p ** filter_upwards [@ENNReal.ae_le_essSup _ _ \u03bc fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)] with x hx ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 x : \u03b1 hx : \u2191\u2016f x\u2016\u208a \u2264 essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc \u22a2 \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2264     essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p ** exact mul_le_mul_right' (ENNReal.rpow_le_rpow hx ENNReal.toReal_nonneg) _ ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 0 \u2264 1 / ENNReal.toReal p ** rw [one_div_nonneg] ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 0 \u2264 ENNReal.toReal p ** exact ENNReal.toReal_nonneg ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 (\u222b\u207b (x : \u03b1), essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^       (1 / ENNReal.toReal p) =     essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) ** rw [lintegral_const_mul''] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^       (1 / ENNReal.toReal p) =     essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)  case hf \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 AEMeasurable fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p ** swap ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^       (1 / ENNReal.toReal p) =     essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) ** rw [ENNReal.mul_rpow_of_nonneg] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *       (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =     essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)  case hz \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 0 \u2264 1 / ENNReal.toReal p ** swap ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 (essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *       (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =     essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc * (\u222b\u207b (x : \u03b1), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) ** rw [\u2190 ENNReal.rpow_mul, one_div, mul_inv_cancel, ENNReal.rpow_one] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 ENNReal.toReal p \u2260 0 ** rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 \u00acp = 0 \u2227 \u00acp = \u22a4 ** exact \u27e8hp_zero, hp_top\u27e9 ** case hf \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 AEMeasurable fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p ** exact hg.nnnorm.aemeasurable.coe_nnreal_ennreal.pow aemeasurable_const ** case hz \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 0 \u2264 1 / ENNReal.toReal p ** rw [one_div_nonneg] ** case hz \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p : \u211d\u22650\u221e f : \u03b1 \u2192 E g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hp_top : \u00acp = \u22a4 hp_zero : \u00acp = 0 \u22a2 0 \u2264 ENNReal.toReal p ** exact ENNReal.toReal_nonneg ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM0.Machine.map_step ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081 : \u039b \u2192 \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q q : \u039b T : Tape \u0393 h : { q := q, Tape := T }.q \u2208 S \u22a2 Option.map (Cfg.map f\u2081 g\u2081) (step M { q := q, Tape := T }) =     step (map M f\u2081 f\u2082 g\u2081 g\u2082) (Cfg.map f\u2081 g\u2081 { q := q, Tape := T }) ** unfold step Machine.map Cfg.map ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081 : \u039b \u2192 \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q q : \u039b T : Tape \u0393 h : { q := q, Tape := T }.q \u2208 S \u22a2 Option.map       (fun x =>         match x with         | { q := q, Tape := T } => { q := g\u2081 q, Tape := Tape.map f\u2081 T })       (match { q := q, Tape := T } with       | { q := q, Tape := T } =>         Option.map           (fun x =>             match x with             | (q', a) =>               { q := q',                 Tape :=                   match a with                   | Stmt.move d => Tape.move d T                   | Stmt.write a => Tape.write a T })           (M q T.head)) =     match       match { q := q, Tape := T } with       | { q := q, Tape := T } => { q := g\u2081 q, Tape := Tape.map f\u2081 T } with     | { q := q, Tape := T } =>       Option.map         (fun x =>           match x with           | (q', a) =>             { q := q',               Tape :=                 match a with                 | Stmt.move d => Tape.move d T                 | Stmt.write a => Tape.write a T })         (match q, T.head with         | q, l => Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (M (g\u2082 q) (PointedMap.f f\u2082 l))) ** simp only [Turing.Tape.map_fst, g\u2082\u2081 q h, f\u2082\u2081 _] ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081 : \u039b \u2192 \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q q : \u039b T : Tape \u0393 h : { q := q, Tape := T }.q \u2208 S \u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })       (Option.map         (fun x =>           { q := x.1,             Tape :=               match x.2 with               | Stmt.move d => Tape.move d T               | Stmt.write a => Tape.write a T })         (M q T.head)) =     Option.map       (fun x =>         { q := x.1,           Tape :=             match x.2 with             | Stmt.move d => Tape.move d (Tape.map f\u2081 T)             | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })       (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (M q T.head)) ** rcases M q T.1 with (_ | \u27e8q', d | a\u27e9) ** case none \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081 : \u039b \u2192 \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q q : \u039b T : Tape \u0393 h : { q := q, Tape := T }.q \u2208 S \u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })       (Option.map         (fun x =>           { q := x.1,             Tape :=               match x.2 with               | Stmt.move d => Tape.move d T               | Stmt.write a => Tape.write a T })         none) =     Option.map       (fun x =>         { q := x.1,           Tape :=             match x.2 with             | Stmt.move d => Tape.move d (Tape.map f\u2081 T)             | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })       (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) none) ** rfl ** case some.mk.move \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081 : \u039b \u2192 \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q q : \u039b T : Tape \u0393 h : { q := q, Tape := T }.q \u2208 S q' : \u039b d : Dir \u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })       (Option.map         (fun x =>           { q := x.1,             Tape :=               match x.2 with               | Stmt.move d => Tape.move d T               | Stmt.write a => Tape.write a T })         (some (q', Stmt.move d))) =     Option.map       (fun x =>         { q := x.1,           Tape :=             match x.2 with             | Stmt.move d => Tape.move d (Tape.map f\u2081 T)             | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })       (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (some (q', Stmt.move d))) ** simp only [step, Cfg.map, Option.map_some', Tape.map_move f\u2081] ** case some.mk.move \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081 : \u039b \u2192 \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q q : \u039b T : Tape \u0393 h : { q := q, Tape := T }.q \u2208 S q' : \u039b d : Dir \u22a2 some { q := g\u2081 q', Tape := Tape.move d (Tape.map f\u2081 T) } =     some       { q := (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.move d)).1,         Tape :=           match (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.move d)).2 with           | Stmt.move d => Tape.move d (Tape.map f\u2081 T)           | Stmt.write a => Tape.write a (Tape.map f\u2081 T) } ** rfl ** case some.mk.write \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081 : \u039b \u2192 \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q q : \u039b T : Tape \u0393 h : { q := q, Tape := T }.q \u2208 S q' : \u039b a : \u0393 \u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })       (Option.map         (fun x =>           { q := x.1,             Tape :=               match x.2 with               | Stmt.move d => Tape.move d T               | Stmt.write a => Tape.write a T })         (some (q', Stmt.write a))) =     Option.map       (fun x =>         { q := x.1,           Tape :=             match x.2 with             | Stmt.move d => Tape.move d (Tape.map f\u2081 T)             | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })       (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (some (q', Stmt.write a))) ** simp only [step, Cfg.map, Option.map_some', Tape.map_write] ** case some.mk.write \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081 : \u039b \u2192 \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q q : \u039b T : Tape \u0393 h : { q := q, Tape := T }.q \u2208 S q' : \u039b a : \u0393 \u22a2 some { q := g\u2081 q', Tape := Tape.write (PointedMap.f f\u2081 a) (Tape.map f\u2081 T) } =     some       { q := (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.write a)).1,         Tape :=           match (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.write a)).2 with           | Stmt.move d => Tape.move d (Tape.map f\u2081 T)           | Stmt.write a => Tape.write a (Tape.map f\u2081 T) } ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_pi_closedBall ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u03b9 \u2192 \u211d r : \u211d hr : 0 \u2264 r \u22a2 \u2191\u2191volume (Metric.closedBall a r) = ofReal ((2 * r) ^ Fintype.card \u03b9) ** simp only [MeasureTheory.volume_pi_closedBall a hr, volume_closedBall, Finset.prod_const] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u03b9 \u2192 \u211d r : \u211d hr : 0 \u2264 r \u22a2 ofReal (2 * r) ^ Finset.card Finset.univ = ofReal ((2 * r) ^ Fintype.card \u03b9) ** exact (ENNReal.ofReal_pow (mul_nonneg zero_le_two hr) _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mem\u2112p_two_iff_integrable_sq ** \u03b1 : Type u_1 F : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup F f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc \u22a2 Mem\u2112p f 2 \u2194 Integrable fun x => f x ^ 2 ** convert mem\u2112p_two_iff_integrable_sq_norm hf using 3 ** case h.e'_2.h.e'_5.h \u03b1 : Type u_1 F : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup F f : \u03b1 \u2192 \u211d hf : AEStronglyMeasurable f \u03bc x\u271d : \u03b1 \u22a2 f x\u271d ^ 2 = \u2016f x\u271d\u2016 ^ 2 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.fix_fwd ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 f : \u03b1 \u2192. \u03b2 \u2295 \u03b1 b : \u03b2 a a' : \u03b1 hb : b \u2208 fix f a ha' : Sum.inr a' \u2208 f a \u22a2 b \u2208 fix f a' ** rwa [\u2190 fix_fwd_eq ha'] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.natAbs_valMinAbs_neg ** n : \u2115 a : ZMod n \u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a) ** by_cases h2a : 2 * a.val = n ** case pos n : \u2115 a : ZMod n h2a : 2 * val a = n \u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a) ** rw [a.neg_eq_self_iff.2 (Or.inr h2a)] ** case neg n : \u2115 a : ZMod n h2a : \u00ac2 * val a = n \u22a2 Int.natAbs (valMinAbs (-a)) = Int.natAbs (valMinAbs a) ** rw [valMinAbs_neg_of_ne_half h2a, Int.natAbs_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.sub_apply ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc \u22a2 \u2191\u2191(\u03bc - \u03bd) s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s ** let measure_sub : Measure \u03b1 := MeasureTheory.Measure.ofMeasurable\n  (fun (t : Set \u03b1) (_ : MeasurableSet t) => \u03bc t - \u03bd t) (by simp)\n  (by\n    intro g h_meas h_disj; simp only; rw [ENNReal.tsum_sub]\n    repeat' rw [\u2190 MeasureTheory.measure_iUnion h_disj h_meas]\n    exacts [MeasureTheory.measure_ne_top _ _, fun i => h\u2082 _ (h_meas _)]) ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) \u22a2 \u2191\u2191(\u03bc - \u03bd) s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s ** have h_measure_sub_add : \u03bd + measure_sub = \u03bc := by\n  ext1 t h_t_measurable_set\n  simp only [Pi.add_apply, coe_add]\n  rw [MeasureTheory.Measure.ofMeasurable_apply _ h_t_measurable_set, add_comm,\n    tsub_add_cancel_of_le (h\u2082 t h_t_measurable_set)] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc h_measure_sub_eq : \u03bc - \u03bd = measure_sub \u22a2 \u2191\u2191(\u03bc - \u03bd) s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s ** rw [h_measure_sub_eq] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc h_measure_sub_eq : \u03bc - \u03bd = measure_sub \u22a2 \u2191\u2191measure_sub s = \u2191\u2191\u03bc s - \u2191\u2191\u03bd s ** apply Measure.ofMeasurable_apply _ h\u2081 ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc \u22a2 (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) \u2205 (_ : MeasurableSet \u2205) = 0 ** simp ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc \u22a2 \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (h : \u2200 (i : \u2115), MeasurableSet (f i)),     Pairwise (Disjoint on f) \u2192       (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) =         \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (f i) (_ : MeasurableSet (f i)) ** intro g h_meas h_disj ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc g : \u2115 \u2192 Set \u03b1 h_meas : \u2200 (i : \u2115), MeasurableSet (g i) h_disj : Pairwise (Disjoint on g) \u22a2 (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =     \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i)) ** simp only ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc g : \u2115 \u2192 Set \u03b1 h_meas : \u2200 (i : \u2115), MeasurableSet (g i) h_disj : Pairwise (Disjoint on g) \u22a2 \u2191\u2191\u03bc (\u22c3 i, g i) - \u2191\u2191\u03bd (\u22c3 i, g i) = \u2211' (i : \u2115), (\u2191\u2191\u03bc (g i) - \u2191\u2191\u03bd (g i)) ** rw [ENNReal.tsum_sub] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc g : \u2115 \u2192 Set \u03b1 h_meas : \u2200 (i : \u2115), MeasurableSet (g i) h_disj : Pairwise (Disjoint on g) \u22a2 \u2191\u2191\u03bc (\u22c3 i, g i) - \u2191\u2191\u03bd (\u22c3 i, g i) = \u2211' (i : \u2115), \u2191\u2191\u03bc (g i) - \u2211' (i : \u2115), \u2191\u2191\u03bd (g i)  case h\u2081 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc g : \u2115 \u2192 Set \u03b1 h_meas : \u2200 (i : \u2115), MeasurableSet (g i) h_disj : Pairwise (Disjoint on g) \u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bd (g i) \u2260 \u22a4  case h\u2082 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc g : \u2115 \u2192 Set \u03b1 h_meas : \u2200 (i : \u2115), MeasurableSet (g i) h_disj : Pairwise (Disjoint on g) \u22a2 (fun i => \u2191\u2191\u03bd (g i)) \u2264 fun i => \u2191\u2191\u03bc (g i) ** repeat' rw [\u2190 MeasureTheory.measure_iUnion h_disj h_meas] ** case h\u2081 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc g : \u2115 \u2192 Set \u03b1 h_meas : \u2200 (i : \u2115), MeasurableSet (g i) h_disj : Pairwise (Disjoint on g) \u22a2 \u2191\u2191\u03bd (\u22c3 i, g i) \u2260 \u22a4  case h\u2082 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc g : \u2115 \u2192 Set \u03b1 h_meas : \u2200 (i : \u2115), MeasurableSet (g i) h_disj : Pairwise (Disjoint on g) \u22a2 (fun i => \u2191\u2191\u03bd (g i)) \u2264 fun i => \u2191\u2191\u03bc (g i) ** exacts [MeasureTheory.measure_ne_top _ _, fun i => h\u2082 _ (h_meas _)] ** case h\u2081 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc g : \u2115 \u2192 Set \u03b1 h_meas : \u2200 (i : \u2115), MeasurableSet (g i) h_disj : Pairwise (Disjoint on g) \u22a2 \u2211' (i : \u2115), \u2191\u2191\u03bd (g i) \u2260 \u22a4 ** rw [\u2190 MeasureTheory.measure_iUnion h_disj h_meas] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) \u22a2 \u03bd + measure_sub = \u03bc ** ext1 t h_t_measurable_set ** case h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) t : Set \u03b1 h_t_measurable_set : MeasurableSet t \u22a2 \u2191\u2191(\u03bd + measure_sub) t = \u2191\u2191\u03bc t ** simp only [Pi.add_apply, coe_add] ** case h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) t : Set \u03b1 h_t_measurable_set : MeasurableSet t \u22a2 \u2191\u2191\u03bd t +       \u2191\u2191(ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)               (_ :                 \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),                   Pairwise (Disjoint on g) \u2192                     (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =                       \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))))         t =     \u2191\u2191\u03bc t ** rw [MeasureTheory.Measure.ofMeasurable_apply _ h_t_measurable_set, add_comm,\n  tsub_add_cancel_of_le (h\u2082 t h_t_measurable_set)] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc \u22a2 \u03bc - \u03bd = measure_sub ** rw [MeasureTheory.Measure.sub_def] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc \u22a2 sInf {d | \u03bc \u2264 d + \u03bd} = measure_sub ** apply le_antisymm ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc \u22a2 measure_sub \u2264 sInf {d | \u03bc \u2264 d + \u03bd} ** apply le_sInf ** case a.a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc \u22a2 \u2200 (b : Measure \u03b1), b \u2208 {d | \u03bc \u2264 d + \u03bd} \u2192 measure_sub \u2264 b ** intro d h_d ** case a.a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc d : Measure \u03b1 h_d : d \u2208 {d | \u03bc \u2264 d + \u03bd} \u22a2 measure_sub \u2264 d ** rw [\u2190 h_measure_sub_add, mem_setOf_eq, add_comm d] at h_d ** case a.a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc d : Measure \u03b1 h_d : \u03bd + measure_sub \u2264 \u03bd + d \u22a2 measure_sub \u2264 d ** apply Measure.le_of_add_le_add_left h_d ** case a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc \u22a2 sInf {d | \u03bc \u2264 d + \u03bd} \u2264 measure_sub ** apply sInf_le ** case a.a \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bd h\u2081 : MeasurableSet s h\u2082 : \u03bd \u2264 \u03bc measure_sub : Measure \u03b1 :=   ofMeasurable (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (_ : \u2191\u2191\u03bc \u2205 - \u2191\u2191\u03bd \u2205 = 0)     (_ :       \u2200 \u2983g : \u2115 \u2192 Set \u03b1\u2984 (h_meas : \u2200 (i : \u2115), MeasurableSet (g i)),         Pairwise (Disjoint on g) \u2192           (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (\u22c3 i, g i) (_ : MeasurableSet (\u22c3 b, g b)) =             \u2211' (i : \u2115), (fun t x => \u2191\u2191\u03bc t - \u2191\u2191\u03bd t) (g i) (_ : MeasurableSet (g i))) h_measure_sub_add : \u03bd + measure_sub = \u03bc \u22a2 measure_sub \u2208 {d | \u03bc \u2264 d + \u03bd} ** simp [le_refl, add_comm, h_measure_sub_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.pow_right_comm ** a m n : Nat \u22a2 (a ^ m) ^ n = (a ^ n) ^ m ** rw [\u2190Nat.pow_mul, Nat.pow_mul'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.absolutelyContinuous_ennreal_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e \u22a2 s \u226a\u1d65 \u03bc \u2194 totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc ** constructor <;> intro h ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : s \u226a\u1d65 \u03bc \u22a2 totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc ** refine' Measure.AbsolutelyContinuous.mk fun S hS\u2081 hS\u2082 => _ ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : s \u226a\u1d65 \u03bc S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0 \u22a2 \u2191\u2191(totalVariation s) S = 0 ** obtain \u27e8i, hi\u2081, hi\u2082, hi\u2083, hpos, hneg\u27e9 := s.toJordanDecomposition_spec ** case mp.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : s \u226a\u1d65 \u03bc S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0 i : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i hi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c hpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082 hneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083 \u22a2 \u2191\u2191(totalVariation s) S = 0 ** rw [totalVariation, Measure.add_apply, hpos, hneg, toMeasureOfZeroLE_apply _ _ _ hS\u2081,\n  toMeasureOfLEZero_apply _ _ _ hS\u2081] ** case mp.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : s \u226a\u1d65 \u03bc S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0 i : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i hi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c hpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082 hneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083 \u22a2 \u2191{ val := \u2191s (i \u2229 S), property := (_ : 0 \u2264 \u2191s (i \u2229 S)) } +       \u2191{ val := -\u2191s (i\u1d9c \u2229 S), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 S)) } =     0 ** rw [\u2190 VectorMeasure.AbsolutelyContinuous.ennrealToMeasure] at h ** case mp.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : \u2200 \u2983s_1 : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s_1 = 0 \u2192 \u2191s s_1 = 0 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0 i : Set \u03b1 hi\u2081 : MeasurableSet i hi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i hi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c hpos : (toJordanDecomposition s).posPart = toMeasureOfZeroLE s i hi\u2081 hi\u2082 hneg : (toJordanDecomposition s).negPart = toMeasureOfLEZero s i\u1d9c (_ : MeasurableSet i\u1d9c) hi\u2083 \u22a2 \u2191{ val := \u2191s (i \u2229 S), property := (_ : 0 \u2264 \u2191s (i \u2229 S)) } +       \u2191{ val := -\u2191s (i\u1d9c \u2229 S), property := (_ : 0 \u2264 -\u2191s (i\u1d9c \u2229 S)) } =     0 ** simp [h (measure_mono_null (i.inter_subset_right S) hS\u2082),\n  h (measure_mono_null (i\u1d9c.inter_subset_right S) hS\u2082), \u2190 NNReal.eq_iff] ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc \u22a2 s \u226a\u1d65 \u03bc ** refine' VectorMeasure.AbsolutelyContinuous.mk fun S hS\u2081 hS\u2082 => _ ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : \u2191\u03bc S = 0 \u22a2 \u2191s S = 0 ** rw [\u2190 VectorMeasure.ennrealToMeasure_apply hS\u2081] at hS\u2082 ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e h : totalVariation s \u226a VectorMeasure.ennrealToMeasure \u03bc S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) S = 0 \u22a2 \u2191s S = 0 ** exact null_of_totalVariation_zero s (h hS\u2082) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun' ** \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d h_meas_f : AEMeasurable f h_meas_g : AEMeasurable g h_indep_fun : IndepFun f g \u22a2 \u222b\u207b (\u03c9 : \u03a9), (f * g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc ** have fg_ae : f * g =\u1d50[\u03bc] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk ** \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d h_meas_f : AEMeasurable f h_meas_g : AEMeasurable g h_indep_fun : IndepFun f g fg_ae : f * g =\u1d50[\u03bc] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g \u22a2 \u222b\u207b (\u03c9 : \u03a9), (f * g) \u03c9 \u2202\u03bc = (\u222b\u207b (\u03c9 : \u03a9), f \u03c9 \u2202\u03bc) * \u222b\u207b (\u03c9 : \u03a9), g \u03c9 \u2202\u03bc ** rw [lintegral_congr_ae h_meas_f.ae_eq_mk, lintegral_congr_ae h_meas_g.ae_eq_mk,\n  lintegral_congr_ae fg_ae] ** \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d h_meas_f : AEMeasurable f h_meas_g : AEMeasurable g h_indep_fun : IndepFun f g fg_ae : f * g =\u1d50[\u03bc] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g \u22a2 \u222b\u207b (a : \u03a9), (AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g) a \u2202\u03bc =     (\u222b\u207b (a : \u03a9), AEMeasurable.mk f h_meas_f a \u2202\u03bc) * \u222b\u207b (a : \u03a9), AEMeasurable.mk g h_meas_g a \u2202\u03bc ** apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun h_meas_f.measurable_mk\n    h_meas_g.measurable_mk ** \u03a9 : Type u_1 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f g : \u03a9 \u2192 \u211d\u22650\u221e X Y : \u03a9 \u2192 \u211d h_meas_f : AEMeasurable f h_meas_g : AEMeasurable g h_indep_fun : IndepFun f g fg_ae : f * g =\u1d50[\u03bc] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g \u22a2 IndepFun (AEMeasurable.mk f h_meas_f) (AEMeasurable.mk g h_meas_g) ** exact h_indep_fun.ae_eq h_meas_f.ae_eq_mk h_meas_g.ae_eq_mk ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.one_mem_div_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u2074 : DecidableEq \u03b1 inst\u271d\u00b3 : DecidableEq \u03b2 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : DivisionMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : F s t : Finset \u03b1 a b : \u03b1 \u22a2 1 \u2208 s / t \u2194 \u00acDisjoint s t ** rw [\u2190 mem_coe, \u2190 disjoint_coe, coe_div, Set.one_mem_div_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.evariance_def' ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 \u22a2 evariance X \u2119 = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) ** by_cases h\u2112 : Mem\u2112p X 2 ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : Mem\u2112p X 2 \u22a2 evariance X \u2119 = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) ** rw [\u2190 h\u2112.ofReal_variance_eq, variance_def' h\u2112, ENNReal.ofReal_sub _ (sq_nonneg _)] ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : Mem\u2112p X 2 \u22a2 ENNReal.ofReal (\u222b (a : \u03a9), (X ^ 2) a) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) =     (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) ** congr ** case pos.e_a \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : Mem\u2112p X 2 \u22a2 ENNReal.ofReal (\u222b (a : \u03a9), (X ^ 2) a) = \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) ** rw [lintegral_coe_eq_integral] ** case pos.e_a \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : Mem\u2112p X 2 \u22a2 ENNReal.ofReal (\u222b (a : \u03a9), (X ^ 2) a) = ENNReal.ofReal (\u222b (a : \u03a9), \u2191(\u2016X a\u2016\u208a ^ 2)) ** congr 2 with \u03c9 ** case pos.e_a.e_r.e_f.h \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : Mem\u2112p X 2 \u03c9 : \u03a9 \u22a2 (X ^ 2) \u03c9 = \u2191(\u2016X \u03c9\u2016\u208a ^ 2) ** simp only [Pi.pow_apply, NNReal.coe_pow, coe_nnnorm, Real.norm_eq_abs, Even.pow_abs even_two] ** case pos.e_a.hfi \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : Mem\u2112p X 2 \u22a2 Integrable fun x => \u2191(\u2016X x\u2016\u208a ^ 2) ** exact h\u2112.abs.integrable_sq ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : \u00acMem\u2112p X 2 \u22a2 evariance X \u2119 = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) ** symm ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : \u00acMem\u2112p X 2 \u22a2 (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2)) - ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) = evariance X \u2119 ** rw [evariance_eq_top hX h\u2112, ENNReal.sub_eq_top_iff] ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : \u00acMem\u2112p X 2 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4 \u2227 ENNReal.ofReal ((\u222b (a : \u03a9), X a) ^ 2) \u2260 \u22a4 ** refine' \u27e8_, ENNReal.ofReal_ne_top\u27e9 ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : \u00acMem\u2112p X 2 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4 ** rw [Mem\u2112p, not_and] at h\u2112 ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : AEStronglyMeasurable X \u2119 \u2192 \u00acsnorm X 2 \u2119 < \u22a4 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4 ** specialize h\u2112 hX ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : \u00acsnorm X 2 \u2119 < \u22a4 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4 ** simp only [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,\n  coe_two, one_div, ENNReal.rpow_eq_top_iff, inv_lt_zero, inv_pos, and_true_iff,\n  or_iff_not_imp_left, not_and_or, zero_lt_two] at h\u2112 ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : AEStronglyMeasurable X \u2119 h\u2112 : (\u00ac\u00ac\u222b\u207b (x : \u03a9), \u2191\u2016X x\u2016\u208a ^ 2 = 0 \u2192 0 \u2264 2) \u2192 \u222b\u207b (x : \u03a9), \u2191\u2016X x\u2016\u208a ^ 2 = \u22a4 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2016X \u03c9\u2016\u208a ^ 2) = \u22a4 ** exact_mod_cast h\u2112 fun _ => zero_le_two ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsFundamentalDomain.measure_zero_of_invariant ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u2070 : Group G inst\u271d\u2079 : Group H inst\u271d\u2078 : MulAction G \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MulAction H \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : NormedAddCommGroup E s t\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace G inst\u271d\u00b2 : MeasurableSMul G \u03b1 inst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d : Countable G \u03bd : Measure \u03b1 h : IsFundamentalDomain G s t : Set \u03b1 ht : \u2200 (g : G), g \u2022 t = t hts : \u2191\u2191\u03bc (t \u2229 s) = 0 \u22a2 \u2191\u2191\u03bc t = 0 ** rw [measure_eq_tsum h] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u2070 : Group G inst\u271d\u2079 : Group H inst\u271d\u2078 : MulAction G \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MulAction H \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : NormedAddCommGroup E s t\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace G inst\u271d\u00b2 : MeasurableSMul G \u03b1 inst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d : Countable G \u03bd : Measure \u03b1 h : IsFundamentalDomain G s t : Set \u03b1 ht : \u2200 (g : G), g \u2022 t = t hts : \u2191\u2191\u03bc (t \u2229 s) = 0 \u22a2 \u2211' (g : G), \u2191\u2191\u03bc (g \u2022 t \u2229 s) = 0 ** simp [ht, hts] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.lcm_self ** i : \u2124 \u22a2 lcm i i = natAbs i ** rw [Int.lcm] ** i : \u2124 \u22a2 Nat.lcm (natAbs i) (natAbs i) = natAbs i ** apply Nat.lcm_self ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.vars_map ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S f : R \u2192+* S \u22a2 vars (\u2191(map f) p) \u2286 vars p ** classical simp [vars_def, degrees_map] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S f : R \u2192+* S \u22a2 vars (\u2191(map f) p) \u2286 vars p ** simp [vars_def, degrees_map] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : PseudoMetrizableSpace \u03b2 f : \u03b1 \u2192 \u03b2 a b : \u03b1 \u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (\u0399 a b)) \u2194     AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc a b)) \u2227 AEStronglyMeasurable f (Measure.restrict \u03bc (Ioc b a)) ** rw [uIoc_eq_union, aestronglyMeasurable_union_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.StronglyMeasurable.integral_kernel_prod_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x ** borelize E ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x ** haveI : TopologicalSpace.SeparableSpace (range (uncurry f) \u222a {0} : Set E) :=\n  hf.separableSpace_range_union_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x ** let s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=\n  SimpleFunc.approxOn _ hf.measurable (range (uncurry f) \u222a {0}) 0 (by simp) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x ** let s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x ** let f' : \u2115 \u2192 \u03b1 \u2192 E := fun n =>\n  {x | Integrable (f x) (\u03ba x)}.indicator fun x => (s' n x).integral (\u03ba x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x ** have hf' : \u2200 n, StronglyMeasurable (f' n) := by\n  intro n; refine' StronglyMeasurable.indicator _ (measurableSet_kernel_integrable hf)\n  have : \u2200 x, ((s' n x).range.filter fun x => x \u2260 0) \u2286 (s n).range := by\n    intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y\n    simp_rw [SimpleFunc.mem_range]; rintro \u27e8z, rfl\u27e9; exact \u27e8(x, z), rfl\u27e9\n  simp only [SimpleFunc.integral_eq_sum_of_subset (this _)]\n  refine' Finset.stronglyMeasurable_sum _ fun x _ => _\n  refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _\n  simp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp]\n  apply kernel.measurable_kernel_prod_mk_left\n  exact (s n).measurableSet_fiber x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) h2f' : Tendsto f' atTop (\ud835\udcdd fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x) \u22a2 StronglyMeasurable fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x ** exact stronglyMeasurable_of_tendsto _ hf' h2f' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) \u22a2 0 \u2208 range (uncurry f) \u222a {0} ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) \u22a2 \u2200 (n : \u2115), StronglyMeasurable (f' n) ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 \u22a2 StronglyMeasurable (f' n) ** refine' StronglyMeasurable.indicator _ (measurableSet_kernel_integrable hf) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 \u22a2 StronglyMeasurable fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) ** have : \u2200 x, ((s' n x).range.filter fun x => x \u2260 0) \u2286 (s n).range := by\n  intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y\n  simp_rw [SimpleFunc.mem_range]; rintro \u27e8z, rfl\u27e9; exact \u27e8(x, z), rfl\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 this : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n) \u22a2 StronglyMeasurable fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) ** simp only [SimpleFunc.integral_eq_sum_of_subset (this _)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 this : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n) \u22a2 StronglyMeasurable fun x =>     Finset.sum       (SimpleFunc.range         (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0           (_ : 0 \u2208 range (uncurry f) \u222a {0}) n))       fun x_1 =>       ENNReal.toReal           (\u2191\u2191(\u2191\u03ba x)             (\u2191(SimpleFunc.comp                   (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0                     (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)                   (Prod.mk x) (_ : Measurable (Prod.mk x))) \u207b\u00b9'               {x_1})) \u2022         x_1 ** refine' Finset.stronglyMeasurable_sum _ fun x _ => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 this : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n) x : E x\u271d :   x \u2208     SimpleFunc.range       (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0         (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u22a2 StronglyMeasurable fun x_1 =>     ENNReal.toReal         (\u2191\u2191(\u2191\u03ba x_1)           (\u2191(SimpleFunc.comp                 (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0                   (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)                 (Prod.mk x_1) (_ : Measurable (Prod.mk x_1))) \u207b\u00b9'             {x})) \u2022       x ** refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 this : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n) x : E x\u271d :   x \u2208     SimpleFunc.range       (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0         (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u22a2 Measurable fun x_1 =>     \u2191\u2191(\u2191\u03ba x_1)       (\u2191(SimpleFunc.comp             (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0               (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)             (Prod.mk x_1) (_ : Measurable (Prod.mk x_1))) \u207b\u00b9'         {x}) ** simp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 this : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n) x : E x\u271d :   x \u2208     SimpleFunc.range       (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0         (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u22a2 Measurable fun x_1 =>     \u2191\u2191(\u2191\u03ba x_1)       (Prod.mk x_1 \u207b\u00b9'         (\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0               (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u207b\u00b9'           {x})) ** apply kernel.measurable_kernel_prod_mk_left ** case ht \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 this : \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n) x : E x\u271d :   x \u2208     SimpleFunc.range       (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0         (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u22a2 MeasurableSet     (\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0           (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u207b\u00b9'       {x}) ** exact (s n).measurableSet_fiber x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 \u22a2 \u2200 (x : \u03b1), Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n) ** intro x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 x : \u03b1 \u22a2 Finset.filter (fun x => x \u2260 0) (SimpleFunc.range (s' n x)) \u2286 SimpleFunc.range (s n) ** refine' Finset.Subset.trans (Finset.filter_subset _ _) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 x : \u03b1 \u22a2 SimpleFunc.range (s' n x) \u2286 SimpleFunc.range (s n) ** intro y ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 x : \u03b1 y : E \u22a2 y \u2208 SimpleFunc.range (s' n x) \u2192 y \u2208 SimpleFunc.range (s n) ** simp_rw [SimpleFunc.mem_range] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 x : \u03b1 y : E \u22a2 y \u2208       range         \u2191(SimpleFunc.comp             (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0               (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)             (Prod.mk x) (_ : Measurable (Prod.mk x))) \u2192     y \u2208       range         \u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0             (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) ** rintro \u27e8z, rfl\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) n : \u2115 x : \u03b1 z : \u03b2 \u22a2 \u2191(SimpleFunc.comp           (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0             (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)           (Prod.mk x) (_ : Measurable (Prod.mk x)))       z \u2208     range       \u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0           (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) ** exact \u27e8(x, z), rfl\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) \u22a2 Tendsto f' atTop (\ud835\udcdd fun x => \u222b (y : \u03b2), f x y \u2202\u2191\u03ba x) ** rw [tendsto_pi_nhds] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) \u22a2 \u2200 (x : \u03b1), Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x)) ** intro x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 \u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x)) ** by_cases hfx : Integrable (f x) (\u03ba x) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) \u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x)) ** have : \u2200 n, Integrable (s' n x) (\u03ba x) := by\n  intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable\n  apply eventually_of_forall; intro y\n  simp_rw [SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) this : \u2200 (n : \u2115), Integrable \u2191(s' n x) \u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x)) ** simp only [ hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,\n  mem_setOf_eq] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) this : \u2200 (n : \u2115), Integrable \u2191(s' n x) \u22a2 Tendsto     (fun i =>       \u222b (x_1 : \u03b2),         \u2191(SimpleFunc.comp               (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0                 (_ : 0 \u2208 range (uncurry f) \u222a {0}) i)               (Prod.mk x) (_ : Measurable (Prod.mk x)))           x_1 \u2202\u2191\u03ba x)     atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x)) ** refine'\n  tendsto_integral_of_dominated_convergence (fun y => \u2016f x y\u2016 + \u2016f x y\u2016)\n    (fun n => (s' n x).aestronglyMeasurable) (hfx.norm.add hfx.norm) _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) \u22a2 \u2200 (n : \u2115), Integrable \u2191(s' n x) ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) n : \u2115 \u22a2 Integrable \u2191(s' n x) ** apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) n : \u2115 \u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u2191\u03ba x, \u2016\u2191(s' n x) a\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) a ** apply eventually_of_forall ** case hp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) n : \u2115 \u22a2 \u2200 (x_1 : \u03b2), \u2016\u2191(s' n x) x_1\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) x_1 ** intro y ** case hp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) n : \u2115 y : \u03b2 \u22a2 \u2016\u2191(s' n x) y\u2016 \u2264 ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) y ** simp_rw [SimpleFunc.coe_comp] ** case hp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) n : \u2115 y : \u03b2 \u22a2 \u2016(\u2191(SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0               (_ : 0 \u2208 range (uncurry f) \u222a {0}) n) \u2218           Prod.mk x)         y\u2016 \u2264     ((fun a => \u2016f x a\u2016) + fun a => \u2016f x a\u2016) y ** exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) this : \u2200 (n : \u2115), Integrable \u2191(s' n x) \u22a2 \u2200 (n : \u2115),     \u2200\u1d50 (a : \u03b2) \u2202\u2191\u03ba x,       \u2016\u2191(SimpleFunc.comp                 (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0                   (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)                 (Prod.mk x) (_ : Measurable (Prod.mk x)))             a\u2016 \u2264         (fun y => \u2016f x y\u2016 + \u2016f x y\u2016) a ** exact fun n => eventually_of_forall fun y =>\n  SimpleFunc.norm_approxOn_zero_le hf.measurable (by simp) (x, y) n ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) this : \u2200 (n : \u2115), Integrable \u2191(s' n x) n : \u2115 y : \u03b2 \u22a2 0 \u2208 range (uncurry f) \u222a {0} ** simp ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) this : \u2200 (n : \u2115), Integrable \u2191(s' n x) \u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u2191\u03ba x,     Tendsto       (fun n =>         \u2191(SimpleFunc.comp               (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0                 (_ : 0 \u2208 range (uncurry f) \u222a {0}) n)               (Prod.mk x) (_ : Measurable (Prod.mk x)))           a)       atTop (\ud835\udcdd (f x a)) ** refine' eventually_of_forall fun y => SimpleFunc.tendsto_approxOn hf.measurable (by simp) _ ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) this : \u2200 (n : \u2115), Integrable \u2191(s' n x) y : \u03b2 \u22a2 uncurry f (x, y) \u2208 closure (range (uncurry f) \u222a {0}) ** apply subset_closure ** case pos.refine'_2.a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) this : \u2200 (n : \u2115), Integrable \u2191(s' n x) y : \u03b2 \u22a2 uncurry f (x, y) \u2208 range (uncurry f) \u222a {0} ** simp [-uncurry_apply_pair] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b2 : MeasurableSpace E := borel E this\u271d\u00b9 : BorelSpace E this\u271d : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : Integrable (f x) this : \u2200 (n : \u2115), Integrable \u2191(s' n x) y : \u03b2 \u22a2 0 \u2208 range (uncurry f) \u222a {0} ** simp ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 E : Type u_4 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E this : TopologicalSpace.SeparableSpace \u2191(range (uncurry f) \u222a {0}) s : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) E :=   SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) \u222a {0}) 0     (_ : 0 \u2208 range (uncurry f) \u222a {0}) s' : \u2115 \u2192 \u03b1 \u2192 SimpleFunc \u03b2 E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x)) f' : \u2115 \u2192 \u03b1 \u2192 E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral (\u2191\u03ba x) (s' n x) hf' : \u2200 (n : \u2115), StronglyMeasurable (f' n) x : \u03b1 hfx : \u00acIntegrable (f x) \u22a2 Tendsto (fun i => f' i x) atTop (\ud835\udcdd (\u222b (y : \u03b2), f x y \u2202\u2191\u03ba x)) ** simp [hfx, integral_undef] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.measurableSet_iInf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u_6 s\u271d t u : Set \u03b1 \u03b9 : Sort u_7 m : \u03b9 \u2192 MeasurableSpace \u03b1 s : Set \u03b1 \u22a2 MeasurableSet s \u2194 \u2200 (i : \u03b9), MeasurableSet s ** rw [iInf, measurableSet_sInf, forall_range_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Finite.exists_injOn_of_encard_le ** \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t \u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s ** obtain (rfl | h | \u27e8a, has, -\u27e9) := s.eq_empty_or_encard_eq_top_or_encard_diff_singleton_lt ** case inr.inr.intro.intro \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s \u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s ** obtain \u27e8b, hbt\u27e9 := encard_pos.1 ((encard_pos.2 \u27e8_, has\u27e9).trans_le hle) ** case inr.inr.intro.intro.intro \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t \u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s ** have hle' : (s \\ {a}).encard \u2264 (t \\ {b}).encard ** case inr.inr.intro.intro.intro \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) \u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s ** obtain \u27e8f\u2080, hf\u2080s, hinj\u27e9 := exists_injOn_of_encard_le (hs.diff {a}) hle' ** case inr.inr.intro.intro.intro.intro.intro \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hf\u2080s : s \\ {a} \u2286 f\u2080 \u207b\u00b9' (t \\ {b}) hinj : InjOn f\u2080 (s \\ {a}) \u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s ** simp only [preimage_diff, subset_def, mem_diff, mem_singleton_iff, mem_preimage, and_imp] at hf\u2080s ** case inr.inr.intro.intro.intro.intro.intro \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b \u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s ** use Function.update f\u2080 a b ** case h \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b \u22a2 s \u2286 Function.update f\u2080 a b \u207b\u00b9' t \u2227 InjOn (Function.update f\u2080 a b) s ** rw [\u2190insert_eq_of_mem has, \u2190insert_diff_singleton, injOn_insert (fun h \u21a6 h.2 rfl)] ** case h \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b \u22a2 insert a (s \\ {a}) \u2286 Function.update f\u2080 a b \u207b\u00b9' t \u2227     InjOn (Function.update f\u2080 a b) (s \\ {a}) \u2227 \u00acFunction.update f\u2080 a b a \u2208 Function.update f\u2080 a b '' (s \\ {a}) ** simp only [mem_diff, mem_singleton_iff, not_true, and_false, insert_diff_singleton, subset_def,\n  mem_insert_iff, mem_preimage, ne_eq, Function.update_apply, forall_eq_or_imp, ite_true, and_imp,\n  mem_image, ite_eq_left_iff, not_exists, not_and, not_forall, exists_prop, and_iff_right hbt] ** case h \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b \u22a2 (\u2200 (a_1 : \u03b1), a_1 \u2208 s \u2192 (if a_1 = a then b else f\u2080 a_1) \u2208 t) \u2227     InjOn (Function.update f\u2080 a b) (s \\ {a}) \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 \u00acx = a \u2227 \u00acf\u2080 x = b ** refine \u27e8?_, ?_, fun x hxs hxa \u21a6 \u27e8hxa, (hf\u2080s x hxs hxa).2\u27e9\u27e9 ** case h.refine_2 \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b \u22a2 InjOn (Function.update f\u2080 a b) (s \\ {a}) ** exact InjOn.congr hinj (fun x \u27e8_, hxa\u27e9 \u21a6 by rwa [Function.update_noteq]) ** case inl \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d t\u271d\u00b9 s : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 t : Set \u03b2 hs : Set.Finite \u2205 hle : encard \u2205 \u2264 encard t \u22a2 \u2203 f, \u2205 \u2286 f \u207b\u00b9' t \u2227 InjOn f \u2205 ** simp ** case inr.inl \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t h : encard s = \u22a4 \u22a2 \u2203 f, s \u2286 f \u207b\u00b9' t \u2227 InjOn f s ** exact (encard_ne_top_iff.mpr hs h).elim ** case hle' \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t \u22a2 encard (s \\ {a}) \u2264 encard (t \\ {b}) ** rwa [\u2190WithTop.add_le_add_iff_right WithTop.one_ne_top,\nencard_diff_singleton_add_one has, encard_diff_singleton_add_one hbt] ** case h.refine_1 \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b \u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 s \u2192 (if a_1 = a then b else f\u2080 a_1) \u2208 t ** rintro x hx ** case h.refine_1 \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b x : \u03b1 hx : x \u2208 s \u22a2 (if x = a then b else f\u2080 x) \u2208 t ** split_ifs with h ** case pos \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b x : \u03b1 hx : x \u2208 s h : x = a \u22a2 b \u2208 t  case neg \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b x : \u03b1 hx : x \u2208 s h : \u00acx = a \u22a2 f\u2080 x \u2208 t ** assumption ** case neg \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b x : \u03b1 hx : x \u2208 s h : \u00acx = a \u22a2 f\u2080 x \u2208 t ** exact (hf\u2080s x hx h).1 ** \u03b1 : Type u_2 \u03b2 : Type u_1 s\u271d\u00b9 t\u271d\u00b9 s\u271d : Set \u03b1 t\u271d : Set \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Nonempty \u03b2 s : Set \u03b1 t : Set \u03b2 hs : Set.Finite s hle : encard s \u2264 encard t a : \u03b1 has : a \u2208 s b : \u03b2 hbt : b \u2208 t hle' : encard (s \\ {a}) \u2264 encard (t \\ {b}) f\u2080 : \u03b1 \u2192 \u03b2 hinj : InjOn f\u2080 (s \\ {a}) hf\u2080s : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx = a \u2192 f\u2080 x \u2208 t \u2227 \u00acf\u2080 x = b x : \u03b1 x\u271d : x \u2208 s \\ {a} left\u271d : x \u2208 s hxa : \u00acx \u2208 {a} \u22a2 f\u2080 x = Function.update f\u2080 a b x ** rwa [Function.update_noteq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_le_snorm_of_exponent_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e hpq : p \u2264 q inst\u271d : IsProbabilityMeasure \u03bc f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc \u22a2 snorm f q \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p - 1 / ENNReal.toReal q) = snorm f q \u03bc ** simp [measure_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.sum_comm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 \u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } \u22a2 (kernel.sum fun n => kernel.sum (\u03ba n)) = kernel.sum fun m => kernel.sum fun n => \u03ba n m ** ext a s ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 \u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } a : \u03b1 s : Set \u03b2 a\u271d : MeasurableSet s \u22a2 \u2191\u2191(\u2191(kernel.sum fun n => kernel.sum (\u03ba n)) a) s = \u2191\u2191(\u2191(kernel.sum fun m => kernel.sum fun n => \u03ba n m) a) s ** simp_rw [sum_apply] ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 \u03ba : \u03b9 \u2192 \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } a : \u03b1 s : Set \u03b2 a\u271d : MeasurableSet s \u22a2 \u2191\u2191(Measure.sum fun n => Measure.sum fun n_1 => \u2191(\u03ba n n_1) a) s =     \u2191\u2191(Measure.sum fun n => Measure.sum fun n_1 => \u2191(\u03ba n_1 n) a) s ** rw [Measure.sum_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.norm_condexpIndL1Fin_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G \u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** have : 0 \u2264 \u222b a : \u03b1, \u2016condexpIndL1Fin hm hs h\u03bcs x a\u2016 \u2202\u03bc :=\n  integral_nonneg fun a => norm_nonneg _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc \u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** rw [L1.norm_eq_integral_norm, \u2190 ENNReal.toReal_ofReal (norm_nonneg x), \u2190 ENNReal.toReal_mul, \u2190\n  ENNReal.toReal_ofReal this,\n  ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top h\u03bcs ENNReal.ofReal_ne_top),\n  ofReal_integral_norm_eq_lintegral_nnnorm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc \u22a2 \u222b\u207b (x_1 : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * ENNReal.ofReal \u2016x\u2016  \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc \u22a2 Integrable fun a => \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a ** swap ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc \u22a2 \u222b\u207b (x_1 : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * ENNReal.ofReal \u2016x\u2016 ** have h_eq :\n  \u222b\u207b a, \u2016condexpIndL1Fin hm hs h\u03bcs x a\u2016\u208a \u2202\u03bc = \u222b\u207b a, \u2016condexpIndSMul hm hs h\u03bcs x a\u2016\u208a \u2202\u03bc := by\n  refine' lintegral_congr_ae _\n  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h\u03bcs x).mono fun z hz => _\n  dsimp only\n  rw [hz] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc h_eq : \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc \u22a2 \u222b\u207b (x_1 : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * ENNReal.ofReal \u2016x\u2016 ** rw [h_eq, ofReal_norm_eq_coe_nnnorm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc h_eq : \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a ** exact lintegral_nnnorm_condexpIndSMul_le hm hs h\u03bcs x ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc \u22a2 Integrable fun a => \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a ** rw [\u2190 mem\u2112p_one_iff_integrable] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc \u22a2 Mem\u2112p (fun a => \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a) 1 ** exact Lp.mem\u2112p _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc ** refine' lintegral_congr_ae _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc \u22a2 (fun a => \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a ** refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h\u03bcs x).mono fun z hz => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc z : \u03b1 hz : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) z = \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) z \u22a2 (fun a => \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016\u208a) z = (fun a => \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a) z ** dsimp only ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : G this : 0 \u2264 \u222b (a : \u03b1), \u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) a\u2016 \u2202\u03bc z : \u03b1 hz : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) z = \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) z \u22a2 \u2191\u2016\u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) z\u2016\u208a = \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) z\u2016\u208a ** rw [hz] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.BorelCantelli.predictablePart_process_ae_eq ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 \u2131\u271d : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 s\u271d : \u2115 \u2192 Set \u03a9 \u2131 : Filtration \u2115 m0 \u03bc : Measure \u03a9 s : \u2115 \u2192 Set \u03a9 n : \u2115 \u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k] ** have := martingalePart_process_ae_eq \u2131 \u03bc s n ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 \u2131\u271d : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 s\u271d : \u2115 \u2192 Set \u03a9 \u2131 : Filtration \u2115 m0 \u03bc : Measure \u03a9 s : \u2115 \u2192 Set \u03a9 n : \u2115 this :   martingalePart (process s) \u2131 \u03bc n =     \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k] ** simp_rw [martingalePart, process, Finset.sum_sub_distrib] at this ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 \u2131\u271d : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 s\u271d : \u2115 \u2192 Set \u03a9 \u2131 : Filtration \u2115 m0 \u03bc : Measure \u03a9 s : \u2115 \u2192 Set \u03a9 n : \u2115 this :   \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 - predictablePart (process s) \u2131 \u03bc n =     \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 - \u2211 x in Finset.range n, \u03bc[Set.indicator (s (x + 1)) 1|\u2191\u2131 x] \u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k] ** exact sub_right_injective this ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec'.comp\u2081 ** n : \u2115 f : \u2115 \u2192. \u2115 g : Vector \u2115 n \u2192 \u2115 hf : Partrec' fun v => f (Vector.head v) hg : Partrec' \u2191g \u22a2 Partrec' fun v => f (g v) ** simpa using hf.comp' (Partrec'.cons hg Partrec'.nil) ** Qed",
        "informal": ""
    },
    {
        "formal": "torusIntegral_dim0 ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin 0 \u2192 \u2102) \u2192 E c : Fin 0 \u2192 \u2102 R : Fin 0 \u2192 \u211d \u22a2 (\u222f (x : Fin 0 \u2192 \u2102) in T(c, R), f x) = f c ** simp only [torusIntegral, Fin.prod_univ_zero, one_smul,\n  Subsingleton.elim (fun _ : Fin 0 => 2 * \u03c0) 0, Icc_self, Measure.restrict_singleton, volume_pi,\n  integral_smul_measure, integral_dirac, Measure.pi_of_empty (fun _ : Fin 0 \u21a6 volume) 0,\n  Measure.dirac_apply_of_mem (mem_singleton _), Subsingleton.elim (torusMap c R 0) c] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.replicateTR_loop_eq ** \u03b1\u271d : Type u_1 a : \u03b1\u271d acc : List \u03b1\u271d n : Nat \u22a2 replicateTR.loop a (n + 1) acc = replicate (n + 1) a ++ acc ** rw [\u2190 replicateTR_loop_replicate_eq _ 1 n, replicate, replicate,\nreplicateTR.loop, replicateTR_loop_eq n, replicateTR_loop_eq n, append_assoc] ** \u03b1\u271d : Type u_1 a : \u03b1\u271d acc : List \u03b1\u271d n : Nat \u22a2 replicate n a ++ a :: acc = replicate n a ++ ([a] ++ acc) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condCdf'_eq_condCdfRat ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a \u22a2 condCdf' \u03c1 a \u2191r = condCdfRat \u03c1 a r ** rw [\u2190 inf_gt_condCdfRat \u03c1 a r, condCdf'] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a \u22a2 \u2a05 r_1, condCdfRat \u03c1 a \u2191r_1 = \u2a05 r_1, condCdfRat \u03c1 a \u2191r_1 ** refine' Equiv.iInf_congr _ _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a \u22a2 { r' // \u2191r < \u2191r' } \u2243 \u2191(Ioi r) ** exact\n  { toFun := fun t => \u27e8t.1, by exact_mod_cast t.2\u27e9\n    invFun := fun t => \u27e8t.1, by exact_mod_cast t.2\u27e9\n    left_inv := fun t => by simp only [Subtype.coe_eta]\n    right_inv := fun t => by simp only [Subtype.coe_eta] } ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a t : { r' // \u2191r < \u2191r' } \u22a2 \u2191t \u2208 Ioi r ** exact_mod_cast t.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a t : \u2191(Ioi r) \u22a2 \u2191r < \u2191\u2191t ** exact_mod_cast t.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a t : { r' // \u2191r < \u2191r' } \u22a2 (fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) ((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t) = t ** simp only [Subtype.coe_eta] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a t : \u2191(Ioi r) \u22a2 (fun t => { val := \u2191t, property := (_ : r < \u2191t) }) ((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t) = t ** simp only [Subtype.coe_eta] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a \u22a2 \u2200 (x : { r' // \u2191r < \u2191r' }),     condCdfRat \u03c1 a         \u2191(\u2191{ toFun := fun t => { val := \u2191t, property := (_ : r < \u2191t) },                 invFun := fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) },                 left_inv :=                   (_ :                     \u2200 (t : { r' // \u2191r < \u2191r' }),                       { val := \u2191t, property := (_ : \u2191r < \u2191\u2191((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t)) } =                         t),                 right_inv :=                   (_ :                     \u2200 (t : \u2191(Ioi r)),                       { val := \u2191t, property := (_ : r < \u2191((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t)) } =                         t) }             x) =       condCdfRat \u03c1 a \u2191x ** intro t ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a t : { r' // \u2191r < \u2191r' } \u22a2 condCdfRat \u03c1 a       \u2191(\u2191{ toFun := fun t => { val := \u2191t, property := (_ : r < \u2191t) },               invFun := fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) },               left_inv :=                 (_ :                   \u2200 (t : { r' // \u2191r < \u2191r' }),                     { val := \u2191t, property := (_ : \u2191r < \u2191\u2191((fun t => { val := \u2191t, property := (_ : r < \u2191t) }) t)) } = t),               right_inv :=                 (_ :                   \u2200 (t : \u2191(Ioi r)),                     { val := \u2191t, property := (_ : r < \u2191((fun t => { val := \u2191t, property := (_ : \u2191r < \u2191\u2191t) }) t)) } =                       t) }           t) =     condCdfRat \u03c1 a \u2191t ** simp only [Equiv.coe_fn_mk, Subtype.coe_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.inv_smul_integral_comp_add_div ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c\u271d d\u271d : \u211d f : \u211d \u2192 E c d : \u211d \u22a2 c\u207b\u00b9 \u2022 \u222b (x : \u211d) in a..b, f (d + x / c) = \u222b (x : \u211d) in d + a / c..d + b / c, f x ** by_cases hc : c = 0 <;> simp [hc, integral_comp_add_div] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.natAbs_inj_of_nonneg_of_nonneg ** a\u271d b\u271d : \u2124 n : \u2115 a b : \u2124 ha : 0 \u2264 a hb : 0 \u2264 b \u22a2 natAbs a = natAbs b \u2194 a = b ** rw [\u2190 sq_eq_sq ha hb, \u2190 natAbs_eq_iff_sq_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.union_ae_eq_right_iff_ae_subset ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 \u22a2 s \u222a t =\u1d50[\u03bc] t \u2194 s \u2264\u1d50[\u03bc] t ** rw [union_comm, union_ae_eq_left_iff_ae_subset] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_pi_ball ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u03b9 \u2192 \u211d r : \u211d hr : 0 < r \u22a2 \u2191\u2191volume (Metric.ball a r) = ofReal ((2 * r) ^ Fintype.card \u03b9) ** simp only [MeasureTheory.volume_pi_ball a hr, volume_ball, Finset.prod_const] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u03b9 \u2192 \u211d r : \u211d hr : 0 < r \u22a2 ofReal (2 * r) ^ Finset.card Finset.univ = ofReal ((2 * r) ^ Fintype.card \u03b9) ** exact (ENNReal.ofReal_pow (mul_nonneg zero_le_two hr.le) _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Coprime.gcd_mul_left_cancel ** k n m : Nat H : Coprime k n \u22a2 Coprime (gcd (k * m) n) k ** rw [Coprime, Nat.gcd_assoc, H.symm.gcd_eq_one, gcd_one_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "blimsup_cthickening_ae_eq_blimsup_thickening ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) hr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i \u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p ** refine' eventuallyLE_antisymm_iff.mpr \u27e8_, HasSubset.Subset.eventuallyLE (_ : _ \u2264 _)\u27e9 ** case refine'_1 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) hr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i \u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p ** rw [eventuallyLE_congr (blimsup_cthickening_mul_ae_eq \u03bc p s (@one_half_pos \u211d _) r hr).symm\n  EventuallyEq.rfl] ** case refine'_1 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) hr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i \u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p ** apply HasSubset.Subset.eventuallyLE ** case refine'_1.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) hr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i \u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2286 blimsup (fun i => thickening (r i) (s i)) atTop p ** change _ \u2264 _ ** case refine'_1.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) hr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i \u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264 blimsup (fun i => thickening (r i) (s i)) atTop p ** refine' mono_blimsup' (hr'.mono fun i hi pi => cthickening_subset_thickening' (hi pi) _ (s i)) ** case refine'_1.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) hr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i i : \u2115 hi : p i \u2192 0 < r i pi : p i \u22a2 1 / 2 * r i < r i ** nlinarith [hi pi] ** case refine'_2 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) hr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i \u22a2 blimsup (fun i => thickening (r i) (s i)) atTop p \u2264 blimsup (fun i => cthickening (r i) (s i)) atTop p ** exact mono_blimsup fun i _ => thickening_subset_cthickening _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Basis.parallelepiped_eq_map ** \u03b9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : Fintype \u03b9 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E b : Basis \u03b9 \u211d E \u22a2 parallelepiped b =     PositiveCompacts.map \u2191(LinearEquiv.symm (equivFun b)) (_ : Continuous \u2191(ContinuousLinearEquiv.symm (equivFunL b)))       (_ : IsOpenMap \u2191(ContinuousLinearEquiv.symm (equivFunL b))) (PositiveCompacts.piIcc01 \u03b9) ** classical\nrw [\u2190 Basis.parallelepiped_basisFun, \u2190 Basis.parallelepiped_map]\ncongr\next; simp only [map_apply, Pi.basisFun_apply, equivFun_symm_apply, LinearMap.stdBasis_apply',\n  Finset.sum_univ_ite] ** \u03b9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : Fintype \u03b9 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E b : Basis \u03b9 \u211d E \u22a2 parallelepiped b =     PositiveCompacts.map \u2191(LinearEquiv.symm (equivFun b)) (_ : Continuous \u2191(ContinuousLinearEquiv.symm (equivFunL b)))       (_ : IsOpenMap \u2191(ContinuousLinearEquiv.symm (equivFunL b))) (PositiveCompacts.piIcc01 \u03b9) ** rw [\u2190 Basis.parallelepiped_basisFun, \u2190 Basis.parallelepiped_map] ** \u03b9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : Fintype \u03b9 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E b : Basis \u03b9 \u211d E \u22a2 parallelepiped b = parallelepiped (Basis.map (Pi.basisFun \u211d \u03b9) (LinearEquiv.symm (equivFun b))) ** congr ** case e_b \u03b9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : Fintype \u03b9 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E b : Basis \u03b9 \u211d E \u22a2 b = Basis.map (Pi.basisFun \u211d \u03b9) (LinearEquiv.symm (equivFun b)) ** ext ** case e_b.a \u03b9 : Type u_1 E : Type u_2 inst\u271d\u00b2 : Fintype \u03b9 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E b : Basis \u03b9 \u211d E i\u271d : \u03b9 \u22a2 \u2191b i\u271d = \u2191(Basis.map (Pi.basisFun \u211d \u03b9) (LinearEquiv.symm (equivFun b))) i\u271d ** simp only [map_apply, Pi.basisFun_apply, equivFun_symm_apply, LinearMap.stdBasis_apply',\nFinset.sum_univ_ite] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le_Ico ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** refine' le_of_forall_pos_le_add fun \u03b5 \u03b5pos => _ ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5 ** rcases exists_lt_lowerSemicontinuous_integral_lt \u03c6 \u03c6int \u03b5pos with\n  \u27e8G', f_lt_G', G'cont, G'int, G'lt_top, hG'\u27e9 ** case intro.intro.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 \u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5 ** set s := {t | g t - g a \u2264 \u222b u in a..t, (G' u).toReal} \u2229 Icc a b ** case intro.intro.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b \u22a2 g b - g a \u2264 (\u222b (y : \u211d) in a..b, \u03c6 y) + \u03b5 ** have s_closed : IsClosed s := by\n  have : ContinuousOn (fun t => (g t - g a, \u222b u in a..t, (G' u).toReal)) (Icc a b) := by\n    rw [\u2190 uIcc_of_le hab] at G'int hcont \u22a2\n    exact (hcont.sub continuousOn_const).prod (continuousOn_primitive_interval G'int)\n  simp only [inter_comm]\n  exact this.preimage_closed_of_closed isClosed_Icc OrderClosedTopology.isClosed_le' ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b \u22a2 IsClosed s ** have : ContinuousOn (fun t => (g t - g a, \u222b u in a..t, (G' u).toReal)) (Icc a b) := by\n  rw [\u2190 uIcc_of_le hab] at G'int hcont \u22a2\n  exact (hcont.sub continuousOn_const).prod (continuousOn_primitive_interval G'int) ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b this : ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) (Icc a b) \u22a2 IsClosed s ** simp only [inter_comm] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b this : ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) (Icc a b) \u22a2 IsClosed (Icc a b \u2229 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)}) ** exact this.preimage_closed_of_closed isClosed_Icc OrderClosedTopology.isClosed_le' ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b \u22a2 ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) (Icc a b) ** rw [\u2190 uIcc_of_le hab] at G'int hcont \u22a2 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g [[a, b]] hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b \u22a2 ContinuousOn (fun t => (g t - g a, \u222b (u : \u211d) in a..t, EReal.toReal (G' u))) [[a, b]] ** exact (hcont.sub continuousOn_const).prod (continuousOn_primitive_interval G'int) ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s \u22a2 Icc a b \u2286 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} ** refine s_closed.Icc_subset_of_forall_exists_gt\n  (by simp only [integral_same, mem_setOf_eq, sub_self, le_rfl]) fun t ht v t_lt_v => ?_ ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t \u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v) ** obtain \u27e8y, g'_lt_y', y_lt_G'\u27e9 : \u2203 y : \u211d, (g' t : EReal) < y \u2227 (y : EReal) < G' t :=\n  EReal.lt_iff_exists_real_btwn.1 ((EReal.coe_le_coe_iff.2 (h\u03c6g t ht.2)).trans_lt (f_lt_G' t)) ** case intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) \u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v) ** have I2 : \u2200\u1da0 u in \ud835\udcdd[>] t, g u - g t \u2264 (u - t) * y := by\n  have g'_lt_y : g' t < y := EReal.coe_lt_coe_iff.1 g'_lt_y'\n  filter_upwards [(hderiv t \u27e8ht.2.1, ht.2.2\u27e9).limsup_slope_le' (not_mem_Ioi.2 le_rfl) g'_lt_y,\n    self_mem_nhdsWithin] with u hu t_lt_u\n  have := mul_le_mul_of_nonneg_left hu.le (sub_pos.2 t_lt_u.out).le\n  rwa [\u2190 smul_eq_mul, sub_smul_slope] at this ** case intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y \u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v) ** have I3 : \u2200\u1da0 u in \ud835\udcdd[>] t, g u - g t \u2264 \u222b w in t..u, (G' w).toReal := by\n  filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1 ** case intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) \u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v) ** have I4 : \u2200\u1da0 u in \ud835\udcdd[>] t, u \u2208 Ioc t (min v b) := by\n  refine' mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.2 \u27e8min v b, _, Subset.rfl\u27e9\n  simp only [lt_min_iff, mem_Ioi]\n  exact \u27e8t_lt_v, ht.2.2\u27e9 ** case intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) \u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v) ** rcases (I3.and I4).exists with \u27e8x, hx, h'x\u27e9 ** case intro.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) x : \u211d hx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w) h'x : x \u2208 Ioc t (min v b) \u22a2 Set.Nonempty ({t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ioc t v) ** refine' \u27e8x, _, Ioc_subset_Ioc le_rfl (min_le_left _ _) h'x\u27e9 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s \u22a2 a \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} ** simp only [integral_same, mem_setOf_eq, sub_self, le_rfl] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t \u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) ** have B : \u2200\u1da0 u in \ud835\udcdd t, (y : EReal) < G' u := G'cont.lowerSemicontinuousAt _ _ y_lt_G' ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u \u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) ** rcases mem_nhds_iff_exists_Ioo_subset.1 B with \u27e8m, M, \u27e8hm, hM\u27e9, H\u27e9 ** case intro.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M \u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) ** have : Ioo t (min M b) \u2208 \ud835\udcdd[>] t := Ioo_mem_nhdsWithin_Ioi' (lt_min hM ht.right.right) ** case intro.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t \u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) ** filter_upwards [this] with u hu ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) \u22a2 (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) ** have I : Icc t u \u2286 Icc a b := Icc_subset_Icc ht.2.1 (hu.2.le.trans (min_le_right _ _)) ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b \u22a2 (u - t) * y = \u222b (x : \u211d) in Icc t u, y ** simp only [hu.left.le, MeasureTheory.integral_const, Algebra.id.smul_eq_mul, sub_nonneg,\n  MeasurableSet.univ, Real.volume_Icc, Measure.restrict_apply, univ_inter,\n  ENNReal.toReal_ofReal] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b \u22a2 \u222b (x : \u211d) in Icc t u, y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) ** rw [intervalIntegral.integral_of_le hu.1.le, \u2190 integral_Icc_eq_integral_Ioc] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b \u22a2 \u222b (x : \u211d) in Icc t u, y \u2264 \u222b (t : \u211d) in Icc t u, EReal.toReal (G' t) ** apply set_integral_mono_ae_restrict ** case hf \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b \u22a2 IntegrableOn (fun a => y) (Icc t u) ** simp only [integrableOn_const, Real.volume_Icc, ENNReal.ofReal_lt_top, or_true_iff] ** case hg \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b \u22a2 IntegrableOn (fun a => EReal.toReal (G' a)) (Icc t u) ** exact IntegrableOn.mono_set G'int I ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b \u22a2 (fun a => y) \u2264\u1d50[Measure.restrict volume (Icc t u)] fun a => EReal.toReal (G' a) ** have C1 : \u2200\u1d50 x : \u211d \u2202volume.restrict (Icc t u), G' x < \u221e :=\n  ae_mono (Measure.restrict_mono I le_rfl) G'lt_top ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b C1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4 \u22a2 (fun a => y) \u2264\u1d50[Measure.restrict volume (Icc t u)] fun a => EReal.toReal (G' a) ** have C2 : \u2200\u1d50 x : \u211d \u2202volume.restrict (Icc t u), x \u2208 Icc t u :=\n  ae_restrict_mem measurableSet_Icc ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b C1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4 C2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u \u22a2 (fun a => y) \u2264\u1d50[Measure.restrict volume (Icc t u)] fun a => EReal.toReal (G' a) ** filter_upwards [C1, C2] with x G'x hx ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b C1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4 C2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u x : \u211d G'x : G' x < \u2191\u22a4 hx : x \u2208 Icc t u \u22a2 y \u2264 EReal.toReal (G' x) ** apply EReal.coe_le_coe_iff.1 ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b C1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4 C2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u x : \u211d G'x : G' x < \u2191\u22a4 hx : x \u2208 Icc t u \u22a2 \u2191y \u2264 \u2191(EReal.toReal (G' x)) ** have : x \u2208 Ioo m M := by\n  simp only [hm.trans_le hx.left,\n    (hx.right.trans_lt hu.right).trans_le (min_le_left M b), mem_Ioo, and_self_iff] ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this\u271d : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b C1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4 C2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u x : \u211d G'x : G' x < \u2191\u22a4 hx : x \u2208 Icc t u this : x \u2208 Ioo m M \u22a2 \u2191y \u2264 \u2191(EReal.toReal (G' x)) ** refine (H this).out.le.trans_eq ?_ ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this\u271d : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b C1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4 C2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u x : \u211d G'x : G' x < \u2191\u22a4 hx : x \u2208 Icc t u this : x \u2208 Ioo m M \u22a2 G' x = \u2191(EReal.toReal (G' x)) ** exact (EReal.coe_toReal G'x.ne (f_lt_G' x).ne_bot).symm ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t B : \u2200\u1da0 (u : \u211d) in \ud835\udcdd t, \u2191y < G' u m M : \u211d H : Ioo m M \u2286 {x | (fun u => \u2191y < G' u) x} hm : m < t hM : t < M this : Ioo t (min M b) \u2208 \ud835\udcdd[Ioi t] t u : \u211d hu : u \u2208 Ioo t (min M b) I : Icc t u \u2286 Icc a b C1 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), G' x < \u2191\u22a4 C2 : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc t u), x \u2208 Icc t u x : \u211d G'x : G' x < \u2191\u22a4 hx : x \u2208 Icc t u \u22a2 x \u2208 Ioo m M ** simp only [hm.trans_le hx.left,\n  (hx.right.trans_lt hu.right).trans_le (min_le_left M b), mem_Ioo, and_self_iff] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) \u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y ** have g'_lt_y : g' t < y := EReal.coe_lt_coe_iff.1 g'_lt_y' ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) g'_lt_y : g' t < y \u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y ** filter_upwards [(hderiv t \u27e8ht.2.1, ht.2.2\u27e9).limsup_slope_le' (not_mem_Ioi.2 le_rfl) g'_lt_y,\n  self_mem_nhdsWithin] with u hu t_lt_u ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) g'_lt_y : g' t < y u : \u211d hu : slope g t u < y t_lt_u : u \u2208 Ioi t \u22a2 g u - g t \u2264 (u - t) * y ** have := mul_le_mul_of_nonneg_left hu.le (sub_pos.2 t_lt_u.out).le ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) g'_lt_y : g' t < y u : \u211d hu : slope g t u < y t_lt_u : u \u2208 Ioi t this : (u - t) * slope g t u \u2264 (u - t) * y \u22a2 g u - g t \u2264 (u - t) * y ** rwa [\u2190 smul_eq_mul, sub_smul_slope] at this ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y \u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) ** filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) \u22a2 \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) ** refine' mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.2 \u27e8min v b, _, Subset.rfl\u27e9 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) \u22a2 min v b \u2208 Ioi t ** simp only [lt_min_iff, mem_Ioi] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) \u22a2 t < v \u2227 t < b ** exact \u27e8t_lt_v, ht.2.2\u27e9 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) x : \u211d hx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w) h'x : x \u2208 Ioc t (min v b) \u22a2 g x - g a = g t - g a + (g x - g t) ** abel ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) x : \u211d hx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w) h'x : x \u2208 Ioc t (min v b) \u22a2 (\u222b (w : \u211d) in a..t, EReal.toReal (G' w)) + \u222b (w : \u211d) in t..x, EReal.toReal (G' w) =     \u222b (w : \u211d) in a..x, EReal.toReal (G' w) ** apply integral_add_adjacent_intervals ** case hab \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) x : \u211d hx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w) h'x : x \u2208 Ioc t (min v b) \u22a2 IntervalIntegrable (fun x => EReal.toReal (G' x)) volume a t ** rw [intervalIntegrable_iff_integrable_Ioc_of_le ht.2.1] ** case hab \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) x : \u211d hx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w) h'x : x \u2208 Ioc t (min v b) \u22a2 IntegrableOn (fun x => EReal.toReal (G' x)) (Ioc a t) ** exact IntegrableOn.mono_set G'int\n  (Ioc_subset_Icc_self.trans (Icc_subset_Icc le_rfl ht.2.2.le)) ** case hbc \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) x : \u211d hx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w) h'x : x \u2208 Ioc t (min v b) \u22a2 IntervalIntegrable (fun x => EReal.toReal (G' x)) volume t x ** rw [intervalIntegrable_iff_integrable_Ioc_of_le h'x.1.le] ** case hbc \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) x : \u211d hx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w) h'x : x \u2208 Ioc t (min v b) \u22a2 IntegrableOn (fun x => EReal.toReal (G' x)) (Ioc t x) ** apply IntegrableOn.mono_set G'int ** case hbc \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Ico a b v : \u211d t_lt_v : v \u2208 Ioi t y : \u211d g'_lt_y' : \u2191(g' t) < \u2191y y_lt_G' : \u2191y < G' t I1 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, (u - t) * y \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I2 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 (u - t) * y I3 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, g u - g t \u2264 \u222b (w : \u211d) in t..u, EReal.toReal (G' w) I4 : \u2200\u1da0 (u : \u211d) in \ud835\udcdd[Ioi t] t, u \u2208 Ioc t (min v b) x : \u211d hx : g x - g t \u2264 \u222b (w : \u211d) in t..x, EReal.toReal (G' w) h'x : x \u2208 Ioc t (min v b) \u22a2 Ioc t x \u2286 Icc a b ** exact Ioc_subset_Icc_self.trans (Icc_subset_Icc ht.2.1 (h'x.2.trans (min_le_right _ _))) ** case h.e'_4.h.e'_5 \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s main : Icc a b \u2286 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u22a2 \u222b (y : \u211d) in a..b, \u03c6 y = \u222b (x : \u211d) in Icc a b, \u03c6 x ** rw [intervalIntegral.integral_of_le hab] ** case h.e'_4.h.e'_5 \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 g' x \u2264 \u03c6 x \u03b5 : \u211d \u03b5pos : 0 < \u03b5 G' : \u211d \u2192 EReal f_lt_G' : \u2200 (x : \u211d), \u2191(\u03c6 x) < G' x G'cont : LowerSemicontinuous G' G'int : Integrable fun x => EReal.toReal (G' x) G'lt_top : \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Icc a b), G' x < \u22a4 hG' : \u222b (x : \u211d) in Icc a b, EReal.toReal (G' x) < (\u222b (x : \u211d) in Icc a b, \u03c6 x) + \u03b5 s : Set \u211d := {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u2229 Icc a b s_closed : IsClosed s main : Icc a b \u2286 {t | g t - g a \u2264 \u222b (u : \u211d) in a..t, EReal.toReal (G' u)} \u22a2 \u222b (x : \u211d) in Ioc a b, \u03c6 x = \u222b (x : \u211d) in Icc a b, \u03c6 x ** simp only [integral_Icc_eq_integral_Ioc', Real.volume_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "Multiset.toFinset_bind_dedup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : DecidableEq \u03b1 s t : Multiset \u03b1 inst\u271d : DecidableEq \u03b2 m : Multiset \u03b1 f : \u03b1 \u2192 Multiset \u03b2 \u22a2 toFinset (bind (dedup m) f) = toFinset (bind m f) ** simp_rw [toFinset, dedup_bind_dedup] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_tsum ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 \u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc ** by_cases hG : CompleteSpace G ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G \u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc  case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : \u00acCompleteSpace G \u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G \u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc ** have hf'' : \u2200 i, AEMeasurable (fun x => (\u2016f i x\u2016\u208a : \u211d\u22650\u221e)) \u03bc := fun i => (hf i).ennnorm ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a \u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc ** have hhh : \u2200\u1d50 a : \u03b1 \u2202\u03bc, Summable fun n => (\u2016f n a\u2016\u208a : \u211d) := by\n  rw [\u2190 lintegral_tsum hf''] at hf'\n  refine' (ae_lt_top' (AEMeasurable.ennreal_tsum hf'') hf').mono _\n  intro x hx\n  rw [\u2190 ENNReal.tsum_coe_ne_top_iff_summable_coe]\n  exact hx.ne ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc ** convert (MeasureTheory.hasSum_integral_of_dominated_convergence (fun i a => \u2016f i a\u2016\u208a) hf _ hhh\n        \u27e8_, _\u27e9 _).tsum_eq.symm ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : \u00acCompleteSpace G \u22a2 \u222b (a : \u03b1), \u2211' (i : \u03b9), f i a \u2202\u03bc = \u2211' (i : \u03b9), \u222b (a : \u03b1), f i a \u2202\u03bc ** simp [integral, hG] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a ** rw [\u2190 lintegral_tsum hf''] at hf' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a ** refine' (ae_lt_top' (AEMeasurable.ennreal_tsum hf'') hf').mono _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a \u22a2 \u2200 (x : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i x\u2016\u208a < \u22a4 \u2192 Summable fun n => \u2191\u2016f n x\u2016\u208a ** intro x hx ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a x : \u03b1 hx : \u2211' (i : \u03b9), \u2191\u2016f i x\u2016\u208a < \u22a4 \u22a2 Summable fun n => \u2191\u2016f n x\u2016\u208a ** rw [\u2190 ENNReal.tsum_coe_ne_top_iff_summable_coe] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u222b\u207b (a : \u03b1), \u2211' (i : \u03b9), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a x : \u03b1 hx : \u2211' (i : \u03b9), \u2191\u2016f i x\u2016\u208a < \u22a4 \u22a2 \u2211' (a : \u03b9), \u2191\u2016f a x\u2016\u208a \u2260 \u22a4 ** exact hx.ne ** case pos.convert_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 \u2200 (n : \u03b9), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016f n a\u2016 \u2264 (fun i a => \u2191\u2016f i a\u2016\u208a) n a ** intro n ** case pos.convert_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a n : \u03b9 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016f n a\u2016 \u2264 (fun i a => \u2191\u2016f i a\u2016\u208a) n a ** filter_upwards with x ** case pos.convert_2.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a n : \u03b9 x : \u03b1 \u22a2 \u2016f n x\u2016 \u2264 \u2191\u2016f n x\u2016\u208a ** rfl ** case pos.convert_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 AEStronglyMeasurable (fun a => \u2211' (n : \u03b9), (fun i a => \u2191\u2016f i a\u2016\u208a) n a) \u03bc ** simp_rw [\u2190 NNReal.coe_tsum] ** case pos.convert_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 AEStronglyMeasurable (fun a => \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a)) \u03bc ** rw [aestronglyMeasurable_iff_aemeasurable] ** case pos.convert_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 AEMeasurable fun a => \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a) ** apply AEMeasurable.coe_nnreal_real ** case pos.convert_3.hf \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 AEMeasurable fun x => \u2211' (a : \u03b9), \u2016f a x\u2016\u208a ** apply AEMeasurable.nnreal_tsum ** case pos.convert_3.hf.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 \u2200 (i : \u03b9), AEMeasurable fun x => \u2016f i x\u2016\u208a ** exact fun i => (hf i).nnnorm.aemeasurable ** case pos.convert_4 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 HasFiniteIntegral fun a => \u2211' (n : \u03b9), (fun i a => \u2191\u2016f i a\u2016\u208a) n a ** dsimp [HasFiniteIntegral] ** case pos.convert_4 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a \u2202\u03bc < \u22a4 ** have : \u222b\u207b a, \u2211' n, \u2016f n a\u2016\u208a \u2202\u03bc < \u22a4 := by rwa [lintegral_tsum hf'', lt_top_iff_ne_top] ** case pos.convert_4 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a this : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a \u2202\u03bc < \u22a4 ** convert this using 1 ** case h.e'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a this : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc ** apply lintegral_congr_ae ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a this : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4 \u22a2 (fun a => \u2191\u2016\u2211' (n : \u03b9), \u2016f n a\u2016\u2016\u208a) =\u1d50[\u03bc] fun a => \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a ** simp_rw [\u2190 coe_nnnorm, \u2190 NNReal.coe_tsum, NNReal.nnnorm_eq] ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a this : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4 \u22a2 (fun a => \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a)) =\u1d50[\u03bc] fun a => \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a ** filter_upwards [hhh] with a ha ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a this : \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4 a : \u03b1 ha : Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 \u2191(\u2211' (a_1 : \u03b9), \u2016f a_1 a\u2016\u208a) = \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a ** exact ENNReal.coe_tsum (NNReal.summable_coe.mp ha) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 \u222b\u207b (a : \u03b1), \u2211' (n : \u03b9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc < \u22a4 ** rwa [lintegral_tsum hf'', lt_top_iff_ne_top] ** case pos.convert_5 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasSum (fun n => f n a) (\u2211' (i : \u03b9), f i a) ** filter_upwards [hhh] with x hx ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf' : \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), \u2191\u2016f i a\u2016\u208a \u2202\u03bc \u2260 \u22a4 hG : CompleteSpace G hf'' : \u2200 (i : \u03b9), AEMeasurable fun x => \u2191\u2016f i x\u2016\u208a hhh : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Summable fun n => \u2191\u2016f n a\u2016\u208a x : \u03b1 hx : Summable fun n => \u2191\u2016f n x\u2016\u208a \u22a2 HasSum (fun n => f n x) (\u2211' (i : \u03b9), f i x) ** exact (summable_of_summable_norm hx).hasSum ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_sigmaMk_preimage_sigmaMap ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) \u22a2 Sigma.mk i '' (g i \u207b\u00b9' s) = Sigma.map f g \u207b\u00b9' (Sigma.mk (f i) '' s) ** refine' (image_sigmaMk_preimage_sigmaMap_subset f g i s).antisymm _ ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) \u22a2 Sigma.map f g \u207b\u00b9' (Sigma.mk (f i) '' s) \u2286 Sigma.mk i '' (g i \u207b\u00b9' s) ** rintro \u27e8j, x\u27e9 \u27e8y, hys, hxy\u27e9 ** case mk.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j\u271d : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) j : \u03b9 x : \u03b1 j y : \u03b2 (f i) hys : y \u2208 s hxy : { fst := f i, snd := y } = Sigma.map f g { fst := j, snd := x } \u22a2 { fst := j, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** simp only [hf.eq_iff, Sigma.map, Sigma.ext_iff] at hxy ** case mk.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j\u271d : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) j : \u03b9 x : \u03b1 j y : \u03b2 (f i) hys : y \u2208 s hxy : i = j \u2227 HEq y (g j x) \u22a2 { fst := j, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** rcases hxy with \u27e8rfl, hxy\u27e9 ** case mk.intro.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) y : \u03b2 (f i) hys : y \u2208 s x : \u03b1 i hxy : HEq y (g i x) \u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** rw [heq_iff_eq] at hxy ** case mk.intro.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) y : \u03b2 (f i) hys : y \u2208 s x : \u03b1 i hxy : y = g i x \u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** subst y ** case mk.intro.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) x : \u03b1 i hys : g i x \u2208 s \u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** exact \u27e8x, hys, rfl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.natAbs_le_iff_sq_le ** a\u271d b\u271d : \u2124 n : \u2115 a b : \u2124 \u22a2 natAbs a \u2264 natAbs b \u2194 a ^ 2 \u2264 b ^ 2 ** rw [sq, sq] ** a\u271d b\u271d : \u2124 n : \u2115 a b : \u2124 \u22a2 natAbs a \u2264 natAbs b \u2194 a * a \u2264 b * b ** exact natAbs_le_iff_mul_self_le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrable_condexp ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 \u22a2 Integrable (\u03bc[f|m]) ** by_cases hm : m \u2264 m0 ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 \u22a2 Integrable (\u03bc[f|m])  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : \u00acm \u2264 m0 \u22a2 Integrable (\u03bc[f|m]) ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 \u22a2 Integrable (\u03bc[f|m]) ** by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm) ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 Integrable (\u03bc[f|m])  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 Integrable (\u03bc[f|m]) ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 Integrable (\u03bc[f|m]) ** haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 Integrable (\u03bc[f|m]) ** exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).symm ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : \u00acm \u2264 m0 \u22a2 Integrable (\u03bc[f|m]) ** rw [condexp_of_not_le hm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : \u00acm \u2264 m0 \u22a2 Integrable 0 ** exact integrable_zero _ _ _ ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 Integrable (\u03bc[f|m]) ** rw [condexp_of_not_sigmaFinite hm h\u03bcm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 Integrable 0 ** exact integrable_zero _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Martingale.condexp_stoppedValue_stopping_time_ae_eq_restrict_le ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n \u22a2 \u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3] =\u1d50[Measure.restrict \u03bc {x | \u03c4 x \u2264 \u03c3 x}] stoppedValue f \u03c4 ** rw [ae_eq_restrict_iff_indicator_ae_eq\n  (h\u03c4.measurableSpace_le _ (h\u03c4.measurableSet_le_stopping_time h\u03c3))] ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n \u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (\u03bc[stoppedValue f \u03c4|IsStoppingTime.measurableSpace h\u03c3]) =\u1d50[\u03bc]     Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue f \u03c4) ** refine' (condexp_indicator (integrable_stoppedValue \u03b9 h\u03c4 h.integrable h\u03c4_le)\n  (h\u03c4.measurableSet_stopping_time_le h\u03c3)).symm.trans _ ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n \u22a2 \u03bc[Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)|IsStoppingTime.measurableSpace h\u03c3] =\u1d50[\u03bc]     Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue f \u03c4) ** have h_int :\n    Integrable ({\u03c9 : \u03a9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}.indicator (stoppedValue (fun n : \u03b9 => f n) \u03c4)) \u03bc := by\n  refine' (integrable_stoppedValue \u03b9 h\u03c4 h.integrable h\u03c4_le).indicator _\n  exact h\u03c4.measurableSpace_le _ (h\u03c4.measurableSet_le_stopping_time h\u03c3) ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) h_meas :   AEStronglyMeasurable' (IsStoppingTime.measurableSpace h\u03c3)     (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u03bc \u22a2 \u03bc[Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)|IsStoppingTime.measurableSpace h\u03c3] =\u1d50[\u03bc]     Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue f \u03c4) ** exact condexp_of_aestronglyMeasurable' h\u03c3.measurableSpace_le h_meas h_int ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n \u22a2 Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) ** refine' (integrable_stoppedValue \u03b9 h\u03c4 h.integrable h\u03c4_le).indicator _ ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} ** exact h\u03c4.measurableSpace_le _ (h\u03c4.measurableSet_le_stopping_time h\u03c3) ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u22a2 AEStronglyMeasurable' (IsStoppingTime.measurableSpace h\u03c3)     (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u03bc ** refine' StronglyMeasurable.aeStronglyMeasurable' _ ** \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u22a2 StronglyMeasurable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) ** refine' StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on\n  (h\u03c4.measurableSet_le_stopping_time h\u03c3) _ _ _ ** case refine'_1 \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u22a2 \u2200 (t : Set \u03a9), MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) \u2192 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) ** intro t ht ** case refine'_1 \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) t : Set \u03a9 ht : MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) \u22a2 MeasurableSet ({\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2229 t) ** rw [Set.inter_comm _ t] at ht \u22a2 ** case refine'_1 \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) t : Set \u03a9 ht : MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}) \u22a2 MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}) ** rw [h\u03c4.measurableSet_inter_le_iff h\u03c3, IsStoppingTime.measurableSet_min_iff h\u03c4 h\u03c3] at ht ** case refine'_1 \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) t : Set \u03a9 ht : MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}) \u2227 MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}) \u22a2 MeasurableSet (t \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9}) ** exact ht.2 ** case refine'_2 \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u22a2 StronglyMeasurable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) ** refine' StronglyMeasurable.indicator _ (h\u03c4.measurableSet_le_stopping_time h\u03c3) ** case refine'_2 \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u22a2 StronglyMeasurable (stoppedValue (fun n => f n) \u03c4) ** refine' Measurable.stronglyMeasurable _ ** case refine'_2 \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u22a2 Measurable (stoppedValue (fun n => f n) \u03c4) ** exact measurable_stoppedValue h.adapted.progMeasurable_of_discrete h\u03c4 ** case refine'_3 \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) \u22a2 \u2200 (x : \u03a9), \u00acx \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4) x = 0 ** intro x hx ** case refine'_3 \u03a9 : Type u_1 E : Type u_2 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \u211d E inst\u271d\u00b9\u00b9 : CompleteSpace E \u03b9 : Type u_3 inst\u271d\u00b9\u2070 : LinearOrder \u03b9 inst\u271d\u2079 : LocallyFiniteOrder \u03b9 inst\u271d\u2078 : OrderBot \u03b9 inst\u271d\u2077 : TopologicalSpace \u03b9 inst\u271d\u2076 : DiscreteTopology \u03b9 inst\u271d\u2075 : MeasurableSpace \u03b9 inst\u271d\u2074 : BorelSpace \u03b9 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : SecondCountableTopology E \u2131 : Filtration \u03b9 m \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 f : \u03b9 \u2192 \u03a9 \u2192 E i n : \u03b9 h : Martingale f \u2131 \u03bc h\u03c4 : IsStoppingTime \u2131 \u03c4 h\u03c3 : IsStoppingTime \u2131 \u03c3 inst\u271d : SigmaFinite (Measure.trim \u03bc (_ : IsStoppingTime.measurableSpace h\u03c3 \u2264 m)) h\u03c4_le : \u2200 (x : \u03a9), \u03c4 x \u2264 n h_int : Integrable (Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4)) x : \u03a9 hx : \u00acx \u2208 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} \u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 \u2264 \u03c3 \u03c9} (stoppedValue (fun n => f n) \u03c4) x = 0 ** simp only [hx, Set.indicator_of_not_mem, not_false_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableEquiv.map_apply_eq_iff_map_symm_apply_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 e : \u03b1 \u2243\u1d50 \u03b2 \u22a2 Measure.map (\u2191e) \u03bc = \u03bd \u2194 Measure.map (\u2191(symm e)) \u03bd = \u03bc ** rw [\u2190 (map_measurableEquiv_injective e).eq_iff, map_map_symm, eq_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.pi_pi ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b2 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) s : (i : \u03b9) \u2192 Set (\u03b1 i) \u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** haveI : Encodable \u03b9 := Fintype.toEncodable \u03b9 ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b2 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) s : (i : \u03b9) \u2192 Set (\u03b1 i) this : Encodable \u03b9 \u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** rw [\u2190 pi'_eq_pi, pi'_pi] ** Qed",
        "informal": ""
    },
    {
        "formal": "ContinuousLinearMap.add_compLpL ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : Fact (1 \u2264 p) L L' : E \u2192L[\ud835\udd5c] F \u22a2 compLpL p \u03bc (L + L') = compLpL p \u03bc L + compLpL p \u03bc L' ** ext1 f ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : Fact (1 \u2264 p) L L' : E \u2192L[\ud835\udd5c] F f : { x // x \u2208 Lp E p } \u22a2 \u2191(compLpL p \u03bc (L + L')) f = \u2191(compLpL p \u03bc L + compLpL p \u03bc L') f ** exact add_compLp L L' f ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f \u22a2 \u03bc[fun a => f (X a, Y a)|MeasurableSpace.comap X m\u03b2] =\u1d50[\u03bc] fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a) ** have hf_int' : Integrable (fun a => f (X a, Y a)) \u03bc :=\n  (integrable_map_measure hf_int.1 (hX.aemeasurable.prod_mk hY)).mp hf_int ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) \u22a2 \u03bc[fun a => f (X a, Y a)|MeasurableSpace.comap X m\u03b2] =\u1d50[\u03bc] fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a) ** refine' (ae_eq_condexp_of_forall_set_integral_eq hX.comap_le hf_int' (fun s _ _ => _) _ _).symm ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s\u271d : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) s : Set \u03b1 x\u271d\u00b9 : MeasurableSet s x\u271d : \u2191\u2191\u03bc s < \u22a4 \u22a2 IntegrableOn (fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a)) s ** exact (hf_int.integral_condDistrib hX.aemeasurable hY).integrableOn ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) \u22a2 \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u222b (y : \u03a9), f (X x, y) \u2202\u2191(condDistrib Y X \u03bc) (X x) \u2202\u03bc = \u222b (x : \u03b1) in s, f (X x, Y x) \u2202\u03bc ** rintro s \u27e8t, ht, rfl\u27e9 _ ** case refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t\u271d : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) t : Set \u03b2 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4 \u22a2 \u222b (x : \u03b1) in X \u207b\u00b9' t, \u222b (y : \u03a9), f (X x, y) \u2202\u2191(condDistrib Y X \u03bc) (X x) \u2202\u03bc = \u222b (x : \u03b1) in X \u207b\u00b9' t, f (X x, Y x) \u2202\u03bc ** change \u222b a in X \u207b\u00b9' t, ((fun x' => \u222b y, f (x', y) \u2202(condDistrib Y X \u03bc) x') \u2218 X) a \u2202\u03bc =\n  \u222b a in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc ** case refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t\u271d : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) t : Set \u03b2 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4 \u22a2 \u222b (a : \u03b1) in X \u207b\u00b9' t, ((fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x') \u2218 X) a \u2202\u03bc =     \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc ** simp only [Function.comp_apply] ** case refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t\u271d : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) t : Set \u03b2 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4 \u22a2 \u222b (a : \u03b1) in X \u207b\u00b9' t, \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a) \u2202\u03bc = \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc ** rw [\u2190 integral_map hX.aemeasurable (f := fun x' => \u222b y, f (x', y) \u2202(condDistrib Y X \u03bc) x')] ** case refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t\u271d : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) t : Set \u03b2 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4 \u22a2 \u222b (y : \u03b2), \u222b (y_1 : \u03a9), f (y, y_1) \u2202\u2191(condDistrib Y X \u03bc) y \u2202Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)) =     \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc  case refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t\u271d : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) t : Set \u03b2 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4 \u22a2 AEStronglyMeasurable (fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x')     (Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t))) ** swap ** case refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t\u271d : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) t : Set \u03b2 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4 \u22a2 \u222b (y : \u03b2), \u222b (y_1 : \u03a9), f (y, y_1) \u2202\u2191(condDistrib Y X \u03bc) y \u2202Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t)) =     \u222b (a : \u03b1) in X \u207b\u00b9' t, f (X a, Y a) \u2202\u03bc ** rw [\u2190 Measure.restrict_map hX ht, \u2190 Measure.fst_map_prod_mk\u2080 hY, condDistrib,\n  set_integral_condKernel_univ_right ht hf_int.integrableOn,\n  set_integral_map (ht.prod MeasurableSet.univ) hf_int.1 (hX.aemeasurable.prod_mk hY),\n  mk_preimage_prod, preimage_univ, inter_univ] ** case refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t\u271d : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) t : Set \u03b2 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4 \u22a2 AEStronglyMeasurable (fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x')     (Measure.map X (Measure.restrict \u03bc (X \u207b\u00b9' t))) ** rw [\u2190 Measure.restrict_map hX ht] ** case refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t\u271d : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) t : Set \u03b2 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc (X \u207b\u00b9' t) < \u22a4 \u22a2 AEStronglyMeasurable (fun x' => \u222b (y : \u03a9), f (x', y) \u2202\u2191(condDistrib Y X \u03bc) x') (Measure.restrict (Measure.map X \u03bc) t) ** exact (hf_int.1.integral_condDistrib_map hY).restrict ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf_int : Integrable f hf_int' : Integrable fun a => f (X a, Y a) \u22a2 AEStronglyMeasurable' (MeasurableSpace.comap X m\u03b2) (fun a => \u222b (y : \u03a9), f (X a, y) \u2202\u2191(condDistrib Y X \u03bc) (X a)) \u03bc ** exact aestronglyMeasurable'_integral_condDistrib hX.aemeasurable hY hf_int.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.eval_eq_rfindOpt ** c : Code n x : \u2115 \u22a2 x \u2208 eval c n \u2194 x \u2208 rfindOpt fun k => evaln k c n ** refine' evaln_complete.trans (Nat.rfindOpt_mono _).symm ** c : Code n x : \u2115 \u22a2 \u2200 {a m n_1 : \u2115}, m \u2264 n_1 \u2192 a \u2208 evaln m c n \u2192 a \u2208 evaln n_1 c n ** intro a m n hl ** c : Code n\u271d x a m n : \u2115 hl : m \u2264 n \u22a2 a \u2208 evaln m c n\u271d \u2192 a \u2208 evaln n c n\u271d ** apply evaln_mono hl ** Qed",
        "informal": ""
    },
    {
        "formal": "List.pairwise_middle ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u2200 {x y : \u03b1}, R x y \u2192 R y x a : \u03b1 l\u2081 l\u2082 : List \u03b1 \u22a2 Pairwise R (l\u2081 ++ a :: l\u2082) \u2194 Pairwise R (a :: (l\u2081 ++ l\u2082)) ** show Pairwise R (l\u2081 ++ ([a] ++ l\u2082)) \u2194 Pairwise R ([a] ++ l\u2081 ++ l\u2082) ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u2200 {x y : \u03b1}, R x y \u2192 R y x a : \u03b1 l\u2081 l\u2082 : List \u03b1 \u22a2 Pairwise R (l\u2081 ++ ([a] ++ l\u2082)) \u2194 Pairwise R ([a] ++ l\u2081 ++ l\u2082) ** rw [\u2190 append_assoc, pairwise_append, @pairwise_append _ _ ([a] ++ l\u2081), pairwise_append_comm s] ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u2200 {x y : \u03b1}, R x y \u2192 R y x a : \u03b1 l\u2081 l\u2082 : List \u03b1 \u22a2 (Pairwise (fun {x y} => R x y) ([a] ++ l\u2081) \u2227       Pairwise R l\u2082 \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 l\u2081 ++ [a] \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 R a_1 b) \u2194     Pairwise R ([a] ++ l\u2081) \u2227 Pairwise R l\u2082 \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 [a] ++ l\u2081 \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 R a_1 b ** simp only [mem_append, or_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Holor.slice_eq ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 x y : Holor \u03b1 (d :: ds) h : slice x = slice y t : HolorIndex (d :: ds) i : \u2115 is : List \u2115 hiis : \u2191t = i :: is \u22a2 Forall\u2082 (fun x x_1 => x < x_1) (i :: is) (d :: ds) ** rw [\u2190 hiis] ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 x y : Holor \u03b1 (d :: ds) h : slice x = slice y t : HolorIndex (d :: ds) i : \u2115 is : List \u2115 hiis : \u2191t = i :: is \u22a2 Forall\u2082 (fun x x_1 => x < x_1) (\u2191t) (d :: ds) ** exact t.2 ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 x y : Holor \u03b1 (d :: ds) h : slice x = slice y t : HolorIndex (d :: ds) i : \u2115 is : List \u2115 hiis : \u2191t = i :: is hiisdds : Forall\u2082 (fun x x_1 => x < x_1) (i :: is) (d :: ds) hid : i < d hisds : Forall\u2082 (fun x x_1 => x < x_1) is ds \u22a2 slice x i hid { val := is, property := hisds } = slice y i hid { val := is, property := hisds } ** rw [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_congr_left' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f : \u03b1 \u2192 E \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f : \u03b1 \u2192 E hf : Integrable f \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f ** simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left' T T' hT hT' h] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f ** simp_rw [setToFun_undef _ hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.lcm_zero_right ** i : \u2124 \u22a2 lcm i 0 = 0 ** rw [Int.lcm] ** i : \u2124 \u22a2 Nat.lcm (natAbs i) (natAbs 0) = 0 ** apply Nat.lcm_zero_right ** Qed",
        "informal": ""
    },
    {
        "formal": "String.revFind_of_valid ** p : Char \u2192 Bool s : String \u22a2 revFind s p =     Option.map (fun x => { byteIdx := utf8Len x }) (List.tail? (List.dropWhile (fun x => !p x) (List.reverse s.data))) ** simpa using revFindAux_of_valid p s.1.reverse [] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.zipWith_eq_zipWithTR ** \u22a2 @zipWith = @zipWithTR ** funext \u03b1 \u03b2 \u03b3 f as bs ** case h.h.h.h.h.h \u03b1 : Type u_3 \u03b2 : Type u_2 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 as : List \u03b1 bs : List \u03b2 \u22a2 zipWith f as bs = zipWithTR f as bs ** exact (go as bs #[]).symm ** \u03b1 : Type u_3 \u03b2 : Type u_2 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 as : List \u03b1 bs : List \u03b2 head\u271d : \u03b1 tail\u271d : List \u03b1 acc : Array \u03b3 \u22a2 zipWithTR.go f (head\u271d :: tail\u271d) [] acc = acc.data ++ zipWith f (head\u271d :: tail\u271d) [] ** simp [zipWithTR.go, zipWith] ** \u03b1 : Type u_3 \u03b2 : Type u_2 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 as\u271d : List \u03b1 bs\u271d : List \u03b2 a : \u03b1 as : List \u03b1 b : \u03b2 bs : List \u03b2 acc : Array \u03b3 \u22a2 zipWithTR.go f (a :: as) (b :: bs) acc = acc.data ++ zipWith f (a :: as) (b :: bs) ** simp [zipWithTR.go, zipWith, go as bs] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.mapAccumr\u2082_eq_map\u2082 ** \u03b1 : Type n : \u2115 \u03b2 : Type xs : Vector \u03b1 n ys : Vector \u03b2 n \u03c3 \u03b3 : Type f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3 s\u2080 : \u03c3 S : Set \u03c3 h\u2080 : s\u2080 \u2208 S closure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S out : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2 \u22a2 (mapAccumr\u2082 f xs ys s\u2080).2 = map\u2082 (fun x x_1 => (f x x_1 s\u2080).2) xs ys ** rw[Vector.map\u2082_eq_mapAccumr\u2082] ** \u03b1 : Type n : \u2115 \u03b2 : Type xs : Vector \u03b1 n ys : Vector \u03b2 n \u03c3 \u03b3 : Type f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3 s\u2080 : \u03c3 S : Set \u03c3 h\u2080 : s\u2080 \u2208 S closure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S out : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2 \u22a2 (mapAccumr\u2082 f xs ys s\u2080).2 = (mapAccumr\u2082 (fun x y x_1 => ((), (f x y s\u2080).2)) xs ys ()).2 ** apply mapAccumr\u2082_bisim_tail ** case h \u03b1 : Type n : \u2115 \u03b2 : Type xs : Vector \u03b1 n ys : Vector \u03b2 n \u03c3 \u03b3 : Type f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3 s\u2080 : \u03c3 S : Set \u03c3 h\u2080 : s\u2080 \u2208 S closure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S out : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2 \u22a2 \u2203 R,     R s\u2080 () \u2227       \u2200 {s : \u03c3} {q : Unit} (a : \u03b1) (b : \u03b2),         R s q \u2192 R (f a b s).1 ((), (f a b s\u2080).2).1 \u2227 (f a b s).2 = ((), (f a b s\u2080).2).2 ** use fun s _ => s \u2208 S, h\u2080 ** case right \u03b1 : Type n : \u2115 \u03b2 : Type xs : Vector \u03b1 n ys : Vector \u03b2 n \u03c3 \u03b3 : Type f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3 s\u2080 : \u03c3 S : Set \u03c3 h\u2080 : s\u2080 \u2208 S closure : \u2200 (a : \u03b1) (b : \u03b2) (s : \u03c3), s \u2208 S \u2192 (f a b s).1 \u2208 S out : \u2200 (a : \u03b1) (b : \u03b2) (s s' : \u03c3), s \u2208 S \u2192 s' \u2208 S \u2192 (f a b s).2 = (f a b s').2 \u22a2 \u2200 {s : \u03c3} {q : Unit} (a : \u03b1) (b : \u03b2), s \u2208 S \u2192 (f a b s).1 \u2208 S \u2227 (f a b s).2 = ((), (f a b s\u2080).2).2 ** exact @fun s _q a b h => \u27e8closure a b s h, out a b s s\u2080 h h\u2080\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "parallelepiped_comp_equiv ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 \u22a2 parallelepiped (v \u2218 \u2191e) = parallelepiped v ** simp only [parallelepiped] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 \u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1 ** let K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a : \u03b9' => \u211d) e ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this : Icc 0 1 = \u2191K '' Icc 0 1 \u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1 ** rw [this, \u2190 image_comp] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this : Icc 0 1 = \u2191K '' Icc 0 1 \u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1 ** congr 1 with x ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this : Icc 0 1 = \u2191K '' Icc 0 1 x : E \u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1 ** have := fun z : \u03b9' \u2192 \u211d => e.symm.sum_comp fun i => z i \u2022 v (e i) ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this\u271d : Icc 0 1 = \u2191K '' Icc 0 1 x : E this : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 i : \u03b9, z (\u2191e.symm i) \u2022 v (\u2191e (\u2191e.symm i)) = \u2211 i : \u03b9', z i \u2022 v (\u2191e i) \u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1 ** simp_rw [Equiv.apply_symm_apply] at this ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e this\u271d : Icc 0 1 = \u2191K '' Icc 0 1 x : E this : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 x : \u03b9, z (\u2191e.symm x) \u2022 v x = \u2211 x : \u03b9', z x \u2022 v (\u2191e x) \u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1 ** simp_rw [Function.comp_apply, mem_image, mem_Icc, Equiv.piCongrLeft'_apply, this] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e \u22a2 Icc 0 1 = \u2191K '' Icc 0 1 ** rw [\u2190 Equiv.preimage_eq_iff_eq_image] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e \u22a2 \u2191K \u207b\u00b9' Icc 0 1 = Icc 0 1 ** ext x ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e x : \u03b9' \u2192 \u211d \u22a2 x \u2208 \u2191K \u207b\u00b9' Icc 0 1 \u2194 x \u2208 Icc 0 1 ** simp only [mem_preimage, mem_Icc, Pi.le_def, Pi.zero_apply, Equiv.piCongrLeft'_apply,\n  Pi.one_apply] ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e x : \u03b9' \u2192 \u211d \u22a2 ((\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1) \u2194 (\u2200 (i : \u03b9'), 0 \u2264 x i) \u2227 \u2200 (i : \u03b9'), x i \u2264 1 ** refine'\n  \u27e8fun h => \u27e8fun i => _, fun i => _\u27e9, fun h =>\n    \u27e8fun i => h.1 (e.symm i), fun i => h.2 (e.symm i)\u27e9\u27e9 ** case h.refine'_1 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e x : \u03b9' \u2192 \u211d h : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1 i : \u03b9' \u22a2 0 \u2264 x i ** simpa only [Equiv.symm_apply_apply] using h.1 (e i) ** case h.refine'_2 \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E e : \u03b9' \u2243 \u03b9 K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e x : \u03b9' \u2192 \u211d h : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1 i : \u03b9' \u22a2 x i \u2264 1 ** simpa only [Equiv.symm_apply_apply] using h.2 (e i) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.image_of_measurable_injOn ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2075 : T2Space \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 s : Set \u03b3 f : \u03b3 \u2192 \u03b2 inst\u271d\u00b3 : OpensMeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 inst\u271d\u00b9 : StandardBorelSpace \u03b3 inst\u271d : SecondCountableTopology \u03b2 hs : MeasurableSet s f_meas : Measurable f f_inj : InjOn f s \u22a2 MeasurableSet (f '' s) ** letI := upgradeStandardBorel \u03b3 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2075 : T2Space \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 s : Set \u03b3 f : \u03b3 \u2192 \u03b2 inst\u271d\u00b3 : OpensMeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 inst\u271d\u00b9 : StandardBorelSpace \u03b3 inst\u271d : SecondCountableTopology \u03b2 hs : MeasurableSet s f_meas : Measurable f f_inj : InjOn f s this : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3 \u22a2 MeasurableSet (f '' s) ** let t\u03b3 : TopologicalSpace \u03b3 := inferInstance ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2075 : T2Space \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 s : Set \u03b3 f : \u03b3 \u2192 \u03b2 inst\u271d\u00b3 : OpensMeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 inst\u271d\u00b9 : StandardBorelSpace \u03b3 inst\u271d : SecondCountableTopology \u03b2 hs : MeasurableSet s f_meas : Measurable f f_inj : InjOn f s this : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3 t\u03b3 : TopologicalSpace \u03b3 := inferInstance \u22a2 MeasurableSet (f '' s) ** obtain \u27e8t', t't, f_cont, t'_polish\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b3, t' \u2264 t\u03b3 \u2227 @Continuous \u03b3 \u03b2 t' t\u03b2 f \u2227 @PolishSpace \u03b3 t' :=\n  f_meas.exists_continuous ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2075 : T2Space \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 s : Set \u03b3 f : \u03b3 \u2192 \u03b2 inst\u271d\u00b3 : OpensMeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 inst\u271d\u00b9 : StandardBorelSpace \u03b3 inst\u271d : SecondCountableTopology \u03b2 hs : MeasurableSet s f_meas : Measurable f f_inj : InjOn f s this : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3 t\u03b3 : TopologicalSpace \u03b3 := inferInstance t' : TopologicalSpace \u03b3 t't : t' \u2264 t\u03b3 f_cont : Continuous f t'_polish : PolishSpace \u03b3 \u22a2 MeasurableSet (f '' s) ** have M : MeasurableSet[@borel \u03b3 t'] s :=\n  @Continuous.measurable \u03b3 \u03b3 t' (@borel \u03b3 t')\n    (@BorelSpace.opensMeasurable \u03b3 t' (@borel \u03b3 t') (@BorelSpace.mk _ _ (borel \u03b3) rfl))\n    t\u03b3 _ _ _ (continuous_id_of_le t't) s hs ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u2075 : T2Space \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2 s : Set \u03b3 f : \u03b3 \u2192 \u03b2 inst\u271d\u00b3 : OpensMeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 inst\u271d\u00b9 : StandardBorelSpace \u03b3 inst\u271d : SecondCountableTopology \u03b2 hs : MeasurableSet s f_meas : Measurable f f_inj : InjOn f s this : UpgradedStandardBorel \u03b3 := upgradeStandardBorel \u03b3 t\u03b3 : TopologicalSpace \u03b3 := inferInstance t' : TopologicalSpace \u03b3 t't : t' \u2264 t\u03b3 f_cont : Continuous f t'_polish : PolishSpace \u03b3 M : MeasurableSet s \u22a2 MeasurableSet (f '' s) ** exact\n  @MeasurableSet.image_of_continuousOn_injOn \u03b3\n    \u03b2 _ _ _  s f _ t' t'_polish (@borel \u03b3 t') (@BorelSpace.mk _ _ (borel \u03b3) rfl)\n    M (@Continuous.continuousOn \u03b3 \u03b2 t' t\u03b2 f s f_cont) f_inj ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.nat_rec ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2 hf : Primrec f hg : Primrec\u2082 g n : \u2115 \u22a2 Nat.unpaired       (fun z n =>         Nat.casesOn n 0 fun y =>           Nat.unpaired             (fun z n =>               Nat.rec (encode (Option.map f (decode z)))                 (fun y IH =>                   encode                     (Option.map (fun p => g p.1 p.2)                       (decode                         (Nat.pair (Nat.unpair (Nat.pair z (Nat.pair y IH))).1                           (Nat.pair (Nat.unpair (Nat.unpair (Nat.pair z (Nat.pair y IH))).2).1                             (Nat.pred (Nat.unpair (Nat.unpair (Nat.pair z (Nat.pair y IH))).2).2))))))                 n)             (Nat.pair (Nat.unpair (Nat.pair z y)).2 (Nat.unpair (Nat.unpair (Nat.pair z y)).1).2))       (Nat.pair (id n) (encode (decode (Nat.unpair n).1))) =     Nat.unpaired       (fun m n =>         encode           (Option.bind (decode m) fun a => Option.map (fun n => Nat.rec (f a) (fun n IH => g a (n, IH)) n) (decode n)))       n ** simp only [Nat.unpaired, id_eq, Nat.unpair_pair, decode_prod_val, decode_nat,\n  Option.some_bind, Option.map_map, Option.map_some'] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2 hf : Primrec f hg : Primrec\u2082 g n : \u2115 \u22a2 Nat.rec 0       (fun n_1 n_ih =>         Nat.rec (encode (Option.map f (decode n_1)))           (fun y IH =>             encode               (Option.map (fun p => g p.1 p.2)                 (Option.bind (decode n_1) fun a => Option.map (Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))))           (Nat.unpair n).2)       (encode (decode (Nat.unpair n).1)) =     encode       (Option.bind (decode (Nat.unpair n).1) fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2)) ** cases' @decode \u03b1 _ n.unpair.1 with a ** case some \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2 hf : Primrec f hg : Primrec\u2082 g n : \u2115 a : \u03b1 \u22a2 Nat.rec 0       (fun n_1 n_ih =>         Nat.rec (encode (Option.map f (decode n_1)))           (fun y IH =>             encode               (Option.map (fun p => g p.1 p.2)                 (Option.bind (decode n_1) fun a => Option.map (Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))))           (Nat.unpair n).2)       (encode (some a)) =     encode (Option.bind (some a) fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2)) ** simp only [encode_some, encodek, Option.map_some', Option.some_bind, Option.map_map] ** case some \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2 hf : Primrec f hg : Primrec\u2082 g n : \u2115 a : \u03b1 \u22a2 Nat.rec (Nat.succ (encode (f a)))       (fun y IH => encode (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH))))       (Nat.unpair n).2 =     Nat.succ (encode (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2)) ** induction' n.unpair.2 with m <;> simp [encodek] ** case some.succ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2 hf : Primrec f hg : Primrec\u2082 g n : \u2115 a : \u03b1 m : \u2115 n_ih\u271d :   Nat.rec (Nat.succ (encode (f a)))       (fun y IH => encode (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))) m =     Nat.succ (encode (Nat.rec (f a) (fun n IH => g a (n, IH)) m)) \u22a2 encode       (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk m)         (decode           (Nat.pred             (Nat.rec (Nat.succ (encode (f a)))               (fun y IH => encode (Option.map ((fun p => g p.1 p.2) \u2218 Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH))))               m)))) =     Nat.succ (encode (g a (m, Nat.rec (f a) (fun n IH => g a (n, IH)) m))) ** simp [*, encodek] ** case none \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u2115 \u00d7 \u03b2 \u2192 \u03b2 hf : Primrec f hg : Primrec\u2082 g n : \u2115 \u22a2 Nat.rec 0       (fun n_1 n_ih =>         Nat.rec (encode (Option.map f (decode n_1)))           (fun y IH =>             encode               (Option.map (fun p => g p.1 p.2)                 (Option.bind (decode n_1) fun a => Option.map (Prod.mk a \u2218 Prod.mk y) (decode (Nat.pred IH)))))           (Nat.unpair n).2)       (encode none) =     encode (Option.bind none fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2)) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_submodule ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc \u2191s = 0 ** obtain \u27e8x, hx\u27e9 : \u2203 x, x \u2209 s := by\n  simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs ** case intro E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s \u22a2 \u2191\u2191\u03bc \u2191s = 0 ** obtain \u27e8c, cpos, cone\u27e9 : \u2203 c : \u211d, 0 < c \u2227 c < 1 := \u27e81 / 2, by norm_num, by norm_num\u27e9 ** case intro.intro.intro E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 \u22a2 \u2191\u2191\u03bc \u2191s = 0 ** have A : IsBounded (range fun n : \u2115 => c ^ n \u2022 x) :=\n  have : Tendsto (fun n : \u2115 => c ^ n \u2022 x) atTop (\ud835\udcdd ((0 : \u211d) \u2022 x)) :=\n    (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x\n  isBounded_range_of_tendsto _ this ** case intro.intro.intro E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A : Bornology.IsBounded (range fun n => c ^ n \u2022 x) \u22a2 \u2191\u2191\u03bc \u2191s = 0 ** apply addHaar_eq_zero_of_disjoint_translates \u03bc _ A _\n  (Submodule.closed_of_finiteDimensional s).measurableSet ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A : Bornology.IsBounded (range fun n => c ^ n \u2022 x) \u22a2 Pairwise (Disjoint on fun n => {c ^ n \u2022 x} + \u2191s) ** intro m n hmn ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n \u22a2 (Disjoint on fun n => {c ^ n \u2022 x} + \u2191s) m n ** simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,\n  SetLike.mem_coe] ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n \u22a2 \u2200 \u2983a : E\u2984, -(c ^ m \u2022 x) + a \u2208 s \u2192 \u00ac-(c ^ n \u2022 x) + a \u2208 s ** intro y hym hyn ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n y : E hym : -(c ^ m \u2022 x) + y \u2208 s hyn : -(c ^ n \u2022 x) + y \u2208 s \u22a2 False ** have A : (c ^ n - c ^ m) \u2022 x \u2208 s := by\n  convert s.sub_mem hym hyn using 1\n  simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub] ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n y : E hym : -(c ^ m \u2022 x) + y \u2208 s hyn : -(c ^ n \u2022 x) + y \u2208 s A : (c ^ n - c ^ m) \u2022 x \u2208 s \u22a2 False ** have H : c ^ n - c ^ m \u2260 0 := by\n  simpa only [sub_eq_zero, Ne.def] using (strictAnti_pow cpos cone).injective.ne hmn.symm ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n y : E hym : -(c ^ m \u2022 x) + y \u2208 s hyn : -(c ^ n \u2022 x) + y \u2208 s A : (c ^ n - c ^ m) \u2022 x \u2208 s H : c ^ n - c ^ m \u2260 0 \u22a2 False ** have : x \u2208 s := by\n  convert s.smul_mem (c ^ n - c ^ m)\u207b\u00b9 A\n  rw [smul_smul, inv_mul_cancel H, one_smul] ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n y : E hym : -(c ^ m \u2022 x) + y \u2208 s hyn : -(c ^ n \u2022 x) + y \u2208 s A : (c ^ n - c ^ m) \u2022 x \u2208 s H : c ^ n - c ^ m \u2260 0 this : x \u2208 s \u22a2 False ** exact hx this ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 \u22a2 \u2203 x, \u00acx \u2208 s ** simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s \u22a2 0 < 1 / 2 ** norm_num ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s \u22a2 1 / 2 < 1 ** norm_num ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n y : E hym : -(c ^ m \u2022 x) + y \u2208 s hyn : -(c ^ n \u2022 x) + y \u2208 s \u22a2 (c ^ n - c ^ m) \u2022 x \u2208 s ** convert s.sub_mem hym hyn using 1 ** case h.e'_4 E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n y : E hym : -(c ^ m \u2022 x) + y \u2208 s hyn : -(c ^ n \u2022 x) + y \u2208 s \u22a2 (c ^ n - c ^ m) \u2022 x = -(c ^ m \u2022 x) + y - (-(c ^ n \u2022 x) + y) ** simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub] ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n y : E hym : -(c ^ m \u2022 x) + y \u2208 s hyn : -(c ^ n \u2022 x) + y \u2208 s A : (c ^ n - c ^ m) \u2022 x \u2208 s \u22a2 c ^ n - c ^ m \u2260 0 ** simpa only [sub_eq_zero, Ne.def] using (strictAnti_pow cpos cone).injective.ne hmn.symm ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n y : E hym : -(c ^ m \u2022 x) + y \u2208 s hyn : -(c ^ n \u2022 x) + y \u2208 s A : (c ^ n - c ^ m) \u2022 x \u2208 s H : c ^ n - c ^ m \u2260 0 \u22a2 x \u2208 s ** convert s.smul_mem (c ^ n - c ^ m)\u207b\u00b9 A ** case h.e'_4 E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Submodule \u211d E hs : s \u2260 \u22a4 x : E hx : \u00acx \u2208 s c : \u211d cpos : 0 < c cone : c < 1 A\u271d : Bornology.IsBounded (range fun n => c ^ n \u2022 x) m n : \u2115 hmn : m \u2260 n y : E hym : -(c ^ m \u2022 x) + y \u2208 s hyn : -(c ^ n \u2022 x) + y \u2208 s A : (c ^ n - c ^ m) \u2022 x \u2208 s H : c ^ n - c ^ m \u2260 0 \u22a2 x = (c ^ n - c ^ m)\u207b\u00b9 \u2022 (c ^ n - c ^ m) \u2022 x ** rw [smul_smul, inv_mul_cancel H, one_smul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pairwiseDisjoint_prod_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 \u03ba : Sort u_6 r p q : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : Frame \u03b1 s : Set \u03b9 t : Set \u03b9' f : \u03b9 \u00d7 \u03b9' \u2192 \u03b1 \u22a2 PairwiseDisjoint (s \u00d7\u02e2 t) f \u2194     (PairwiseDisjoint s fun i => \u2a06 i' \u2208 t, f (i, i')) \u2227 PairwiseDisjoint t fun i' => \u2a06 i \u2208 s, f (i, i') ** refine'\n      \u27e8fun h => \u27e8fun i hi j hj hij => _, fun i hi j hj hij => _\u27e9, fun h => h.1.prod_left h.2\u27e9 <;>\n    simp_rw [Function.onFun, iSup_disjoint_iff, disjoint_iSup_iff] <;>\n  intro i' hi' j' hj' ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 \u03ba : Sort u_6 r p q : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : Frame \u03b1 s : Set \u03b9 t : Set \u03b9' f : \u03b9 \u00d7 \u03b9' \u2192 \u03b1 h : PairwiseDisjoint (s \u00d7\u02e2 t) f i : \u03b9 hi : i \u2208 s j : \u03b9 hj : j \u2208 s hij : i \u2260 j i' : \u03b9' hi' : i' \u2208 t j' : \u03b9' hj' : j' \u2208 t \u22a2 Disjoint (f (i, i')) (f (j, j')) ** exact h (mk_mem_prod hi hi') (mk_mem_prod hj hj') (ne_of_apply_ne Prod.fst hij) ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 \u03ba : Sort u_6 r p q : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : Frame \u03b1 s : Set \u03b9 t : Set \u03b9' f : \u03b9 \u00d7 \u03b9' \u2192 \u03b1 h : PairwiseDisjoint (s \u00d7\u02e2 t) f i : \u03b9' hi : i \u2208 t j : \u03b9' hj : j \u2208 t hij : i \u2260 j i' : \u03b9 hi' : i' \u2208 s j' : \u03b9 hj' : j' \u2208 s \u22a2 Disjoint (f (i', i)) (f (j', j)) ** exact h (mk_mem_prod hi' hi) (mk_mem_prod hj' hj) (ne_of_apply_ne Prod.snd hij) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condCdf_le_one ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d \u22a2 \u2191(condCdf \u03c1 a) x \u2264 1 ** obtain \u27e8r, hrx\u27e9 := exists_rat_gt x ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d r : \u211a hrx : x < \u2191r \u22a2 \u2191(condCdf \u03c1 a) x \u2264 1 ** rw [\u2190 StieltjesFunction.iInf_rat_gt_eq] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d r : \u211a hrx : x < \u2191r \u22a2 \u2a05 r, \u2191(condCdf \u03c1 a) \u2191\u2191r \u2264 1 ** simp_rw [condCdf_eq_condCdfRat] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d r : \u211a hrx : x < \u2191r \u22a2 \u2a05 r, condCdfRat \u03c1 a \u2191r \u2264 1 ** refine' ciInf_le_of_le (bddBelow_range_condCdfRat_gt \u03c1 a x) _ (condCdfRat_le_one _ _ _) ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d r : \u211a hrx : x < \u2191r \u22a2 { r' // x < \u2191r' } ** exact \u27e8r, hrx\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_inl_range_inr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f : \u03b9 \u2192 \u03b1 s t : Set \u03b1 \u22a2 Sum.inl \u207b\u00b9' range Sum.inr = \u2205 ** rw [\u2190 image_univ, preimage_inl_image_inr] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.iUnion_pi_of_monotone ** \u03b1\u271d : Type u \u03b2 : Type v \u03b9\u271d : Sort w \u03b3 : Type x \u03b9 : Type u_1 \u03b9' : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b9' inst\u271d : Nonempty \u03b9' \u03b1 : \u03b9 \u2192 Type u_3 I : Set \u03b9 s : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i) hI : Set.Finite I hs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i) \u22a2 (\u22c3 j, pi I fun i => s i j) = pi I fun i => \u22c3 j, s i j ** simp only [pi_def, biInter_eq_iInter, preimage_iUnion] ** \u03b1\u271d : Type u \u03b2 : Type v \u03b9\u271d : Sort w \u03b3 : Type x \u03b9 : Type u_1 \u03b9' : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b9' inst\u271d : Nonempty \u03b9' \u03b1 : \u03b9 \u2192 Type u_3 I : Set \u03b9 s : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i) hI : Set.Finite I hs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i) \u22a2 \u22c3 j, \u22c2 x, eval \u2191x \u207b\u00b9' s (\u2191x) j = \u22c2 x, \u22c3 i, eval \u2191x \u207b\u00b9' s (\u2191x) i ** haveI := hI.fintype.finite ** \u03b1\u271d : Type u \u03b2 : Type v \u03b9\u271d : Sort w \u03b3 : Type x \u03b9 : Type u_1 \u03b9' : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b9' inst\u271d : Nonempty \u03b9' \u03b1 : \u03b9 \u2192 Type u_3 I : Set \u03b9 s : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i) hI : Set.Finite I hs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i) this : Finite \u2191I \u22a2 \u22c3 j, \u22c2 x, eval \u2191x \u207b\u00b9' s (\u2191x) j = \u22c2 x, \u22c3 i, eval \u2191x \u207b\u00b9' s (\u2191x) i ** refine' iUnion_iInter_of_monotone (\u03b9' := \u03b9') (fun (i : I) j\u2081 j\u2082 h => _) ** \u03b1\u271d : Type u \u03b2 : Type v \u03b9\u271d : Sort w \u03b3 : Type x \u03b9 : Type u_1 \u03b9' : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b9' inst\u271d : Nonempty \u03b9' \u03b1 : \u03b9 \u2192 Type u_3 I : Set \u03b9 s : (i : \u03b9) \u2192 \u03b9' \u2192 Set (\u03b1 i) hI : Set.Finite I hs : \u2200 (i : \u03b9), i \u2208 I \u2192 Monotone (s i) this : Finite \u2191I i : \u2191I j\u2081 j\u2082 : \u03b9' h : j\u2081 \u2264 j\u2082 \u22a2 eval \u2191i \u207b\u00b9' s (\u2191i) j\u2081 \u2264 eval \u2191i \u207b\u00b9' s (\u2191i) j\u2082 ** exact preimage_mono <| hs i i.2 h ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.min'_insert ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H\u271d : Finset.Nonempty s\u271d x a : \u03b1 s : Finset \u03b1 H : Finset.Nonempty s \u22a2 IsLeast (\u2191(insert a s)) (min (min' s H) a) ** rw [coe_insert, min_comm] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H\u271d : Finset.Nonempty s\u271d x a : \u03b1 s : Finset \u03b1 H : Finset.Nonempty s \u22a2 IsLeast (insert a \u2191s) (min a (min' s H)) ** exact (isLeast_min' _ _).insert _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_norm_rpow ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q \u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q ** by_cases h0 : p = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 \u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q ** by_cases hp_top : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q ** rw [snorm_eq_snorm' h0 hp_top, snorm_eq_snorm' _ _] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q  \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 p * ENNReal.ofReal q \u2260 0  \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 p * ENNReal.ofReal q \u2260 \u22a4 ** swap ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q  \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 p * ENNReal.ofReal q \u2260 \u22a4 ** swap ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal (p * ENNReal.ofReal q)) \u03bc ^ q ** rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal hq_pos.le] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm' (fun x => \u2016f x\u2016 ^ q) (ENNReal.toReal p) \u03bc = snorm' f (ENNReal.toReal p * q) \u03bc ^ q ** exact snorm'_norm_rpow f p.toReal q hq_pos ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : p = 0 \u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q ** simp [h0, ENNReal.zero_rpow_of_pos hq_pos] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 snorm (fun x => \u2016f x\u2016 ^ q) p \u03bc = snorm f (p * ENNReal.ofReal q) \u03bc ^ q ** simp only [hp_top, snorm_exponent_top, ENNReal.top_mul', hq_pos.not_le, ENNReal.ofReal_eq_zero,\n  if_false, snorm_exponent_top, snormEssSup] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q ** have h_rpow :\n  essSup (fun x : \u03b1 => (\u2016\u2016f x\u2016 ^ q\u2016\u208a : \u211d\u22650\u221e)) \u03bc =\n    essSup (fun x : \u03b1 => (\u2016f x\u2016\u208a : \u211d\u22650\u221e) ^ q) \u03bc := by\n  congr\n  ext1 x\n  conv_rhs => rw [\u2190 nnnorm_norm]\n  rw [ENNReal.coe_rpow_of_nonneg _ hq_pos.le, ENNReal.coe_eq_coe]\n  ext\n  push_cast\n  rw [Real.norm_rpow_of_nonneg (norm_nonneg _)] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 h_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc \u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q ** rw [h_rpow] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 h_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc \u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q ** have h_rpow_mono := ENNReal.strictMono_rpow_of_pos hq_pos ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 h_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc h_rpow_mono : StrictMono fun x => x ^ q \u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q ** have h_rpow_surj := (ENNReal.rpow_left_bijective hq_pos.ne.symm).2 ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 h_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc h_rpow_mono : StrictMono fun x => x ^ q h_rpow_surj : Function.Surjective fun y => y ^ q \u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q ** let iso := h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 h_rpow : essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc h_rpow_mono : StrictMono fun x => x ^ q h_rpow_surj : Function.Surjective fun y => y ^ q iso : \u211d\u22650\u221e \u2243o \u211d\u22650\u221e := StrictMono.orderIsoOfSurjective (fun x => x ^ q) h_rpow_mono h_rpow_surj \u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc ^ q ** exact (iso.essSup_apply (fun x => (\u2016f x\u2016\u208a : \u211d\u22650\u221e)) \u03bc).symm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 essSup (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a ^ q) \u03bc ** congr ** case e_f \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 (fun x => \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a) = fun x => \u2191\u2016f x\u2016\u208a ^ q ** ext1 x ** case e_f.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 x : \u03b1 \u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191\u2016f x\u2016\u208a ^ q ** conv_rhs => rw [\u2190 nnnorm_norm] ** case e_f.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 x : \u03b1 \u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191\u2016\u2016f x\u2016\u2016\u208a ^ q ** rw [ENNReal.coe_rpow_of_nonneg _ hq_pos.le, ENNReal.coe_eq_coe] ** case e_f.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 x : \u03b1 \u22a2 \u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2016\u2016f x\u2016\u2016\u208a ^ q ** ext ** case e_f.h.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 x : \u03b1 \u22a2 \u2191\u2016\u2016f x\u2016 ^ q\u2016\u208a = \u2191(\u2016\u2016f x\u2016\u2016\u208a ^ q) ** push_cast ** case e_f.h.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : p = \u22a4 x : \u03b1 \u22a2 \u2016\u2016f x\u2016 ^ q\u2016 = \u2016\u2016f x\u2016\u2016 ^ q ** rw [Real.norm_rpow_of_nonneg (norm_nonneg _)] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 p * ENNReal.ofReal q \u2260 0 ** refine' mul_ne_zero h0 _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 ENNReal.ofReal q \u2260 0 ** rwa [Ne.def, ENNReal.ofReal_eq_zero, not_le] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F hq_pos : 0 < q h0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 p * ENNReal.ofReal q \u2260 \u22a4 ** exact ENNReal.mul_ne_top hp_top ENNReal.ofReal_ne_top ** Qed",
        "informal": ""
    },
    {
        "formal": "Setoid.eqv_class_mem' ** \u03b1 : Type u_1 c : Set (Set \u03b1) H : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b x : \u03b1 \u22a2 {y | Rel (mkClasses c H) x y} \u2208 c ** convert @Setoid.eqv_class_mem _ _ H x using 3 ** case h.e'_4.h.e'_2.h.a \u03b1 : Type u_1 c : Set (Set \u03b1) H : \u2200 (a : \u03b1), \u2203! b x, a \u2208 b x x\u271d : \u03b1 \u22a2 Rel (mkClasses c H) x x\u271d \u2194 Rel (mkClasses c H) x\u271d x ** rw [Setoid.comm'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.EqOn.piecewise_ite' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 \u03b4 : \u03b1 \u2192 Sort u_6 s : Set \u03b1 f\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i inst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s) f f' g : \u03b1 \u2192 \u03b2 t t' : Set \u03b1 h : EqOn f g (t \u2229 s) h' : EqOn f' g (t' \u2229 s\u1d9c) \u22a2 EqOn (piecewise s f f') g (Set.ite s t t') ** simp [eqOn_piecewise, *] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.map_eq_sum ** \u03b1 : Type u_2 \u03b2 : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 s : Set \u03b1 inst\u271d\u00b9 : Countable \u03b2 inst\u271d : MeasurableSingletonClass \u03b2 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u03b2 hf : Measurable f \u22a2 map f \u03bc = sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b ** ext1 s hs ** case h \u03b1 : Type u_2 \u03b2 : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 s\u271d : Set \u03b1 inst\u271d\u00b9 : Countable \u03b2 inst\u271d : MeasurableSingletonClass \u03b2 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u03b2 hf : Measurable f s : Set \u03b2 hs : MeasurableSet s \u22a2 \u2191\u2191(map f \u03bc) s = \u2191\u2191(sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b) s ** have : \u2200 y \u2208 s, MeasurableSet (f \u207b\u00b9' {y}) := fun y _ => hf (measurableSet_singleton _) ** case h \u03b1 : Type u_2 \u03b2 : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 s\u271d : Set \u03b1 inst\u271d\u00b9 : Countable \u03b2 inst\u271d : MeasurableSingletonClass \u03b2 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u03b2 hf : Measurable f s : Set \u03b2 hs : MeasurableSet s this : \u2200 (y : \u03b2), y \u2208 s \u2192 MeasurableSet (f \u207b\u00b9' {y}) \u22a2 \u2191\u2191(map f \u03bc) s = \u2191\u2191(sum fun b => \u2191\u2191\u03bc (f \u207b\u00b9' {b}) \u2022 dirac b) s ** simp [\u2190 tsum_measure_preimage_singleton (to_countable s) this, *,\n  tsum_subtype s fun b => \u03bc (f \u207b\u00b9' {b}), \u2190 indicator_mul_right s fun b => \u03bc (f \u207b\u00b9' {b})] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.mul_nonneg_of_nonpos_of_nonpos ** a b : Int ha : a \u2264 0 hb : b \u2264 0 \u22a2 0 \u2264 a * b ** have : 0 * b \u2264 a * b := Int.mul_le_mul_of_nonpos_right ha hb ** a b : Int ha : a \u2264 0 hb : b \u2264 0 this : 0 * b \u2264 a * b \u22a2 0 \u2264 a * b ** rwa [Int.zero_mul] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.supported_empty ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 \u22a2 supported R \u2205 = \u22a5 ** simp [supported_eq_adjoin_X] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.withDensity_add_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u22a2 withDensity (\u03ba + \u03b7) f = withDensity \u03ba f + withDensity \u03b7 f ** by_cases hf : Measurable (Function.uncurry f) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (Function.uncurry f) \u22a2 withDensity (\u03ba + \u03b7) f = withDensity \u03ba f + withDensity \u03b7 f ** ext a s ** case pos.h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (Function.uncurry f) a : \u03b1 s : Set \u03b2 a\u271d : MeasurableSet s \u22a2 \u2191\u2191(\u2191(withDensity (\u03ba + \u03b7) f) a) s = \u2191\u2191(\u2191(withDensity \u03ba f + withDensity \u03b7 f) a) s ** simp only [kernel.withDensity_apply _ hf, coeFn_add, Pi.add_apply, withDensity_add_measure,\n  Measure.add_apply] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u00acMeasurable (Function.uncurry f) \u22a2 withDensity (\u03ba + \u03b7) f = withDensity \u03ba f + withDensity \u03b7 f ** simp_rw [withDensity_of_not_measurable _ hf] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u00acMeasurable (Function.uncurry f) \u22a2 0 = 0 + 0 ** rw [zero_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.withDensity_apply_eq_zero ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f \u22a2 \u2191\u2191(withDensity \u03bc f) s = 0 \u2194 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 ** constructor ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f \u22a2 \u2191\u2191(withDensity \u03bc f) s = 0 \u2192 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 ** intro hs ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 \u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 ** let t := toMeasurable (\u03bc.withDensity f) s ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s \u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 ** apply measure_mono_null (inter_subset_inter_right _ (subset_toMeasurable (\u03bc.withDensity f) s)) ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s \u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0 ** have A : \u03bc.withDensity f t = 0 := by rw [measure_toMeasurable, hs] ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s A : \u2191\u2191(withDensity \u03bc f) t = 0 \u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0 ** rw [withDensity_apply f (measurableSet_toMeasurable _ s), lintegral_eq_zero_iff hf,\n  EventuallyEq, ae_restrict_iff, ae_iff] at A ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s A : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = OfNat.ofNat 0 a)} = 0 \u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0  case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s A : \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc (toMeasurable (withDensity \u03bc f) s), f x = OfNat.ofNat 0 x \u22a2 MeasurableSet {x | f x = OfNat.ofNat 0 x} ** swap ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s A : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = OfNat.ofNat 0 a)} = 0 \u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0 ** simp only [Pi.zero_apply, mem_setOf_eq, Filter.mem_mk] at A ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s A : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0 \u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s) = 0 ** convert A using 2 ** case h.e'_2.h.e'_3 \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s A : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0 \u22a2 {x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s = {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} ** ext x ** case h.e'_2.h.e'_3.h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s A : \u2191\u2191\u03bc {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} = 0 x : \u03b1 \u22a2 x \u2208 {x | f x \u2260 0} \u2229 toMeasurable (withDensity \u03bc f) s \u2194 x \u2208 {a | \u00ac(a \u2208 toMeasurable (withDensity \u03bc f) s \u2192 f a = 0)} ** simp only [and_comm, exists_prop, mem_inter_iff, iff_self_iff, mem_setOf_eq, mem_compl_iff,\n  not_forall] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s \u22a2 \u2191\u2191(withDensity \u03bc f) t = 0 ** rw [measure_toMeasurable, hs] ** case mp \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191(withDensity \u03bc f) s = 0 t : Set \u03b1 := toMeasurable (withDensity \u03bc f) s A : \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc (toMeasurable (withDensity \u03bc f) s), f x = OfNat.ofNat 0 x \u22a2 MeasurableSet {x | f x = OfNat.ofNat 0 x} ** exact hf (measurableSet_singleton 0) ** case mpr \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f \u22a2 \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 \u2192 \u2191\u2191(withDensity \u03bc f) s = 0 ** intro hs ** case mpr \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 \u22a2 \u2191\u2191(withDensity \u03bc f) s = 0 ** let t := toMeasurable \u03bc ({ x | f x \u2260 0 } \u2229 s) ** case mpr \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) A : s \u2286 t \u222a {x | f x = 0} \u22a2 \u2191\u2191(withDensity \u03bc f) s = 0 ** apply measure_mono_null A (measure_union_null _ _) ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) \u22a2 s \u2286 t \u222a {x | f x = 0} ** intro x hx ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) x : \u03b1 hx : x \u2208 s \u22a2 x \u2208 t \u222a {x | f x = 0} ** rcases eq_or_ne (f x) 0 with (fx | fx) ** case inl \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) x : \u03b1 hx : x \u2208 s fx : f x = 0 \u22a2 x \u2208 t \u222a {x | f x = 0} ** simp only [fx, mem_union, mem_setOf_eq, eq_self_iff_true, or_true_iff] ** case inr \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) x : \u03b1 hx : x \u2208 s fx : f x \u2260 0 \u22a2 x \u2208 t \u222a {x | f x = 0} ** left ** case inr.h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) x : \u03b1 hx : x \u2208 s fx : f x \u2260 0 \u22a2 x \u2208 t ** apply subset_toMeasurable _ _ ** case inr.h.a \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) x : \u03b1 hx : x \u2208 s fx : f x \u2260 0 \u22a2 x \u2208 {x | f x \u2260 0} \u2229 s ** exact \u27e8fx, hx\u27e9 ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) A : s \u2286 t \u222a {x | f x = 0} \u22a2 \u2191\u2191(withDensity \u03bc f) t = 0 ** apply withDensity_absolutelyContinuous ** case a \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) A : s \u2286 t \u222a {x | f x = 0} \u22a2 \u2191\u2191\u03bc t = 0 ** rwa [measure_toMeasurable] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) A : s \u2286 t \u222a {x | f x = 0} \u22a2 \u2191\u2191(withDensity \u03bc f) {x | f x = 0} = 0 ** have M : MeasurableSet { x : \u03b1 | f x = 0 } := hf (measurableSet_singleton _) ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) A : s \u2286 t \u222a {x | f x = 0} M : MeasurableSet {x | f x = 0} \u22a2 \u2191\u2191(withDensity \u03bc f) {x | f x = 0} = 0 ** rw [withDensity_apply _ M, lintegral_eq_zero_iff hf] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) A : s \u2286 t \u222a {x | f x = 0} M : MeasurableSet {x | f x = 0} \u22a2 f =\u1da0[ae (restrict \u03bc {x | f x = 0})] 0 ** filter_upwards [ae_restrict_mem M] ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : Measurable f hs : \u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 s) = 0 t : Set \u03b1 := toMeasurable \u03bc ({x | f x \u2260 0} \u2229 s) A : s \u2286 t \u222a {x | f x = 0} M : MeasurableSet {x | f x = 0} \u22a2 \u2200 (a : \u03b1), f a = 0 \u2192 f a = OfNat.ofNat 0 a ** simp only [imp_self, Pi.zero_apply, imp_true_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Multiset.noncommFold_eq_fold ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 assoc : IsAssociative \u03b1 op s : Multiset \u03b1 inst\u271d : IsCommutative \u03b1 op a : \u03b1 \u22a2 noncommFold op s (_ : \u2200 (x : \u03b1), x \u2208 {x | x \u2208 s} \u2192 \u2200 (y : \u03b1), y \u2208 {x | x \u2208 s} \u2192 x \u2260 y \u2192 op x y = op y x) a =     fold op a s ** induction s using Quotient.inductionOn ** case h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 assoc : IsAssociative \u03b1 op inst\u271d : IsCommutative \u03b1 op a : \u03b1 a\u271d : List \u03b1 \u22a2 noncommFold op (Quotient.mk (List.isSetoid \u03b1) a\u271d)       (_ :         \u2200 (x : \u03b1),           x \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192             \u2200 (y : \u03b1), y \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192 x \u2260 y \u2192 op x y = op y x)       a =     fold op a (Quotient.mk (List.isSetoid \u03b1) a\u271d) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Ioi_disjUnion_Iio ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : Fintype \u03b1 inst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1 inst\u271d : LocallyFiniteOrderBot \u03b1 a : \u03b1 \u22a2 disjUnion (Ioi a) (Iio a) (_ : Disjoint (Ioi a) (Iio a)) = {a}\u1d9c ** ext ** case a \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : Fintype \u03b1 inst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1 inst\u271d : LocallyFiniteOrderBot \u03b1 a a\u271d : \u03b1 \u22a2 a\u271d \u2208 disjUnion (Ioi a) (Iio a) (_ : Disjoint (Ioi a) (Iio a)) \u2194 a\u271d \u2208 {a}\u1d9c ** simp [eq_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_neg ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C f : \u03b1 \u2192 E \u22a2 setToFun \u03bc T hT (-f) = -setToFun \u03bc T hT f ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C f : \u03b1 \u2192 E hf : Integrable f \u22a2 setToFun \u03bc T hT (-f) = -setToFun \u03bc T hT f ** rw [setToFun_eq hT hf, setToFun_eq hT hf.neg, Integrable.toL1_neg,\n  (L1.setToL1 hT).map_neg] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 setToFun \u03bc T hT (-f) = -setToFun \u03bc T hT f ** rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 \u00acIntegrable (-f) ** rwa [\u2190 integrable_neg_iff] at hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.eventually_mul_left_iff ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : MeasurableSpace H inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MeasurableMul G \u03bc : Measure G inst\u271d : IsMulLeftInvariant \u03bc t : G p : G \u2192 Prop \u22a2 (\u2200\u1d50 (x : G) \u2202\u03bc, p (t * x)) \u2194 \u2200\u1d50 (x : G) \u2202\u03bc, p x ** conv_rhs => rw [Filter.Eventually, \u2190 map_mul_left_ae \u03bc t]; rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.countable_iff_exists_subset_range ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : Nonempty \u03b1 s : Set \u03b1 h : Set.Countable s \u22a2 \u2203 f, s \u2286 range f ** inhabit \u03b1 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : Nonempty \u03b1 s : Set \u03b1 h : Set.Countable s inhabited_h : Inhabited \u03b1 \u22a2 \u2203 f, s \u2286 range f ** exact \u27e8enumerateCountable h default, subset_range_enumerate _ _\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.insertNth_comm ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d a b : \u03b1 i j : Fin (n + 1) h : i \u2264 j l : List \u03b1 hl : List.length l = n \u22a2 insertNth b (Fin.succ j) (insertNth a i { val := l, property := hl }) =     insertNth a (Fin.castSucc i) (insertNth b j { val := l, property := hl }) ** refine' Subtype.eq _ ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d a b : \u03b1 i j : Fin (n + 1) h : i \u2264 j l : List \u03b1 hl : List.length l = n \u22a2 \u2191(insertNth b (Fin.succ j) (insertNth a i { val := l, property := hl })) =     \u2191(insertNth a (Fin.castSucc i) (insertNth b j { val := l, property := hl })) ** simp only [insertNth_val, Fin.val_succ, Fin.castSucc, Fin.coe_castAdd] ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d a b : \u03b1 i j : Fin (n + 1) h : i \u2264 j l : List \u03b1 hl : List.length l = n \u22a2 List.insertNth (\u2191j + 1) b (List.insertNth (\u2191i) a l) = List.insertNth (\u2191i) a (List.insertNth (\u2191j) b l) ** apply List.insertNth_comm ** case x n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d a b : \u03b1 i j : Fin (n + 1) h : i \u2264 j l : List \u03b1 hl : List.length l = n \u22a2 \u2191i \u2264 \u2191j ** assumption ** case x n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d a b : \u03b1 i j : Fin (n + 1) h : i \u2264 j l : List \u03b1 hl : List.length l = n \u22a2 \u2191j \u2264 List.length l ** rw [hl] ** case x n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d a b : \u03b1 i j : Fin (n + 1) h : i \u2264 j l : List \u03b1 hl : List.length l = n \u22a2 \u2191j \u2264 n ** exact Nat.le_of_succ_le_succ j.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Pos.zero_addString_eq ** s : String \u22a2 0 + s = { byteIdx := utf8ByteSize s } ** rw [\u2190 zero_addString_byteIdx] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_eq_of_bijective ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 f : (i : \u2115) \u2192 i < n \u2192 \u03b1 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a hf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s f_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard s = n ** rw [ncard_eq_toFinset_card _ hs] ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 f : (i : \u2115) \u2192 i < n \u2192 \u03b1 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a hf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s f_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j hs : autoParam (Set.Finite s) _auto\u271d \u22a2 Finset.card (Finite.toFinset hs) = n ** apply Finset.card_eq_of_bijective ** case hf \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 f : (i : \u2115) \u2192 i < n \u2192 \u03b1 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a hf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s f_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j hs : autoParam (Set.Finite s) _auto\u271d \u22a2 \u2200 (a : \u03b1), a \u2208 Finite.toFinset hs \u2192 \u2203 i h, ?f i h = a  case hf' \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 f : (i : \u2115) \u2192 i < n \u2192 \u03b1 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a hf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s f_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j hs : autoParam (Set.Finite s) _auto\u271d \u22a2 \u2200 (i : \u2115) (h : i < n), ?f i h \u2208 Finite.toFinset hs  case f_inj \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 f : (i : \u2115) \u2192 i < n \u2192 \u03b1 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a hf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s f_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j hs : autoParam (Set.Finite s) _auto\u271d \u22a2 \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), ?f i hi = ?f j hj \u2192 i = j  case f \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 f : (i : \u2115) \u2192 i < n \u2192 \u03b1 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a hf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s f_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j hs : autoParam (Set.Finite s) _auto\u271d \u22a2 (i : \u2115) \u2192 i < n \u2192 \u03b1 ** all_goals simpa ** case f_inj \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 f : (i : \u2115) \u2192 i < n \u2192 \u03b1 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 i h, f i h = a hf' : \u2200 (i : \u2115) (h : i < n), f i h \u2208 s f_inj : \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j hs : autoParam (Set.Finite s) _auto\u271d \u22a2 \u2200 (i j : \u2115) (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j ** simpa ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.mapEquiv_trans ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e\u271d : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : CommSemiring S\u2081 inst\u271d\u00b9 : CommSemiring S\u2082 inst\u271d : CommSemiring S\u2083 e : S\u2081 \u2243+* S\u2082 f : S\u2082 \u2243+* S\u2083 p : MvPolynomial \u03c3 S\u2081 \u22a2 \u2191(RingEquiv.trans (mapEquiv \u03c3 e) (mapEquiv \u03c3 f)) p = \u2191(mapEquiv \u03c3 (RingEquiv.trans e f)) p ** simp only [RingEquiv.coe_trans, comp_apply, mapEquiv_apply, RingEquiv.coe_ringHom_trans,\n  map_map] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.pimage_union ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : DecidableEq \u03b2 f g : \u03b1 \u2192. \u03b2 inst\u271d\u00b2 : (x : \u03b1) \u2192 Decidable (f x).Dom inst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (g x).Dom s t : Finset \u03b1 b : \u03b2 inst\u271d : DecidableEq \u03b1 \u22a2 \u2191(pimage f (s \u222a t)) = \u2191(pimage f s \u222a pimage f t) ** simp only [coe_pimage, coe_union, \u2190 PFun.image_union] ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.mod_get_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Mod \u03b1 a b : Part \u03b1 hab : (a % b).Dom \u22a2 get (a % b) hab = get a (_ : a.Dom) % get b (_ : b.Dom) ** simp [mod_def] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Mod \u03b1 a b : Part \u03b1 hab : (a % b).Dom \u22a2 get (Part.bind a fun y => map (fun x => y % x) b) (_ : (Part.bind a fun y => map (fun x => y % x) b).Dom) =     get a (_ : a.Dom) % get b (_ : b.Dom) ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates_aux ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E sb : Bornology.IsBounded s hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s \u22a2 \u2191\u2191\u03bc s = 0 ** by_contra h ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E sb : Bornology.IsBounded s hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s h : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 False ** apply lt_irrefl \u221e ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E sb : Bornology.IsBounded s hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s h : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 \u22a4 < \u22a4 ** calc\n  \u221e = \u2211' _ : \u2115, \u03bc s := (ENNReal.tsum_const_eq_top_of_ne_zero h).symm\n  _ = \u2211' n : \u2115, \u03bc ({u n} + s) := by\n    congr 1; ext1 n; simp only [image_add_left, measure_preimage_add, singleton_add]\n  _ = \u03bc (\u22c3 n, {u n} + s) := Eq.symm <| measure_iUnion hs fun n => by\n    simpa only [image_add_left, singleton_add] using measurable_id.const_add _ h's\n  _ = \u03bc (range u + s) := by rw [\u2190 iUnion_add, iUnion_singleton_eq_range]\n  _ < \u221e := (hu.add sb).measure_lt_top ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E sb : Bornology.IsBounded s hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s h : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 \u2211' (x : \u2115), \u2191\u2191\u03bc s = \u2211' (n : \u2115), \u2191\u2191\u03bc ({u n} + s) ** congr 1 ** case e_f E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E sb : Bornology.IsBounded s hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s h : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 (fun x => \u2191\u2191\u03bc s) = fun n => \u2191\u2191\u03bc ({u n} + s) ** ext1 n ** case e_f.h E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E sb : Bornology.IsBounded s hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s h : \u00ac\u2191\u2191\u03bc s = 0 n : \u2115 \u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc ({u n} + s) ** simp only [image_add_left, measure_preimage_add, singleton_add] ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E sb : Bornology.IsBounded s hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s h : \u00ac\u2191\u2191\u03bc s = 0 n : \u2115 \u22a2 MeasurableSet ({u n} + s) ** simpa only [image_add_left, singleton_add] using measurable_id.const_add _ h's ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E sb : Bornology.IsBounded s hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s h : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191\u03bc (\u22c3 n, {u n} + s) = \u2191\u2191\u03bc (range u + s) ** rw [\u2190 iUnion_add, iUnion_singleton_eq_range] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.get_map\u2082 ** \u03b1 : Type u_2 \u03b2 : Type u_3 n : \u2115 xs : Vector \u03b1 n ys : Vector \u03b2 n \u03b3 : Type u_1 v\u2081 : Vector \u03b1 n v\u2082 : Vector \u03b2 n f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 i : Fin n \u22a2 get (map\u2082 f v\u2081 v\u2082) i = f (get v\u2081 i) (get v\u2082 i) ** clear * - v\u2081 v\u2082 ** \u03b1 : Type u_2 \u03b2 : Type u_3 n : \u2115 \u03b3 : Type u_1 v\u2081 : Vector \u03b1 n v\u2082 : Vector \u03b2 n f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 i : Fin n \u22a2 get (map\u2082 f v\u2081 v\u2082) i = f (get v\u2081 i) (get v\u2082 i) ** induction v\u2081, v\u2082 using inductionOn\u2082 ** case nil \u03b1 : Type u_2 \u03b2 : Type u_3 n : \u2115 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 i : Fin 0 \u22a2 get (map\u2082 f nil nil) i = f (get nil i) (get nil i)  case cons \u03b1 : Type u_2 \u03b2 : Type u_3 n : \u2115 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 n\u271d : \u2115 a\u271d\u00b9 : \u03b1 b\u271d : \u03b2 x\u271d : Vector \u03b1 n\u271d y\u271d : Vector \u03b2 n\u271d a\u271d : \u2200 (i : Fin n\u271d), get (map\u2082 f x\u271d y\u271d) i = f (get x\u271d i) (get y\u271d i) i : Fin (Nat.succ n\u271d) \u22a2 get (map\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d)) i = f (get (a\u271d\u00b9 ::\u1d65 x\u271d) i) (get (b\u271d ::\u1d65 y\u271d) i) ** case nil =>\n  exact Fin.elim0 i ** \u03b1 : Type u_2 \u03b2 : Type u_3 n : \u2115 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 i : Fin 0 \u22a2 get (map\u2082 f nil nil) i = f (get nil i) (get nil i) ** exact Fin.elim0 i ** \u03b1 : Type u_2 \u03b2 : Type u_3 n : \u2115 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 n\u271d : \u2115 x : \u03b1 xs : \u03b2 y : Vector \u03b1 n\u271d ys : Vector \u03b2 n\u271d ih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i) i : Fin (Nat.succ n\u271d) \u22a2 get (map\u2082 f (x ::\u1d65 y) (xs ::\u1d65 ys)) i = f (get (x ::\u1d65 y) i) (get (xs ::\u1d65 ys) i) ** rw [map\u2082_cons] ** \u03b1 : Type u_2 \u03b2 : Type u_3 n : \u2115 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 n\u271d : \u2115 x : \u03b1 xs : \u03b2 y : Vector \u03b1 n\u271d ys : Vector \u03b2 n\u271d ih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i) i : Fin (Nat.succ n\u271d) \u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) i = f (get (x ::\u1d65 y) i) (get (xs ::\u1d65 ys) i) ** cases i using Fin.cases ** case zero \u03b1 : Type u_2 \u03b2 : Type u_3 n : \u2115 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 n\u271d : \u2115 x : \u03b1 xs : \u03b2 y : Vector \u03b1 n\u271d ys : Vector \u03b2 n\u271d ih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i) \u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) 0 = f (get (x ::\u1d65 y) 0) (get (xs ::\u1d65 ys) 0) ** simp only [get_zero, head_cons] ** case succ \u03b1 : Type u_2 \u03b2 : Type u_3 n : \u2115 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 n\u271d : \u2115 x : \u03b1 xs : \u03b2 y : Vector \u03b1 n\u271d ys : Vector \u03b2 n\u271d ih : \u2200 (i : Fin n\u271d), get (map\u2082 f y ys) i = f (get y i) (get ys i) i\u271d : Fin n\u271d \u22a2 get (f x xs ::\u1d65 map\u2082 f y ys) (Fin.succ i\u271d) = f (get (x ::\u1d65 y) (Fin.succ i\u271d)) (get (xs ::\u1d65 ys) (Fin.succ i\u271d)) ** simp only [get_cons_succ, ih] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.ToPartrec.Code.exists_code ** n : \u2115 f : Vector \u2115 n \u2192. \u2115 hf : Nat.Partrec' f \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> f v ** induction' hf with n f hf ** case prim n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 hf : Nat.Primrec' f \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> \u2191f v  case comp n : \u2115 f : Vector \u2115 n \u2192. \u2115 m\u271d n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115 a\u271d\u00b9 : Nat.Partrec' f\u271d a\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i) a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v a_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v  case rfind n : \u2115 f : Vector \u2115 n \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f\u271d a_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v ** induction hf ** case prim.zero n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 \u22a2 \u2203 c, \u2200 (v : Vector \u2115 0), eval c \u2191v = pure <$> (\u2191fun x => 0) v  case prim.succ n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 \u22a2 \u2203 c, \u2200 (v : Vector \u2115 1), eval c \u2191v = pure <$> (\u2191fun v => Nat.succ (Vector.head v)) v  case prim.get n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 n\u271d : \u2115 i\u271d : Fin n\u271d \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i\u271d) v  case prim.comp n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 m\u271d n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f\u271d a\u271d : \u2200 (i : Fin n\u271d), Nat.Primrec' (g\u271d i) a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v a_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> \u2191(g\u271d i) v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (\u2191fun a => f\u271d (Vector.ofFn fun i => g\u271d i a)) v  case prim.prec n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 g\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f\u271d a\u271d : Nat.Primrec' g\u271d a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v a_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v \u22a2 \u2203 c,     \u2200 (v : Vector \u2115 (n\u271d + 1)),       eval c \u2191v =         pure <$>           (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v  case comp n : \u2115 f : Vector \u2115 n \u2192. \u2115 m\u271d n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115 a\u271d\u00b9 : Nat.Partrec' f\u271d a\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i) a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v a_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v  case rfind n : \u2115 f : Vector \u2115 n \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f\u271d a_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v ** case prim.zero => exact \u27e8zero', fun \u27e8[], _\u27e9 => rfl\u27e9 ** case prim.succ n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 \u22a2 \u2203 c, \u2200 (v : Vector \u2115 1), eval c \u2191v = pure <$> (\u2191fun v => Nat.succ (Vector.head v)) v  case prim.get n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 n\u271d : \u2115 i\u271d : Fin n\u271d \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i\u271d) v  case prim.comp n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 m\u271d n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f\u271d a\u271d : \u2200 (i : Fin n\u271d), Nat.Primrec' (g\u271d i) a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v a_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> \u2191(g\u271d i) v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (\u2191fun a => f\u271d (Vector.ofFn fun i => g\u271d i a)) v  case prim.prec n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 g\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f\u271d a\u271d : Nat.Primrec' g\u271d a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v a_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v \u22a2 \u2203 c,     \u2200 (v : Vector \u2115 (n\u271d + 1)),       eval c \u2191v =         pure <$>           (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v  case comp n : \u2115 f : Vector \u2115 n \u2192. \u2115 m\u271d n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115 a\u271d\u00b9 : Nat.Partrec' f\u271d a\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i) a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v a_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v  case rfind n : \u2115 f : Vector \u2115 n \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f\u271d a_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v ** case prim.succ => exact \u27e8succ, fun \u27e8[v], _\u27e9 => rfl\u27e9 ** case prim.comp n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 m\u271d n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f\u271d a\u271d : \u2200 (i : Fin n\u271d), Nat.Primrec' (g\u271d i) a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v a_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> \u2191(g\u271d i) v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (\u2191fun a => f\u271d (Vector.ofFn fun i => g\u271d i a)) v  case prim.prec n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 g\u271d : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f\u271d a\u271d : Nat.Primrec' g\u271d a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> \u2191f\u271d v a_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 2)), eval c \u2191v = pure <$> \u2191g\u271d v \u22a2 \u2203 c,     \u2200 (v : Vector \u2115 (n\u271d + 1)),       eval c \u2191v =         pure <$>           (\u2191fun v => Nat.rec (f\u271d (Vector.tail v)) (fun y IH => g\u271d (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v  case comp n : \u2115 f : Vector \u2115 n \u2192. \u2115 m\u271d n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115 a\u271d\u00b9 : Nat.Partrec' f\u271d a\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i) a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v a_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v  case rfind n : \u2115 f : Vector \u2115 n \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f\u271d a_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v ** case prim.comp m n f g hf hg IHf IHg =>\n  simpa [Part.bind_eq_bind] using exists_code.comp IHf IHg ** case comp n : \u2115 f : Vector \u2115 n \u2192. \u2115 m\u271d n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m\u271d \u2192. \u2115 a\u271d\u00b9 : Nat.Partrec' f\u271d a\u271d : \u2200 (i : Fin n\u271d), Nat.Partrec' (g\u271d i) a_ih\u271d\u00b9 : \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> f\u271d v a_ih\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> g\u271d i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m\u271d), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g\u271d i v) >>= f\u271d) v  case rfind n : \u2115 f : Vector \u2115 n \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f\u271d a_ih\u271d : \u2203 c, \u2200 (v : Vector \u2115 (n\u271d + 1)), eval c \u2191v = pure <$> \u2191f\u271d v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n\u271d), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n => Part.some (decide (f\u271d (n ::\u1d65 v) = 0))) v ** case comp m n f g _ _ IHf IHg => exact exists_code.comp IHf IHg ** n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 \u22a2 \u2203 c, \u2200 (v : Vector \u2115 0), eval c \u2191v = pure <$> (\u2191fun x => 0) v ** exact \u27e8zero', fun \u27e8[], _\u27e9 => rfl\u27e9 ** n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 \u22a2 \u2203 c, \u2200 (v : Vector \u2115 1), eval c \u2191v = pure <$> (\u2191fun v => Nat.succ (Vector.head v)) v ** exact \u27e8succ, fun \u27e8[v], _\u27e9 => rfl\u27e9 ** n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 i : Fin n \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v ** refine' Fin.succRec (fun n => _) (fun n i IH => _) i ** case refine'_1 n\u271d\u00b2 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 i : Fin n\u271d n : \u2115 \u22a2 \u2203 c, \u2200 (v : Vector \u2115 (Nat.succ n)), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v 0) v ** exact \u27e8head, fun \u27e8List.cons a as, _\u27e9 => by simp [Bind.bind]; rfl\u27e9 ** n\u271d\u00b2 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 i : Fin n\u271d n : \u2115 x\u271d : Vector \u2115 (Nat.succ n) a : \u2115 as : List \u2115 property\u271d : List.length (a :: as) = Nat.succ n \u22a2 eval head \u2191{ val := a :: as, property := property\u271d } =     pure <$> (\u2191fun v => Vector.get v 0) { val := a :: as, property := property\u271d } ** simp [Bind.bind] ** n\u271d\u00b2 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 i : Fin n\u271d n : \u2115 x\u271d : Vector \u2115 (Nat.succ n) a : \u2115 as : List \u2115 property\u271d : List.length (a :: as) = Nat.succ n \u22a2 Part.some [a] = pure <$> Part.some (Vector.head { val := a :: as, property := property\u271d }) ** rfl ** case refine'_2 n\u271d\u00b2 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 i\u271d : Fin n\u271d n : \u2115 i : Fin n IH : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 (Nat.succ n)), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v (Fin.succ i)) v ** obtain \u27e8c, h\u27e9 := IH ** case refine'_2.intro n\u271d\u00b2 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 i\u271d : Fin n\u271d n : \u2115 i : Fin n c : Code h : \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 (Nat.succ n)), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v (Fin.succ i)) v ** exact \u27e8c.comp tail, fun v => by simpa [\u2190 Vector.get_tail, Bind.bind] using h v.tail\u27e9 ** n\u271d\u00b2 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 i\u271d : Fin n\u271d n : \u2115 i : Fin n c : Code h : \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (\u2191fun v => Vector.get v i) v v : Vector \u2115 (Nat.succ n) \u22a2 eval (comp c tail) \u2191v = pure <$> (\u2191fun v => Vector.get v (Fin.succ i)) v ** simpa [\u2190 Vector.get_tail, Bind.bind] using h v.tail ** n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 m n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Fin n \u2192 Vector \u2115 m \u2192 \u2115 hf : Nat.Primrec' f hg : \u2200 (i : Fin n), Nat.Primrec' (g i) IHf : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> \u2191f v IHg : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> \u2191(g i) v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> (\u2191fun a => f (Vector.ofFn fun i => g i a)) v ** simpa [Part.bind_eq_bind] using exists_code.comp IHf IHg ** n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g IHf : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> \u2191f v IHg : \u2203 c, \u2200 (v : Vector \u2115 (n + 2)), eval c \u2191v = pure <$> \u2191g v \u22a2 \u2203 c,     \u2200 (v : Vector \u2115 (n + 1)),       eval c \u2191v =         pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v ** obtain \u27e8cf, hf\u27e9 := IHf ** case intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g IHg : \u2203 c, \u2200 (v : Vector \u2115 (n + 2)), eval c \u2191v = pure <$> \u2191g v cf : Code hf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = pure <$> \u2191f v \u22a2 \u2203 c,     \u2200 (v : Vector \u2115 (n + 1)),       eval c \u2191v =         pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v ** obtain \u27e8cg, hg\u27e9 := IHg ** case intro.intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf : Code hf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = pure <$> \u2191f v cg : Code hg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = pure <$> \u2191g v \u22a2 \u2203 c,     \u2200 (v : Vector \u2115 (n + 1)),       eval c \u2191v =         pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v ** simp only [Part.map_eq_map, Part.map_some, PFun.coe_val] at hf hg ** case intro.intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code hf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = Part.some (pure (f v)) hg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v)) \u22a2 \u2203 c,     \u2200 (v : Vector \u2115 (n + 1)),       eval c \u2191v =         pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v ** refine' \u27e8prec cf cg, fun v => _\u27e9 ** case intro.intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code hf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = Part.some (pure (f v)) hg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v)) v : Vector \u2115 (n + 1) \u22a2 eval (prec cf cg) \u2191v =     pure <$> (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v)) v ** rw [\u2190 v.cons_head_tail] ** case intro.intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code hf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = Part.some (pure (f v)) hg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v)) v : Vector \u2115 (n + 1) \u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =     pure <$>       (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))         (Vector.head v ::\u1d65 Vector.tail v) ** specialize hf v.tail ** case intro.intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code hg : \u2200 (v : Vector \u2115 (n + 2)), eval cg \u2191v = Part.some (pure (g v)) v : Vector \u2115 (n + 1) hf : eval cf \u2191(Vector.tail v) = Part.some (pure (f (Vector.tail v))) \u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =     pure <$>       (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))         (Vector.head v ::\u1d65 Vector.tail v) ** replace hg := fun a b => hg (a ::\u1d65 b ::\u1d65 v.tail) ** case intro.intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n + 1) hf : eval cf \u2191(Vector.tail v) = Part.some (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg \u2191(a ::\u1d65 b ::\u1d65 Vector.tail v) = Part.some (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) \u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =     pure <$>       (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))         (Vector.head v ::\u1d65 Vector.tail v) ** simp only [Vector.cons_val, Vector.tail_val] at hf hg ** case intro.intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n + 1) hf : eval cf (List.tail \u2191v) = Part.some (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = Part.some (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) \u22a2 eval (prec cf cg) \u2191(Vector.head v ::\u1d65 Vector.tail v) =     pure <$>       (\u2191fun v => Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))         (Vector.head v ::\u1d65 Vector.tail v) ** simp only [Part.map_eq_map, Part.map_some, Vector.cons_val, Vector.tail_cons, Vector.head_cons,\n  PFun.coe_val, Vector.tail_val] ** case intro.intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n + 1) hf : eval cf (List.tail \u2191v) = Part.some (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = Part.some (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) \u22a2 eval (prec cf cg) (Vector.head v :: List.tail \u2191v) =     Part.some (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))) ** simp only [\u2190 Part.pure_eq_some] at hf hg \u22a2 ** case intro.intro n\u271d\u00b9 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192 \u2115 n : \u2115 f : Vector \u2115 n \u2192 \u2115 g : Vector \u2115 (n + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) \u22a2 eval (prec cf cg) (Vector.head v :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (Vector.head v))) ** induction' v.head with n _ <;>\n  simp [prec, hf, Part.bind_assoc, \u2190 Part.bind_some_eq_map, Part.bind_some,\n    show \u2200 x, pure x = [x] from fun _ => rfl, Bind.bind, Functor.map] ** case intro.intro.succ n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) \u22a2 (Part.bind       (PFun.fix         (fun v =>           Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>             Part.some               (if List.headI (List.tail v) = 0 then                 Sum.inl                   (Nat.succ (List.headI v) ::                     Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))               else                 Sum.inr                   (Nat.succ (List.headI v) ::                     Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))         (0 :: n :: f (Vector.tail v) :: List.tail \u2191v))       fun x => Part.some [List.headI (List.tail (List.tail x))]) =     Part.some [g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v)] ** suffices \u2200 a b, a + b = n \u2192\n  (n.succ :: 0 ::\n    g (n ::\u1d65 Nat.rec (f v.tail) (fun y IH => g (y ::\u1d65 IH ::\u1d65 v.tail)) n ::\u1d65 v.tail) ::\n        v.val.tail : List \u2115) \u2208\n    PFun.fix\n      (fun v : List \u2115 => Part.bind (cg.eval (v.headI :: v.tail.tail))\n        (fun x => Part.some (if v.tail.headI = 0\n          then Sum.inl\n            (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail : List \u2115)\n          else Sum.inr\n            (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail))))\n      (a :: b :: Nat.rec (f v.tail) (fun y IH => g (y ::\u1d65 IH ::\u1d65 v.tail)) a :: v.val.tail) by\n  erw [Part.eq_some_iff.2 (this 0 n (zero_add n))]\n  simp only [List.headI, Part.bind_some, List.tail_cons] ** case intro.intro.succ n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) \u22a2 \u2200 (a b : \u2115),     a + b = n \u2192       Nat.succ n ::           0 ::             g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::               List.tail \u2191v \u2208         PFun.fix           (fun v =>             Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>               Part.some                 (if List.headI (List.tail v) = 0 then                   Sum.inl                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))                 else                   Sum.inr                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))           (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ** intro a b e ** case intro.intro.succ n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) a b : \u2115 e : a + b = n \u22a2 Nat.succ n ::       0 ::         g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::           List.tail \u2191v \u2208     PFun.fix       (fun v =>         Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>           Part.some             (if List.headI (List.tail v) = 0 then               Sum.inl                 (Nat.succ (List.headI v) ::                   Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))             else               Sum.inr                 (Nat.succ (List.headI v) ::                   Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))       (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ** induction' b with b IH generalizing a ** n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) this :   \u2200 (a b : \u2115),     a + b = n \u2192       Nat.succ n ::           0 ::             g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::               List.tail \u2191v \u2208         PFun.fix           (fun v =>             Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>               Part.some                 (if List.headI (List.tail v) = 0 then                   Sum.inl                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))                 else                   Sum.inr                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))           (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) \u22a2 (Part.bind       (PFun.fix         (fun v =>           Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>             Part.some               (if List.headI (List.tail v) = 0 then                 Sum.inl                   (Nat.succ (List.headI v) ::                     Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))               else                 Sum.inr                   (Nat.succ (List.headI v) ::                     Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))         (0 :: n :: f (Vector.tail v) :: List.tail \u2191v))       fun x => Part.some [List.headI (List.tail (List.tail x))]) =     Part.some [g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v)] ** erw [Part.eq_some_iff.2 (this 0 n (zero_add n))] ** n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b9 : Nat.Primrec' f a\u271d : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) this :   \u2200 (a b : \u2115),     a + b = n \u2192       Nat.succ n ::           0 ::             g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::               List.tail \u2191v \u2208         PFun.fix           (fun v =>             Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>               Part.some                 (if List.headI (List.tail v) = 0 then                   Sum.inl                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))                 else                   Sum.inr                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))           (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) \u22a2 (Part.bind       (Part.some         (Nat.succ n ::           0 ::             g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::               List.tail \u2191v))       fun x => Part.some [List.headI (List.tail (List.tail x))]) =     Part.some [g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v)] ** simp only [List.headI, Part.bind_some, List.tail_cons] ** case intro.intro.succ.zero n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b2 : Nat.Primrec' f a\u271d\u00b9 : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) a\u271d b : \u2115 e\u271d : a\u271d + b = n a : \u2115 e : a + Nat.zero = n \u22a2 Nat.succ n ::       0 ::         g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::           List.tail \u2191v \u2208     PFun.fix       (fun v =>         Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>           Part.some             (if List.headI (List.tail v) = 0 then               Sum.inl                 (Nat.succ (List.headI v) ::                   Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))             else               Sum.inr                 (Nat.succ (List.headI v) ::                   Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))       (a :: Nat.zero :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ** refine' PFun.mem_fix_iff.2 (Or.inl <| Part.eq_some_iff.1 _) ** case intro.intro.succ.zero n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b2 : Nat.Primrec' f a\u271d\u00b9 : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) a\u271d b : \u2115 e\u271d : a\u271d + b = n a : \u2115 e : a + Nat.zero = n \u22a2 (Part.bind       (eval cg         (List.headI             (a ::               Nat.zero :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ::           List.tail             (List.tail               (a ::                 Nat.zero ::                   Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v))))       fun x =>       Part.some         (if             List.headI                 (List.tail                   (a ::                     Nat.zero ::                       Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) =               0 then           Sum.inl             (Nat.succ                 (List.headI                   (a ::                     Nat.zero ::                       Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::               Nat.pred                   (List.headI                     (List.tail                       (a ::                         Nat.zero ::                           Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                             List.tail \u2191v))) ::                 List.headI x ::                   List.tail                     (List.tail                       (List.tail                         (a ::                           Nat.zero ::                             Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                               List.tail \u2191v))))         else           Sum.inr             (Nat.succ                 (List.headI                   (a ::                     Nat.zero ::                       Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::               Nat.pred                   (List.headI                     (List.tail                       (a ::                         Nat.zero ::                           Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                             List.tail \u2191v))) ::                 List.headI x ::                   List.tail                     (List.tail                       (List.tail                         (a ::                           Nat.zero ::                             Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                               List.tail \u2191v)))))) =     Part.some       (Sum.inl         (Nat.succ n ::           0 ::             g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::               List.tail \u2191v)) ** simp only [hg, \u2190 e, Part.bind_some, List.tail_cons, pure] ** case intro.intro.succ.zero n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b2 : Nat.Primrec' f a\u271d\u00b9 : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) a\u271d b : \u2115 e\u271d : a\u271d + b = n a : \u2115 e : a + Nat.zero = n \u22a2 Part.some       (if           List.headI               (Nat.zero :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) =             0 then         Sum.inl           (Nat.succ               (List.headI                 (a ::                   Nat.zero ::                     Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::             Nat.pred                 (List.headI                   (Nat.zero ::                     Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::               List.headI                   (List.ret                     (g                       (List.headI                           (a ::                             Nat.zero ::                               Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                                 List.tail \u2191v) ::\u1d65                         Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65                           Vector.tail v))) ::                 List.tail \u2191v)       else         Sum.inr           (Nat.succ               (List.headI                 (a ::                   Nat.zero ::                     Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::             Nat.pred                 (List.headI                   (Nat.zero ::                     Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::               List.headI                   (List.ret                     (g                       (List.headI                           (a ::                             Nat.zero ::                               Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                                 List.tail \u2191v) ::\u1d65                         Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65                           Vector.tail v))) ::                 List.tail \u2191v)) =     Part.some       (Sum.inl         (Nat.succ (a + Nat.zero) ::           0 ::             g                 ((a + Nat.zero) ::\u1d65                   Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (a + Nat.zero) ::\u1d65                     Vector.tail v) ::               List.tail \u2191v)) ** rfl ** case intro.intro.succ.succ n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b2 : Nat.Primrec' f a\u271d\u00b9 : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) a\u271d b\u271d : \u2115 e\u271d : a\u271d + b\u271d = n b : \u2115 IH :   \u2200 (a : \u2115),     a + b = n \u2192       Nat.succ n ::           0 ::             g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::               List.tail \u2191v \u2208         PFun.fix           (fun v =>             Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>               Part.some                 (if List.headI (List.tail v) = 0 then                   Sum.inl                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))                 else                   Sum.inr                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))           (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) a : \u2115 e : a + Nat.succ b = n \u22a2 Nat.succ n ::       0 ::         g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::           List.tail \u2191v \u2208     PFun.fix       (fun v =>         Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>           Part.some             (if List.headI (List.tail v) = 0 then               Sum.inl                 (Nat.succ (List.headI v) ::                   Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))             else               Sum.inr                 (Nat.succ (List.headI v) ::                   Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))       (a :: Nat.succ b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ** refine' PFun.mem_fix_iff.2 (Or.inr \u27e8_, _, IH (a + 1) (by rwa [add_right_comm])\u27e9) ** case intro.intro.succ.succ n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b2 : Nat.Primrec' f a\u271d\u00b9 : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) a\u271d b\u271d : \u2115 e\u271d : a\u271d + b\u271d = n b : \u2115 IH :   \u2200 (a : \u2115),     a + b = n \u2192       Nat.succ n ::           0 ::             g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::               List.tail \u2191v \u2208         PFun.fix           (fun v =>             Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>               Part.some                 (if List.headI (List.tail v) = 0 then                   Sum.inl                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))                 else                   Sum.inr                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))           (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) a : \u2115 e : a + Nat.succ b = n \u22a2 Sum.inr       ((a + 1) ::         b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) (a + 1) :: List.tail \u2191v) \u2208     Part.bind       (eval cg         (List.headI             (a ::               Nat.succ b ::                 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) ::           List.tail             (List.tail               (a ::                 Nat.succ b ::                   Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v))))       fun x =>       Part.some         (if             List.headI                 (List.tail                   (a ::                     Nat.succ b ::                       Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) =               0 then           Sum.inl             (Nat.succ                 (List.headI                   (a ::                     Nat.succ b ::                       Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::               Nat.pred                   (List.headI                     (List.tail                       (a ::                         Nat.succ b ::                           Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                             List.tail \u2191v))) ::                 List.headI x ::                   List.tail                     (List.tail                       (List.tail                         (a ::                           Nat.succ b ::                             Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                               List.tail \u2191v))))         else           Sum.inr             (Nat.succ                 (List.headI                   (a ::                     Nat.succ b ::                       Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::               Nat.pred                   (List.headI                     (List.tail                       (a ::                         Nat.succ b ::                           Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                             List.tail \u2191v))) ::                 List.headI x ::                   List.tail                     (List.tail                       (List.tail                         (a ::                           Nat.succ b ::                             Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                               List.tail \u2191v))))) ** simp only [hg, eval, Part.bind_some, Nat.rec_add_one, List.tail_nil, List.tail_cons, pure] ** case intro.intro.succ.succ n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b2 : Nat.Primrec' f a\u271d\u00b9 : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) a\u271d b\u271d : \u2115 e\u271d : a\u271d + b\u271d = n b : \u2115 IH :   \u2200 (a : \u2115),     a + b = n \u2192       Nat.succ n ::           0 ::             g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::               List.tail \u2191v \u2208         PFun.fix           (fun v =>             Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>               Part.some                 (if List.headI (List.tail v) = 0 then                   Sum.inl                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))                 else                   Sum.inr                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))           (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) a : \u2115 e : a + Nat.succ b = n \u22a2 Sum.inr       ((a + 1) ::         b ::           g (a ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65 Vector.tail v) ::             List.tail \u2191v) \u2208     Part.some       (if           List.headI               (Nat.succ b ::                 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) =             0 then         Sum.inl           (Nat.succ               (List.headI                 (a ::                   Nat.succ b ::                     Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::             Nat.pred                 (List.headI                   (Nat.succ b ::                     Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::               List.headI                   (List.ret                     (g                       (List.headI                           (a ::                             Nat.succ b ::                               Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                                 List.tail \u2191v) ::\u1d65                         Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65                           Vector.tail v))) ::                 List.tail \u2191v)       else         Sum.inr           (Nat.succ               (List.headI                 (a ::                   Nat.succ b ::                     Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::             Nat.pred                 (List.headI                   (Nat.succ b ::                     Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v)) ::               List.headI                   (List.ret                     (g                       (List.headI                           (a ::                             Nat.succ b ::                               Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::                                 List.tail \u2191v) ::\u1d65                         Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a ::\u1d65                           Vector.tail v))) ::                 List.tail \u2191v)) ** exact Part.mem_some_iff.2 rfl ** n\u271d\u00b2 : \u2115 f\u271d\u00b9 : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192 \u2115 n\u271d : \u2115 f : Vector \u2115 n\u271d \u2192 \u2115 g : Vector \u2115 (n\u271d + 2) \u2192 \u2115 a\u271d\u00b2 : Nat.Primrec' f a\u271d\u00b9 : Nat.Primrec' g cf cg : Code v : Vector \u2115 (n\u271d + 1) hf : eval cf (List.tail \u2191v) = pure (pure (f (Vector.tail v))) hg : \u2200 (a b : \u2115), eval cg (a :: b :: List.tail \u2191v) = pure (pure (g (a ::\u1d65 b ::\u1d65 Vector.tail v))) n : \u2115 n_ih\u271d :   eval (prec cf cg) (n :: List.tail \u2191v) =     pure (pure (Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n)) a\u271d b\u271d : \u2115 e\u271d : a\u271d + b\u271d = n b : \u2115 IH :   \u2200 (a : \u2115),     a + b = n \u2192       Nat.succ n ::           0 ::             g (n ::\u1d65 Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) n ::\u1d65 Vector.tail v) ::               List.tail \u2191v \u2208         PFun.fix           (fun v =>             Part.bind (eval cg (List.headI v :: List.tail (List.tail v))) fun x =>               Part.some                 (if List.headI (List.tail v) = 0 then                   Sum.inl                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))                 else                   Sum.inr                     (Nat.succ (List.headI v) ::                       Nat.pred (List.headI (List.tail v)) :: List.headI x :: List.tail (List.tail (List.tail v)))))           (a :: b :: Nat.rec (f (Vector.tail v)) (fun y IH => g (y ::\u1d65 IH ::\u1d65 Vector.tail v)) a :: List.tail \u2191v) a : \u2115 e : a + Nat.succ b = n \u22a2 a + 1 + b = n ** rwa [add_right_comm] ** n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 m n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Fin n \u2192 Vector \u2115 m \u2192. \u2115 a\u271d\u00b9 : Nat.Partrec' f a\u271d : \u2200 (i : Fin n), Nat.Partrec' (g i) IHf : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> f v IHg : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> (fun v => (Vector.mOfFn fun i => g i v) >>= f) v ** exact exists_code.comp IHf IHg ** n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f IHf : \u2203 c, \u2200 (v : Vector \u2115 (n + 1)), eval c \u2191v = pure <$> \u2191f v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v ** obtain \u27e8cf, hf\u27e9 := IHf ** case intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code hf : \u2200 (v : Vector \u2115 (n + 1)), eval cf \u2191v = pure <$> \u2191f v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v ** refine' \u27e8rfind cf, fun v => _\u27e9 ** case intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code hf : \u2200 (v : Vector \u2115 (n + 1)), eval cf \u2191v = pure <$> \u2191f v v : Vector \u2115 n \u22a2 eval (rfind cf) \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v ** replace hf := fun a => hf (a ::\u1d65 v) ** case intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf \u2191(a ::\u1d65 v) = pure <$> \u2191f (a ::\u1d65 v) \u22a2 eval (rfind cf) \u2191v = pure <$> (fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) v ** simp only [Part.map_eq_map, Part.map_some, Vector.cons_val, PFun.coe_val,\n  show \u2200 x, pure x = [x] from fun _ => rfl] at hf \u22a2 ** case intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] \u22a2 eval (rfind cf) \u2191v = Part.map pure (Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) ** refine' Part.ext fun x => _ ** case intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] x : List \u2115 \u22a2 x \u2208 eval (rfind cf) \u2191v \u2194 x \u2208 Part.map pure (Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::\u1d65 v) = 0))) ** simp only [rfind, Part.bind_eq_bind, Part.pure_eq_some, Part.map_eq_map, Part.bind_some,\n  exists_prop, cons_eval, comp_eval, fix_eval, tail_eval, succ_eval, zero'_eval,\n  List.headI_nil, List.headI_cons, pred_eval, Part.map_some, Bool.false_eq_decide_iff,\n  Part.mem_bind_iff, List.length, Part.mem_map_iff, Nat.mem_rfind, List.tail_nil,\n  List.tail_cons, Bool.true_eq_decide_iff, Part.mem_some_iff, Part.map_bind] ** case intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] x : List \u2115 \u22a2 (\u2203 a,       a \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             (0 :: \u2191v) \u2227         x = [Nat.pred (List.headI a)]) \u2194     \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = x ** constructor ** case intro.mp n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] x : List \u2115 \u22a2 (\u2203 a,       a \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             (0 :: \u2191v) \u2227         x = [Nat.pred (List.headI a)]) \u2192     \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = x ** rintro \u27e8v', h1, rfl\u27e9 ** case intro.mp.intro.intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' : List \u2115 h1 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) \u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = [Nat.pred (List.headI v')] ** suffices \u2200 v\u2081 : List \u2115, v' \u2208 PFun.fix\n  (fun v => (cf.eval v).bind fun y => Part.some <|\n    if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail)\n      else Sum.inr (v.headI.succ :: v.tail)) v\u2081 \u2192\n  \u2200 n, (v\u2081 = n :: v.val) \u2192 (\u2200 m < n, \u00acf (m ::\u1d65 v) = 0) \u2192\n    \u2203 a : \u2115,\n      (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [v'.headI.pred]\n  by exact this _ h1 0 rfl (by rintro _ \u27e8\u27e9) ** case intro.mp.intro.intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' : List \u2115 h1 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) \u22a2 \u2200 (v\u2081 : List \u2115),     v' \u2208         PFun.fix           (fun v =>             Part.bind (eval cf v) fun y =>               Part.some                 (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                 else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))           v\u2081 \u2192       \u2200 (n_1 : \u2115),         v\u2081 = n_1 :: \u2191v \u2192           (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** clear h1 ** case intro.mp.intro.intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' : List \u2115 \u22a2 \u2200 (v\u2081 : List \u2115),     v' \u2208         PFun.fix           (fun v =>             Part.bind (eval cf v) fun y =>               Part.some                 (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                 else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))           v\u2081 \u2192       \u2200 (n_1 : \u2115),         v\u2081 = n_1 :: \u2191v \u2192           (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** intro v\u2080 h1 ** case intro.mp.intro.intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 h1 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       v\u2080 \u22a2 \u2200 (n_1 : \u2115),     v\u2080 = n_1 :: \u2191v \u2192       (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192         \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** refine' PFun.fixInduction h1 fun v\u2081 h2 IH => _ ** case intro.mp.intro.intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 h1 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       v\u2080 v\u2081 : List \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       v\u2081 IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf v\u2081) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v\u2081) :: List.tail v\u2081)             else Sum.inr (Nat.succ (List.headI v\u2081) :: List.tail v\u2081))) \u2192       \u2200 (n_1 : \u2115),         a'' = n_1 :: \u2191v \u2192           (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] \u22a2 \u2200 (n_1 : \u2115),     v\u2081 = n_1 :: \u2191v \u2192       (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192         \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** clear h1 ** case intro.mp.intro.intro n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 v\u2081 : List \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       v\u2081 IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf v\u2081) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v\u2081) :: List.tail v\u2081)             else Sum.inr (Nat.succ (List.headI v\u2081) :: List.tail v\u2081))) \u2192       \u2200 (n_1 : \u2115),         a'' = n_1 :: \u2191v \u2192           (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] \u22a2 \u2200 (n_1 : \u2115),     v\u2081 = n_1 :: \u2191v \u2192       (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192         \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** rintro n rfl hm ** case intro.mp.intro.intro n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 n : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n : \u2115),         a'' = n :: \u2191v \u2192           (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 \u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** have := PFun.mem_fix_iff.1 h2 ** case intro.mp.intro.intro n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 n : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n : \u2115),         a'' = n :: \u2191v \u2192           (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 this :   (Sum.inl v' \u2208       Part.bind (eval cf (n :: \u2191v)) fun y =>         Part.some           (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))           else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2228     \u2203 a',       (Sum.inr a' \u2208           Part.bind (eval cf (n :: \u2191v)) fun y =>             Part.some               (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))               else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2227         v' \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             a' \u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** simp only [hf, Part.bind_some] at this ** case intro.mp.intro.intro n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 n : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n : \u2115),         a'' = n :: \u2191v \u2192           (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 this :   Sum.inl v' \u2208       Part.some         (if List.headI [f (n ::\u1d65 v)] = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))         else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228     \u2203 a',       Sum.inr a' \u2208           Part.some             (if List.headI [f (n ::\u1d65 v)] = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227         v' \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             a' \u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** split_ifs at this with h ** n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' : List \u2115 h1 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) this :   \u2200 (v\u2081 : List \u2115),     v' \u2208         PFun.fix           (fun v =>             Part.bind (eval cf v) fun y =>               Part.some                 (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                 else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))           v\u2081 \u2192       \u2200 (n_1 : \u2115),         v\u2081 = n_1 :: \u2191v \u2192           (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] \u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = [Nat.pred (List.headI v')] ** exact this _ h1 0 rfl (by rintro _ \u27e8\u27e9) ** n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' : List \u2115 h1 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) this :   \u2200 (v\u2081 : List \u2115),     v' \u2208         PFun.fix           (fun v =>             Part.bind (eval cf v) fun y =>               Part.some                 (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                 else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))           v\u2081 \u2192       \u2200 (n_1 : \u2115),         v\u2081 = n_1 :: \u2191v \u2192           (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] \u22a2 \u2200 (m : \u2115), m < 0 \u2192 \u00acf (m ::\u1d65 v) = 0 ** rintro _ \u27e8\u27e9 ** case pos n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 n : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n : \u2115),         a'' = n :: \u2191v \u2192           (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 h : List.headI [f (n ::\u1d65 v)] = 0 this :   Sum.inl v' \u2208 Part.some (Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227         v' \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             a' \u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** simp only [List.headI_nil, List.headI_cons, exists_false, or_false_iff, Part.mem_some_iff,\n  List.tail_cons, false_and_iff, Sum.inl.injEq] at this ** case pos n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 n : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n : \u2115),         a'' = n :: \u2191v \u2192           (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 h : List.headI [f (n ::\u1d65 v)] = 0 this : v' = Nat.succ n :: \u2191v \u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** subst this ** case pos n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v\u2080 : List \u2115 n : \u2115 hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 h : List.headI [f (n ::\u1d65 v)] = 0 h2 :   Nat.succ n :: \u2191v \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n_1 : \u2115),         a'' = n_1 :: \u2191v \u2192           (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a,               (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI (Nat.succ n :: \u2191v))] \u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI (Nat.succ n :: \u2191v))] ** exact \u27e8_, \u27e8h, @(hm)\u27e9, rfl\u27e9 ** case neg n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 n : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n : \u2115),         a'' = n :: \u2191v \u2192           (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 h : \u00acList.headI [f (n ::\u1d65 v)] = 0 this :   Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227         v' \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             a' \u22a2 \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] ** refine' IH (n.succ::v.val) (by simp_all) _ rfl fun m h' => _ ** case neg n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 n : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n : \u2115),         a'' = n :: \u2191v \u2192           (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 h : \u00acList.headI [f (n ::\u1d65 v)] = 0 this :   Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227         v' \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             a' m : \u2115 h' : m < Nat.succ n \u22a2 \u00acf (m ::\u1d65 v) = 0 ** obtain h | rfl := Nat.lt_succ_iff_lt_or_eq.1 h' ** case neg.inl n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 n : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n : \u2115),         a'' = n :: \u2191v \u2192           (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 h\u271d : \u00acList.headI [f (n ::\u1d65 v)] = 0 this :   Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227         v' \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             a' m : \u2115 h' : m < Nat.succ n h : m < n \u22a2 \u00acf (m ::\u1d65 v) = 0  case neg.inr n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 m : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (m :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (m :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v))             else Sum.inr (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v)))) \u2192       \u2200 (n_1 : \u2115),         a'' = n_1 :: \u2191v \u2192           (\u2200 (m : \u2115), m < n_1 \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m_1 : \u2115), m_1 < m \u2192 \u00acf (m_1 ::\u1d65 v) = 0 h : \u00acList.headI [f (m ::\u1d65 v)] = 0 this :   Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v))) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (m :: \u2191v)) :: List.tail (m :: \u2191v))) \u2227         v' \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             a' h' : m < Nat.succ m \u22a2 \u00acf (m ::\u1d65 v) = 0 ** exacts [hm _ h, h] ** n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] v' v\u2080 : List \u2115 n : \u2115 h2 :   v' \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) IH :   \u2200 (a'' : List \u2115),     (Sum.inr a'' \u2208         Part.bind (eval cf (n :: \u2191v)) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))             else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v)))) \u2192       \u2200 (n : \u2115),         a'' = n :: \u2191v \u2192           (\u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192             \u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 [a] = [Nat.pred (List.headI v')] hm : \u2200 (m : \u2115), m < n \u2192 \u00acf (m ::\u1d65 v) = 0 h : \u00acList.headI [f (n ::\u1d65 v)] = 0 this :   Sum.inl v' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) \u2227         v' \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             a' \u22a2 Sum.inr (Nat.succ n :: \u2191v) \u2208     Part.bind (eval cf (n :: \u2191v)) fun y =>       Part.some         (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))         else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) ** simp_all ** case intro.mpr n\u271d : \u2115 f\u271d : Vector \u2115 n\u271d \u2192. \u2115 n : \u2115 f : Vector \u2115 (n + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] x : List \u2115 \u22a2 (\u2203 a, (f (a ::\u1d65 v) = 0 \u2227 \u2200 {m : \u2115}, m < a \u2192 \u00acf (m ::\u1d65 v) = 0) \u2227 pure a = x) \u2192     \u2203 a,       a \u2208           PFun.fix             (fun v =>               Part.bind (eval cf v) fun y =>                 Part.some                   (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                   else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))             (0 :: \u2191v) \u2227         x = [Nat.pred (List.headI a)] ** rintro \u27e8n, \u27e8hn, hm\u27e9, rfl\u27e9 ** case intro.mpr.intro.intro.intro n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] n : \u2115 hn : f (n ::\u1d65 v) = 0 hm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0 \u22a2 \u2203 a,     a \u2208         PFun.fix           (fun v =>             Part.bind (eval cf v) fun y =>               Part.some                 (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                 else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))           (0 :: \u2191v) \u2227       pure n = [Nat.pred (List.headI a)] ** refine' \u27e8n.succ::v.1, _, rfl\u27e9 ** case intro.mpr.intro.intro.intro n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] n : \u2115 hn : f (n ::\u1d65 v) = 0 hm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0 \u22a2 Nat.succ n :: \u2191v \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) ** have : (n.succ::v.1 : List \u2115) \u2208\n  PFun.fix (fun v =>\n    (cf.eval v).bind fun y =>\n      Part.some <|\n        if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail)\n          else Sum.inr (v.headI.succ :: v.tail))\n      (n::v.val) :=\n  PFun.mem_fix_iff.2 (Or.inl (by simp [hf, hn])) ** case intro.mpr.intro.intro.intro n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] n : \u2115 hn : f (n ::\u1d65 v) = 0 hm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0 this :   Nat.succ n :: \u2191v \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) \u22a2 Nat.succ n :: \u2191v \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) ** generalize (n.succ :: v.1 : List \u2115) = w at this \u22a2 ** case intro.mpr.intro.intro.intro n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] n : \u2115 hn : f (n ::\u1d65 v) = 0 hm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0 w : List \u2115 this :   w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) \u22a2 w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) ** clear hn ** case intro.mpr.intro.intro.intro n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] n : \u2115 hm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0 w : List \u2115 this :   w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) \u22a2 w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) ** induction' n with n IH ** case intro.mpr.intro.intro.intro.succ n\u271d\u00b2 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d\u00b9 hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] n\u271d : \u2115 hm\u271d : \u2200 {m : \u2115}, m < n\u271d \u2192 \u00acf (m ::\u1d65 v) = 0 w : List \u2115 this\u271d :   w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n\u271d :: \u2191v) n : \u2115 IH :   (\u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192     w \u2208         PFun.fix           (fun v =>             Part.bind (eval cf v) fun y =>               Part.some                 (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                 else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))           (n :: \u2191v) \u2192       w \u2208         PFun.fix           (fun v =>             Part.bind (eval cf v) fun y =>               Part.some                 (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                 else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))           (0 :: \u2191v) hm : \u2200 {m : \u2115}, m < Nat.succ n \u2192 \u00acf (m ::\u1d65 v) = 0 this :   w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (Nat.succ n :: \u2191v) \u22a2 w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) ** refine' IH (fun {m} h' => hm (Nat.lt_succ_of_lt h'))\n  (PFun.mem_fix_iff.2 (Or.inr \u27e8_, _, this\u27e9)) ** case intro.mpr.intro.intro.intro.succ n\u271d\u00b2 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b2 \u2192. \u2115 n\u271d\u00b9 : \u2115 f : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d\u00b9 hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] n\u271d : \u2115 hm\u271d : \u2200 {m : \u2115}, m < n\u271d \u2192 \u00acf (m ::\u1d65 v) = 0 w : List \u2115 this\u271d :   w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n\u271d :: \u2191v) n : \u2115 IH :   (\u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0) \u2192     w \u2208         PFun.fix           (fun v =>             Part.bind (eval cf v) fun y =>               Part.some                 (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                 else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))           (n :: \u2191v) \u2192       w \u2208         PFun.fix           (fun v =>             Part.bind (eval cf v) fun y =>               Part.some                 (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)                 else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))           (0 :: \u2191v) hm : \u2200 {m : \u2115}, m < Nat.succ n \u2192 \u00acf (m ::\u1d65 v) = 0 this :   w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (Nat.succ n :: \u2191v) \u22a2 Sum.inr (Nat.succ n :: \u2191v) \u2208     Part.bind (eval cf (n :: \u2191v)) fun y =>       Part.some         (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))         else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) ** simp only [hf, hm n.lt_succ_self, Part.bind_some, List.headI, eq_self_iff_true, if_false,\n  Part.mem_some_iff, and_self_iff, List.tail_cons] ** n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] n : \u2115 hn : f (n ::\u1d65 v) = 0 hm : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0 \u22a2 Sum.inl (Nat.succ n :: \u2191v) \u2208     Part.bind (eval cf (n :: \u2191v)) fun y =>       Part.some         (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))         else Sum.inr (Nat.succ (List.headI (n :: \u2191v)) :: List.tail (n :: \u2191v))) ** simp [hf, hn] ** case intro.mpr.intro.intro.intro.zero n\u271d\u00b9 : \u2115 f\u271d : Vector \u2115 n\u271d\u00b9 \u2192. \u2115 n\u271d : \u2115 f : Vector \u2115 (n\u271d + 1) \u2192 \u2115 a\u271d : Nat.Partrec' \u2191f cf : Code v : Vector \u2115 n\u271d hf : \u2200 (a : \u2115), eval cf (a :: \u2191v) = Part.some [f (a ::\u1d65 v)] n : \u2115 hm\u271d : \u2200 {m : \u2115}, m < n \u2192 \u00acf (m ::\u1d65 v) = 0 w : List \u2115 this\u271d :   w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (n :: \u2191v) hm : \u2200 {m : \u2115}, m < Nat.zero \u2192 \u00acf (m ::\u1d65 v) = 0 this :   w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (Nat.zero :: \u2191v) \u22a2 w \u2208     PFun.fix       (fun v =>         Part.bind (eval cf v) fun y =>           Part.some             (if List.headI y = 0 then Sum.inl (Nat.succ (List.headI v) :: List.tail v)             else Sum.inr (Nat.succ (List.headI v) :: List.tail v)))       (0 :: \u2191v) ** exact this ** Qed",
        "informal": ""
    },
    {
        "formal": "IntervalIntegrable.comp_add_right ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedRing A f g : \u211d \u2192 E a b : \u211d \u03bc : Measure \u211d hf : IntervalIntegrable f volume a b c : \u211d \u22a2 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c) ** wlog h : a \u2264 b generalizing a b ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedRing A f g : \u211d \u2192 E a\u271d b\u271d : \u211d \u03bc : Measure \u211d c a b : \u211d hf : IntervalIntegrable f volume a b h : a \u2264 b \u22a2 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c) ** rw [intervalIntegrable_iff'] at hf \u22a2 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedRing A f g : \u211d \u2192 E a\u271d b\u271d : \u211d \u03bc : Measure \u211d c a b : \u211d hf : IntegrableOn f [[a, b]] h : a \u2264 b \u22a2 IntegrableOn (fun x => f (x + c)) [[a - c, b - c]] ** have A : MeasurableEmbedding fun x => x + c :=\n  (Homeomorph.addRight c).closedEmbedding.measurableEmbedding ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedRing A\u271d f g : \u211d \u2192 E a\u271d b\u271d : \u211d \u03bc : Measure \u211d c a b : \u211d hf : IntegrableOn f [[a, b]] h : a \u2264 b A : MeasurableEmbedding fun x => x + c \u22a2 IntegrableOn (fun x => f (x + c)) [[a - c, b - c]] ** convert (MeasurableEmbedding.integrableOn_map_iff A).mp hf using 1 ** case h.e'_6 \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedRing A\u271d f g : \u211d \u2192 E a\u271d b\u271d : \u211d \u03bc : Measure \u211d c a b : \u211d hf : IntegrableOn f [[a, b]] h : a \u2264 b A : MeasurableEmbedding fun x => x + c \u22a2 [[a - c, b - c]] = (fun x => x + c) \u207b\u00b9' [[a, b]] ** rw [preimage_add_const_uIcc] ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedRing A f g : \u211d \u2192 E a b : \u211d \u03bc : Measure \u211d hf : IntervalIntegrable f volume a b c : \u211d this :   \u2200 {a b : \u211d}, IntervalIntegrable f volume a b \u2192 a \u2264 b \u2192 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c) h : \u00aca \u2264 b \u22a2 IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c) ** exact IntervalIntegrable.symm (this hf.symm (le_of_not_le h)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.addHaarMeasure_eq_volume_pi ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 \u22a2 addHaarMeasure (piIcc01 \u03b9) = volume ** convert (addHaarMeasure_unique volume (piIcc01 \u03b9)).symm ** case h.e'_2 \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 \u22a2 addHaarMeasure (piIcc01 \u03b9) = \u2191\u2191volume \u2191(piIcc01 \u03b9) \u2022 addHaarMeasure (piIcc01 \u03b9) ** simp only [piIcc01, volume_pi_pi fun _ => Icc (0 : \u211d) 1, PositiveCompacts.coe_mk,\n  Compacts.coe_mk, Finset.prod_const_one, ENNReal.ofReal_one, Real.volume_Icc, one_smul, sub_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.norm_integral_le_lintegral_norm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G \u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc) ** by_cases hG : CompleteSpace G ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hG : CompleteSpace G \u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc) ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hG : CompleteSpace G hf : Integrable f \u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc) ** rw [integral_eq f hf, \u2190 Integrable.norm_toL1_eq_lintegral_norm f hf] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hG : CompleteSpace G hf : Integrable f \u22a2 \u2016L1.integral (Integrable.toL1 f hf)\u2016 \u2264 \u2016Integrable.toL1 f hf\u2016 ** exact L1.norm_integral_le _ ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hG : CompleteSpace G hf : \u00acIntegrable f \u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc) ** rw [integral_undef hf, norm_zero] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hG : CompleteSpace G hf : \u00acIntegrable f \u22a2 0 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc) ** exact toReal_nonneg ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hG : \u00acCompleteSpace G \u22a2 \u2016\u222b (a : \u03b1), f a \u2202\u03bc\u2016 \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc) ** simp [integral, hG] ** Qed",
        "informal": ""
    },
    {
        "formal": "LipschitzWith.lipschitzWith_compLp ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G g\u271d : E \u2192 F c : \u211d\u22650 inst\u271d : Fact (1 \u2264 p) hg : LipschitzWith c g\u271d g0 : g\u271d 0 = 0 f g : { x // x \u2208 Lp E p } \u22a2 dist (compLp hg g0 f) (compLp hg g0 g) \u2264 \u2191c * dist f g ** simp [dist_eq_norm, norm_compLp_sub_le] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurableSet_eq_fun ** G : Type u_1 \u03b1 : Type u_2 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : Div G m\u271d : MeasurableSpace \u03b1 f\u271d g\u271d : \u03b1 \u2192 G \u03bc : Measure \u03b1 m : MeasurableSpace \u03b1 E : Type u_3 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : AddGroup E inst\u271d\u00b9 : MeasurableSingletonClass E inst\u271d : MeasurableSub\u2082 E f g : \u03b1 \u2192 E hf : Measurable f hg : Measurable g \u22a2 MeasurableSet {x | f x = g x} ** suffices h_set_eq : { x : \u03b1 | f x = g x } = { x | (f - g) x = (0 : E) } ** case h_set_eq G : Type u_1 \u03b1 : Type u_2 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : Div G m\u271d : MeasurableSpace \u03b1 f\u271d g\u271d : \u03b1 \u2192 G \u03bc : Measure \u03b1 m : MeasurableSpace \u03b1 E : Type u_3 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : AddGroup E inst\u271d\u00b9 : MeasurableSingletonClass E inst\u271d : MeasurableSub\u2082 E f g : \u03b1 \u2192 E hf : Measurable f hg : Measurable g \u22a2 {x | f x = g x} = {x | (f - g) x = 0} ** ext ** case h_set_eq.h G : Type u_1 \u03b1 : Type u_2 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : Div G m\u271d : MeasurableSpace \u03b1 f\u271d g\u271d : \u03b1 \u2192 G \u03bc : Measure \u03b1 m : MeasurableSpace \u03b1 E : Type u_3 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : AddGroup E inst\u271d\u00b9 : MeasurableSingletonClass E inst\u271d : MeasurableSub\u2082 E f g : \u03b1 \u2192 E hf : Measurable f hg : Measurable g x\u271d : \u03b1 \u22a2 x\u271d \u2208 {x | f x = g x} \u2194 x\u271d \u2208 {x | (f - g) x = 0} ** simp_rw [Set.mem_setOf_eq, Pi.sub_apply, sub_eq_zero] ** G : Type u_1 \u03b1 : Type u_2 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : Div G m\u271d : MeasurableSpace \u03b1 f\u271d g\u271d : \u03b1 \u2192 G \u03bc : Measure \u03b1 m : MeasurableSpace \u03b1 E : Type u_3 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : AddGroup E inst\u271d\u00b9 : MeasurableSingletonClass E inst\u271d : MeasurableSub\u2082 E f g : \u03b1 \u2192 E hf : Measurable f hg : Measurable g h_set_eq : {x | f x = g x} = {x | (f - g) x = 0} \u22a2 MeasurableSet {x | f x = g x} ** rw [h_set_eq] ** G : Type u_1 \u03b1 : Type u_2 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : Div G m\u271d : MeasurableSpace \u03b1 f\u271d g\u271d : \u03b1 \u2192 G \u03bc : Measure \u03b1 m : MeasurableSpace \u03b1 E : Type u_3 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : AddGroup E inst\u271d\u00b9 : MeasurableSingletonClass E inst\u271d : MeasurableSub\u2082 E f g : \u03b1 \u2192 E hf : Measurable f hg : Measurable g h_set_eq : {x | f x = g x} = {x | (f - g) x = 0} \u22a2 MeasurableSet {x | (f - g) x = 0} ** exact (hf.sub hg) measurableSet_eq ** Qed",
        "informal": ""
    },
    {
        "formal": "ContinuousLinearMap.add_compLp ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F L L' : E \u2192L[\ud835\udd5c] F f : { x // x \u2208 Lp E p } \u22a2 compLp (L + L') f = compLp L f + compLp L' f ** ext1 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F L L' : E \u2192L[\ud835\udd5c] F f : { x // x \u2208 Lp E p } \u22a2 \u2191\u2191(compLp (L + L') f) =\u1d50[\u03bc] \u2191\u2191(compLp L f + compLp L' f) ** refine' (coeFn_compLp' (L + L') f).trans _ ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F L L' : E \u2192L[\ud835\udd5c] F f : { x // x \u2208 Lp E p } \u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) =\u1d50[\u03bc] \u2191\u2191(compLp L f + compLp L' f) ** refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F L L' : E \u2192L[\ud835\udd5c] F f : { x // x \u2208 Lp E p } \u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) =\u1d50[\u03bc] \u2191\u2191(compLp L f) + \u2191\u2191(compLp L' f) ** refine'\n  EventuallyEq.trans _ (EventuallyEq.add (L.coeFn_compLp' f).symm (L'.coeFn_compLp' f).symm) ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F L L' : E \u2192L[\ud835\udd5c] F f : { x // x \u2208 Lp E p } \u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) =\u1d50[\u03bc] fun x => \u2191L (\u2191\u2191f x) + \u2191L' (\u2191\u2191f x) ** refine' eventually_of_forall fun x => _ ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F L L' : E \u2192L[\ud835\udd5c] F f : { x // x \u2208 Lp E p } x : \u03b1 \u22a2 (fun a => \u2191(L + L') (\u2191\u2191f a)) x = (fun x => \u2191L (\u2191\u2191f x) + \u2191L' (\u2191\u2191f x)) x ** rw [coe_add', Pi.add_def] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 n\u271d : \u2115 \u22a2 MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n\u271d)) ** rw [restrictNonposSeq] ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 n\u271d : \u2115 \u22a2 MeasurableSet     (MeasureTheory.SignedMeasure.someExistsOneDivLT s       (i \\         \u22c3 k,           \u22c3 (H : k \u2264 n\u271d),             let_fun this := (_ : k < Nat.succ n\u271d);             MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) ** exact someExistsOneDivLT_measurableSet ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.setToL1S_mono ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \u211d F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E G'' : Type u_7 G' : Type u_8 inst\u271d\u00b3 : NormedLatticeAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : NormedLatticeAddCommGroup G'' inst\u271d : NormedSpace \u211d G'' T : Set \u03b1 \u2192 G'' \u2192L[\u211d] G' h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x f g : { x // x \u2208 simpleFunc G'' 1 \u03bc } hfg : f \u2264 g \u22a2 setToL1S T f \u2264 setToL1S T g ** rw [\u2190 sub_nonneg] at hfg \u22a2 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \u211d F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E G'' : Type u_7 G' : Type u_8 inst\u271d\u00b3 : NormedLatticeAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : NormedLatticeAddCommGroup G'' inst\u271d : NormedSpace \u211d G'' T : Set \u03b1 \u2192 G'' \u2192L[\u211d] G' h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x f g : { x // x \u2208 simpleFunc G'' 1 \u03bc } hfg\u271d : f \u2264 g hfg : 0 \u2264 g - f \u22a2 0 \u2264 setToL1S T g - setToL1S T f ** rw [\u2190 setToL1S_sub h_zero h_add] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \u211d F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E G'' : Type u_7 G' : Type u_8 inst\u271d\u00b3 : NormedLatticeAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : NormedLatticeAddCommGroup G'' inst\u271d : NormedSpace \u211d G'' T : Set \u03b1 \u2192 G'' \u2192L[\u211d] G' h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x f g : { x // x \u2208 simpleFunc G'' 1 \u03bc } hfg\u271d : f \u2264 g hfg : 0 \u2264 g - f \u22a2 0 \u2264 setToL1S (fun s => T s) (g - f) ** exact setToL1S_nonneg h_zero h_add hT_nonneg hfg ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.variance_def' ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 variance X \u2119 = (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2 ** rw [hX.variance_eq, sub_sq', integral_sub', integral_add'] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 ((\u222b (a : \u03a9), (X ^ 2) a) + \u222b (a : \u03a9), ((fun x => \u222b (x : \u03a9), X x) ^ 2) a) -       \u222b (a : \u03a9), (2 * X * fun x => \u222b (x : \u03a9), X x) a =     (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2  case hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 Integrable (X ^ 2)  case hg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 Integrable ((fun x => \u222b (x : \u03a9), X x) ^ 2)  case hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 Integrable (X ^ 2 + (fun x => \u222b (x : \u03a9), X x) ^ 2)  case hg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 Integrable (2 * X * fun x => \u222b (x : \u03a9), X x) ** rotate_left ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 ((\u222b (a : \u03a9), (X ^ 2) a) + \u222b (a : \u03a9), ((fun x => \u222b (x : \u03a9), X x) ^ 2) a) -       \u222b (a : \u03a9), (2 * X * fun x => \u222b (x : \u03a9), X x) a =     (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2 ** simp only [Pi.pow_apply, integral_const, measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul,\n  Pi.mul_apply, Pi.ofNat_apply, Nat.cast_ofNat, integral_mul_right, integral_mul_left] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 (\u222b (a : \u03a9), X a ^ 2) + (\u222b (x : \u03a9), X x) ^ 2 - (2 * \u222b (x : \u03a9), X x) * \u222b (x : \u03a9), X x =     (\u222b (a : \u03a9), X a ^ 2) - (\u222b (x : \u03a9), X x) ^ 2 ** ring ** case hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 Integrable (X ^ 2) ** exact hX.integrable_sq ** case hg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 Integrable ((fun x => \u222b (x : \u03a9), X x) ^ 2) ** convert @integrable_const \u03a9 \u211d (_) \u2119 _ _ (\ud835\udd3c[X] ^ 2) ** case hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 Integrable (X ^ 2 + (fun x => \u222b (x : \u03a9), X x) ^ 2) ** apply hX.integrable_sq.add ** case hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 Integrable ((fun x => \u222b (x : \u03a9), X x) ^ 2) ** convert @integrable_const \u03a9 \u211d (_) \u2119 _ _ (\ud835\udd3c[X] ^ 2) ** case hg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Mem\u2112p X 2 \u22a2 Integrable (2 * X * fun x => \u222b (x : \u03a9), X x) ** exact ((hX.integrable one_le_two).const_mul 2).mul_const' _ ** Qed",
        "informal": ""
    },
    {
        "formal": "List.pmap_eq_pmapImpl ** \u22a2 @pmap = @List.pmapImpl ** funext \u03b1 \u03b2 p f L h' ** case h.h.h.h.h.h \u03b1 : Type u_2 \u03b2 : Type u_1 p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 L : List \u03b1 h' : \u2200 (a : \u03b1), a \u2208 L \u2192 p a \u22a2 pmap f L h' = List.pmapImpl f L h' ** exact go L fun _ hx => hx ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.disjSups_subset_sups ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u2074 : DecidableEq \u03b1 inst\u271d\u00b3 : DecidableEq \u03b2 inst\u271d\u00b2 : SemilatticeSup \u03b1 inst\u271d\u00b9 : OrderBot \u03b1 inst\u271d : DecidableRel Disjoint s s\u2081 s\u2082 t t\u2081 t\u2082 u : Finset \u03b1 a b c : \u03b1 \u22a2 s \u25cb t \u2286 s \u22bb t ** simp_rw [subset_iff, mem_sups, mem_disjSups] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u2074 : DecidableEq \u03b1 inst\u271d\u00b3 : DecidableEq \u03b2 inst\u271d\u00b2 : SemilatticeSup \u03b1 inst\u271d\u00b9 : OrderBot \u03b1 inst\u271d : DecidableRel Disjoint s s\u2081 s\u2082 t t\u2081 t\u2082 u : Finset \u03b1 a b c : \u03b1 \u22a2 \u2200 \u2983x : \u03b1\u2984, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 Disjoint a b \u2227 a \u2294 b = x) \u2192 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = x ** exact fun c \u27e8a, b, ha, hb, _, hc\u27e9 => \u27e8a, b, ha, hb, hc\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.norm_Integral_le_one ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2016\u2191{             toAddHom :=               { toFun := integral,                 map_add' := (_ : \u2200 (f g : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (f + g) = integral f + integral g) },             map_smul' := (_ : \u2200 (c : \u211d) (f : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (c \u2022 f) = c \u2022 integral f) }         f\u2016 \u2264     1 * \u2016f\u2016 ** rw [one_mul] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2016\u2191{             toAddHom :=               { toFun := integral,                 map_add' := (_ : \u2200 (f g : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (f + g) = integral f + integral g) },             map_smul' := (_ : \u2200 (c : \u211d) (f : { x // x \u2208 simpleFunc E 1 \u03bc }), integral (c \u2022 f) = c \u2022 integral f) }         f\u2016 \u2264     \u2016f\u2016 ** exact norm_integral_le_norm f ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.sdiff_insert_insert_of_mem_of_not_mem ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t\u271d u v : Finset \u03b1 a b : \u03b1 s t : Finset \u03b1 x : \u03b1 hxs : x \u2208 s hxt : \u00acx \u2208 t \u22a2 insert x (s \\ insert x t) = s \\ t ** rw [sdiff_insert, insert_erase (mem_sdiff.mpr \u27e8hxs, hxt\u27e9)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.foldrM_eq_reverse_foldlM_data.aux ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d : Monad m f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2 arr : Array \u03b1 init : \u03b2 i : Nat h : i \u2264 size arr \u22a2 List.foldlM (fun x y => f y x) init (List.reverse (List.take i arr.data)) = foldrM.fold f arr 0 i h init ** unfold foldrM.fold ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d : Monad m f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2 arr : Array \u03b1 init : \u03b2 i : Nat h : 0 \u2264 size arr \u22a2 List.foldlM (fun x y => f y x) init (List.reverse (List.take 0 arr.data)) =     if (0 == 0) = true then pure init     else       match 0, h with       | 0, x => pure init       | Nat.succ i, h =>         let_fun this := (_ : i < size arr);         do         let __do_lift \u2190 f arr[i] init         foldrM.fold f arr 0 i (_ : i \u2264 size arr) __do_lift ** simp [List.foldlM, List.take] ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d : Monad m f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2 arr : Array \u03b1 init : \u03b2 i\u271d i : Nat h : i + 1 \u2264 size arr \u22a2 List.foldlM (fun x y => f y x) init (List.reverse (List.take (i + 1) arr.data)) =     if (i + 1 == 0) = true then pure init     else       match i + 1, h with       | 0, x => pure init       | Nat.succ i, h =>         let_fun this := (_ : i < size arr);         do         let __do_lift \u2190 f arr[i] init         foldrM.fold f arr 0 i (_ : i \u2264 size arr) __do_lift ** rw [\u2190 List.take_concat_get _ _ h] ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d : Monad m f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2 arr : Array \u03b1 init : \u03b2 i\u271d i : Nat h : i + 1 \u2264 size arr \u22a2 (do       let init \u2190 f arr.data[i] init       List.foldrM (fun x y => f x y) init (List.take i arr.data)) =     if i + 1 = 0 then pure init     else do       let init \u2190 f arr[i] init       List.foldrM (fun x y => f x y) init (List.take i arr.data) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.ofReal_condCdfRat_ae_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r : \u211a \u22a2 (fun a => ENNReal.ofReal (condCdfRat \u03c1 a r)) =\u1d50[Measure.fst \u03c1] preCdf \u03c1 r ** filter_upwards [condCdfRat_ae_eq \u03c1 r, preCdf_le_one \u03c1] with a ha ha_le_one ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r : \u211a a : \u03b1 ha : condCdfRat \u03c1 a r = ENNReal.toReal (preCdf \u03c1 r a) ha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 \u22a2 ENNReal.ofReal (condCdfRat \u03c1 a r) = preCdf \u03c1 r a ** rw [ha, ENNReal.ofReal_toReal] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r : \u211a a : \u03b1 ha : condCdfRat \u03c1 a r = ENNReal.toReal (preCdf \u03c1 r a) ha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 \u22a2 preCdf \u03c1 r a \u2260 \u22a4 ** exact ((ha_le_one r).trans_lt ENNReal.one_lt_top).ne ** Qed",
        "informal": ""
    },
    {
        "formal": "blimsup_thickening_mul_ae_eq ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) \u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p ** let q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i h\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q h\u2082 : blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q \u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p ** rw [h\u2081, h\u2082] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i h\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q h\u2082 : blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q \u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop q =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop q ** exact blimsup_thickening_mul_ae_eq_aux \u03bc q s hM r hr (eventually_of_forall fun i hi => hi.2) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i \u22a2 blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q ** refine' blimsup_congr' (eventually_of_forall fun i h => _) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i i : \u2115 h : thickening (r i) (s i) \u2260 \u22a5 \u22a2 p i \u2194 q i ** replace hi : 0 < r i ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i i : \u2115 h : thickening (r i) (s i) \u2260 \u22a5 hi : 0 < r i \u22a2 p i \u2194 q i ** simp only [hi, iff_self_and, imp_true_iff] ** case hi \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i i : \u2115 h : thickening (r i) (s i) \u2260 \u22a5 \u22a2 0 < r i ** contrapose! h ** case hi \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i i : \u2115 h : r i \u2264 0 \u22a2 thickening (r i) (s i) = \u22a5 ** apply thickening_of_nonpos h ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i h\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q \u22a2 blimsup (fun i => thickening (M * r i) (s i)) atTop p = blimsup (fun i => thickening (M * r i) (s i)) atTop q ** refine' blimsup_congr' (eventually_of_forall fun i h => _) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i h\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q i : \u2115 h : thickening (M * r i) (s i) \u2260 \u22a5 \u22a2 p i \u2194 q i ** replace h : 0 < r i ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i h\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q i : \u2115 h : 0 < r i \u22a2 p i \u2194 q i ** simp only [h, iff_self_and, imp_true_iff] ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i h\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q i : \u2115 h : thickening (M * r i) (s i) \u2260 \u22a5 \u22a2 0 < r i ** rw [\u2190 zero_lt_mul_left hM] ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i h\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q i : \u2115 h : thickening (M * r i) (s i) \u2260 \u22a5 \u22a2 0 < M * r i ** contrapose! h ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 M : \u211d hM : 0 < M r : \u2115 \u2192 \u211d hr : Tendsto r atTop (\ud835\udcdd 0) q : \u2115 \u2192 Prop := fun i => p i \u2227 0 < r i h\u2081 : blimsup (fun i => thickening (r i) (s i)) atTop p = blimsup (fun i => thickening (r i) (s i)) atTop q i : \u2115 h : M * r i \u2264 0 \u22a2 thickening (M * r i) (s i) = \u22a5 ** apply thickening_of_nonpos h ** Qed",
        "informal": ""
    },
    {
        "formal": "String.prev_of_valid ** cs : List Char c : Char cs' : List Char \u22a2 prev { data := cs ++ c :: cs' } { byteIdx := utf8Len cs + csize c } = { byteIdx := utf8Len cs } ** simp [prev] ** cs : List Char c : Char cs' : List Char \u22a2 (if { byteIdx := utf8Len cs + csize c } = 0 then 0     else utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c }) =     { byteIdx := utf8Len cs } ** refine (if_neg (Pos.ne_of_gt add_csize_pos)).trans ?_ ** cs : List Char c : Char cs' : List Char \u22a2 utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c } = { byteIdx := utf8Len cs } ** rw [utf8PrevAux_of_valid] <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.borel_eq_borel_of_le ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : T2Space \u03b2 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b3 f : \u03b3 \u2192 \u03b2 t t' : TopologicalSpace \u03b3 ht : PolishSpace \u03b3 ht' : PolishSpace \u03b3 hle : t \u2264 t' \u22a2 borel \u03b3 = borel \u03b3 ** refine' le_antisymm _ (borel_anti hle) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : T2Space \u03b2 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b3 f : \u03b3 \u2192 \u03b2 t t' : TopologicalSpace \u03b3 ht : PolishSpace \u03b3 ht' : PolishSpace \u03b3 hle : t \u2264 t' \u22a2 borel \u03b3 \u2264 borel \u03b3 ** intro s hs ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : T2Space \u03b2 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b3 f : \u03b3 \u2192 \u03b2 t t' : TopologicalSpace \u03b3 ht : PolishSpace \u03b3 ht' : PolishSpace \u03b3 hle : t \u2264 t' s : Set \u03b3 hs : MeasurableSet s \u22a2 MeasurableSet s ** have e := @Continuous.measurableEmbedding\n  _ _ t' _ (@borel _ t') _ (@BorelSpace.mk _ _ (borel \u03b3) rfl)\n  t _ (@borel _ t) (@BorelSpace.mk _ t (@borel _ t) rfl) (continuous_id_of_le hle) injective_id ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : T2Space \u03b2 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b3 f : \u03b3 \u2192 \u03b2 t t' : TopologicalSpace \u03b3 ht : PolishSpace \u03b3 ht' : PolishSpace \u03b3 hle : t \u2264 t' s : Set \u03b3 hs : MeasurableSet s e : MeasurableEmbedding id \u22a2 MeasurableSet s ** convert e.measurableSet_image.2 hs ** case h.e'_3 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 t\u03b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : T2Space \u03b2 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b3 f : \u03b3 \u2192 \u03b2 t t' : TopologicalSpace \u03b3 ht : PolishSpace \u03b3 ht' : PolishSpace \u03b3 hle : t \u2264 t' s : Set \u03b3 hs : MeasurableSet s e : MeasurableEmbedding id \u22a2 s = id '' s ** simp only [id_eq, image_id'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.map_id ** n\u271d : \u2115 \u03b1 : Type u_1 n : \u2115 v : Vector \u03b1 n \u22a2 toList (map id v) = toList v ** simp only [List.map_id, Vector.toList_map] ** Qed",
        "informal": ""
    },
    {
        "formal": "PosNum.to_nat_inj ** \u03b1 : Type u_1 m n : PosNum h : \u2191m = \u2191n \u22a2 pos m = pos n ** rw [\u2190 PosNum.of_to_nat, \u2190 PosNum.of_to_nat, h] ** Qed",
        "informal": ""
    },
    {
        "formal": "interval_average_eq_div ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u211d \u2192 \u211d a b : \u211d \u22a2 \u2a0d (x : \u211d) in a..b, f x = (\u222b (x : \u211d) in a..b, f x) / (b - a) ** rw [interval_average_eq, smul_eq_mul, div_eq_inv_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd n : \u2115 \u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 n), f k x) \u2208 measurableLE \u03bc \u03bd ** induction' n with m hm ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd \u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x) \u2208 measurableLE \u03bc \u03bd ** refine' \u27e8_, _\u27e9 ** case zero.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd \u22a2 Measurable fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x ** simp [(hf 0).1] ** case zero.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd \u22a2 \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (x : \u03b1) in A, (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A ** intro A hA ** case zero.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (x : \u03b1) in A, (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.zero), f k x) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A ** simp [(hf 0).2 A hA] ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 m\u271d : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd m : \u2115 hm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd \u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.succ m), f k x) \u2208 measurableLE \u03bc \u03bd ** have :\n  (fun a : \u03b1 => \u2a06 (k : \u2115) (_ : k \u2264 m + 1), f k a) = fun a =>\n    f m.succ a \u2294 \u2a06 (k : \u2115) (_ : k \u2264 m), f k a :=\n  funext fun _ => iSup_succ_eq_sup _ _ _ ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 m\u271d : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd m : \u2115 hm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd this : (fun a => \u2a06 k, \u2a06 (_ : k \u2264 m + 1), f k a) = fun a => f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a \u22a2 (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.succ m), f k x) \u2208 measurableLE \u03bc \u03bd ** refine' \u27e8measurable_iSup fun n => Measurable.iSup_Prop _ (hf n).1, fun A hA => _\u27e9 ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 m\u271d : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd m : \u2115 hm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd this : (fun a => \u2a06 k, \u2a06 (_ : k \u2264 m + 1), f k a) = fun a => f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (x : \u03b1) in A, (fun x => \u2a06 k, \u2a06 (_ : k \u2264 Nat.succ m), f k x) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A ** rw [this] ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 m\u271d : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bc \u03bd m : \u2115 hm : (fun x => \u2a06 k, \u2a06 (_ : k \u2264 m), f k x) \u2208 measurableLE \u03bc \u03bd this : (fun a => \u2a06 k, \u2a06 (_ : k \u2264 m + 1), f k a) = fun a => f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (a : \u03b1) in A, f (Nat.succ m) a \u2294 \u2a06 k, \u2a06 (_ : k \u2264 m), f k a \u2202\u03bc \u2264 \u2191\u2191\u03bd A ** exact (sup_mem_measurableLE (hf m.succ) hm).2 A hA ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toOuterMeasure_apply_eq_one_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 \u22a2 \u2191(toOuterMeasure p) s = 1 \u2194 support p \u2286 s ** refine' (p.toOuterMeasure_apply s).symm \u25b8 \u27e8fun h a hap => _, fun h => _\u27e9 ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 h : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1 a : \u03b1 hap : a \u2208 support p \u22a2 a \u2208 s ** refine' by_contra fun hs => ne_of_lt _ (h.trans p.tsum_coe.symm) ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 h : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1 a : \u03b1 hap : a \u2208 support p hs : \u00aca \u2208 s \u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a ** have hs' : s.indicator p a = 0 := Set.indicator_apply_eq_zero.2 fun hs' => False.elim <| hs hs' ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 h : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1 a : \u03b1 hap : a \u2208 support p hs : \u00aca \u2208 s hs' : Set.indicator s (\u2191p) a = 0 \u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a ** have hsa : s.indicator p a < p a := hs'.symm \u25b8 (p.apply_pos_iff a).2 hap ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 h : \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1 a : \u03b1 hap : a \u2208 support p hs : \u00aca \u2208 s hs' : Set.indicator s (\u2191p) a = 0 hsa : Set.indicator s (\u2191p) a < \u2191p a \u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x < \u2211' (a : \u03b1), \u2191p a ** exact ENNReal.tsum_lt_tsum (p.tsum_coe_indicator_ne_top s)\n  (fun x => Set.indicator_apply_le fun _ => le_rfl) hsa ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 h : support p \u2286 s \u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1 ** suffices : \u2200 (x) (_ : x \u2209 s), p x = 0 ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 h : support p \u2286 s this : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0 \u22a2 \u2211' (x : \u03b1), Set.indicator s (\u2191p) x = 1  case this \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 h : support p \u2286 s \u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0 ** exact _root_.trans (tsum_congr\n  fun a => (Set.indicator_apply s p a).trans (ite_eq_left_iff.2 <| symm \u2218 this a)) p.tsum_coe ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 h : support p \u2286 s \u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 \u2191p x = 0 ** exact fun a ha => (p.apply_eq_zero_iff a).2 <| Set.not_mem_subset h ha ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.eval\u2082_mul_monomial ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 \u22a2 \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e ** apply MvPolynomial.induction_on p ** case h_C R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 \u22a2 \u2200 (a : R) {s : \u03c3 \u2192\u2080 \u2115} {a_1 : R},     eval\u2082 f g (\u2191C a * \u2191(monomial s) a_1) = eval\u2082 f g (\u2191C a) * \u2191f a_1 * Finsupp.prod s fun n e => g n ^ e ** intro a' s a ** case h_C R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a\u271d a'\u271d a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 a' : R s : \u03c3 \u2192\u2080 \u2115 a : R \u22a2 eval\u2082 f g (\u2191C a' * \u2191(monomial s) a) = eval\u2082 f g (\u2191C a') * \u2191f a * Finsupp.prod s fun n e => g n ^ e ** simp [C_mul_monomial, eval\u2082_monomial, f.map_mul] ** case h_add R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 \u22a2 \u2200 (p q : MvPolynomial \u03c3 R),     (\u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e) \u2192       (\u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},           eval\u2082 f g (q * \u2191(monomial s) a) = eval\u2082 f g q * \u2191f a * Finsupp.prod s fun n e => g n ^ e) \u2192         \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},           eval\u2082 f g ((p + q) * \u2191(monomial s) a) = eval\u2082 f g (p + q) * \u2191f a * Finsupp.prod s fun n e => g n ^ e ** intro p q ih_p ih_q ** case h_add R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q\u271d : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 p q : MvPolynomial \u03c3 R ih_p : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e ih_q : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (q * \u2191(monomial s) a) = eval\u2082 f g q * \u2191f a * Finsupp.prod s fun n e => g n ^ e \u22a2 \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},     eval\u2082 f g ((p + q) * \u2191(monomial s) a) = eval\u2082 f g (p + q) * \u2191f a * Finsupp.prod s fun n e => g n ^ e ** simp [add_mul, eval\u2082_add, ih_p, ih_q] ** case h_X R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 \u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),     (\u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e) \u2192       \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R},         eval\u2082 f g (p * X n * \u2191(monomial s) a) = eval\u2082 f g (p * X n) * \u2191f a * Finsupp.prod s fun n e => g n ^ e ** intro p n ih s a ** case h_X R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a\u271d a' a\u2081 a\u2082 : R e : \u2115 n\u271d m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 p : MvPolynomial \u03c3 R n : \u03c3 ih : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e s : \u03c3 \u2192\u2080 \u2115 a : R \u22a2 eval\u2082 f g (p * X n * \u2191(monomial s) a) = eval\u2082 f g (p * X n) * \u2191f a * Finsupp.prod s fun n e => g n ^ e ** exact\n  calc\n    (p * X n * monomial s a).eval\u2082 f g = (p * monomial (Finsupp.single n 1 + s) a).eval\u2082 f g :=\n      by rw [monomial_single_add, pow_one, mul_assoc]\n    _ = (p * monomial (Finsupp.single n 1) 1).eval\u2082 f g * f a * s.prod fun n e => g n ^ e := by\n      simp [ih, prod_single_index, prod_add_index, pow_one, pow_add, mul_assoc, mul_left_comm,\n        f.map_one] ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a\u271d a' a\u2081 a\u2082 : R e : \u2115 n\u271d m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 p : MvPolynomial \u03c3 R n : \u03c3 ih : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e s : \u03c3 \u2192\u2080 \u2115 a : R \u22a2 eval\u2082 f g (p * X n * \u2191(monomial s) a) = eval\u2082 f g (p * \u2191(monomial ((fun\u2080 | n => 1) + s)) a) ** rw [monomial_single_add, pow_one, mul_assoc] ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a\u271d a' a\u2081 a\u2082 : R e : \u2115 n\u271d m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 p : MvPolynomial \u03c3 R n : \u03c3 ih : \u2200 {s : \u03c3 \u2192\u2080 \u2115} {a : R}, eval\u2082 f g (p * \u2191(monomial s) a) = eval\u2082 f g p * \u2191f a * Finsupp.prod s fun n e => g n ^ e s : \u03c3 \u2192\u2080 \u2115 a : R \u22a2 eval\u2082 f g (p * \u2191(monomial ((fun\u2080 | n => 1) + s)) a) =     eval\u2082 f g (p * \u2191(monomial fun\u2080 | n => 1) 1) * \u2191f a * Finsupp.prod s fun n e => g n ^ e ** simp [ih, prod_single_index, prod_add_index, pow_one, pow_add, mul_assoc, mul_left_comm,\n  f.map_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "Dense.borel_eq_generateFrom_Ico_mem_aux ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s \u22a2 borel \u03b1 = MeasurableSpace.generateFrom {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} ** set S : Set (Set \u03b1) := { S | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ico l u = S } ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} \u22a2 borel \u03b1 = MeasurableSpace.generateFrom S ** refine' le_antisymm _ (generateFrom_Ico_mem_le_borel _ _) ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} \u22a2 borel \u03b1 \u2264 MeasurableSpace.generateFrom S ** letI : MeasurableSpace \u03b1 := generateFrom S ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S \u22a2 borel \u03b1 \u2264 MeasurableSpace.generateFrom S ** rw [borel_eq_generateFrom_Iio] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S \u22a2 MeasurableSpace.generateFrom (range Iio) \u2264 MeasurableSpace.generateFrom S ** refine' generateFrom_le (forall_range_iff.2 fun a => _) ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 \u22a2 MeasurableSet (Iio a) ** rcases hd.exists_countable_dense_subset_bot_top with \u27e8t, hts, hc, htd, htb, -\u27e9 ** case intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t \u22a2 MeasurableSet (Iio a) ** by_cases ha : \u2200 b < a, (Ioo b a).Nonempty ** case pos \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) \u22a2 MeasurableSet (Iio a) ** convert_to MeasurableSet (\u22c3 (l \u2208 t) (u \u2208 t) (_ : l < u) (_ : u \u2264 a), Ico l u) ** case h.e'_3 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) \u22a2 Iio a = \u22c3 l \u2208 t, \u22c3 u \u2208 t, \u22c3 (_ : l < u), \u22c3 (_ : u \u2264 a), Ico l u ** ext y ** case h.e'_3.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) y : \u03b1 \u22a2 y \u2208 Iio a \u2194 y \u2208 \u22c3 l \u2208 t, \u22c3 u \u2208 t, \u22c3 (_ : l < u), \u22c3 (_ : u \u2264 a), Ico l u ** simp only [mem_iUnion, mem_Iio, mem_Ico] ** case h.e'_3.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) y : \u03b1 \u22a2 y < a \u2194 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1 ** constructor ** case h.e'_3.h.mp \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) y : \u03b1 \u22a2 y < a \u2192 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1 ** intro hy ** case h.e'_3.h.mp \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) y : \u03b1 hy : y < a \u22a2 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1 ** rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) y with \u27e8l, hlt, hly\u27e9 ** case h.e'_3.h.mp.intro.intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) y : \u03b1 hy : y < a l : \u03b1 hlt : l \u2208 t hly : l \u2264 y \u22a2 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1 ** rcases htd.exists_mem_open isOpen_Ioo (ha y hy) with \u27e8u, hut, hyu, hua\u27e9 ** case h.e'_3.h.mp.intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) y : \u03b1 hy : y < a l : \u03b1 hlt : l \u2208 t hly : l \u2264 y u : \u03b1 hut : u \u2208 t hyu : y < u hua : u < a \u22a2 \u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1 ** exact \u27e8l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu\u27e9 ** case h.e'_3.h.mpr \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) y : \u03b1 \u22a2 (\u2203 i h i_1 h h h, i \u2264 y \u2227 y < i_1) \u2192 y < a ** rintro \u27e8l, -, u, -, -, hua, -, hyu\u27e9 ** case h.e'_3.h.mpr.intro.intro.intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) y l u : \u03b1 hua : u \u2264 a hyu : y < u \u22a2 y < a ** exact hyu.trans_le hua ** case pos \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) \u22a2 MeasurableSet (\u22c3 l \u2208 t, \u22c3 u \u2208 t, \u22c3 (_ : l < u), \u22c3 (_ : u \u2264 a), Ico l u) ** refine' MeasurableSet.biUnion hc fun a ha => MeasurableSet.biUnion hc fun b hb => _ ** case pos \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d\u00b9 b\u271d x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a\u271d : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha\u271d : \u2200 (b : \u03b1), b < a\u271d \u2192 Set.Nonempty (Ioo b a\u271d) a : \u03b1 ha : a \u2208 t b : \u03b1 hb : b \u2208 t \u22a2 MeasurableSet (\u22c3 (_ : a < b), \u22c3 (_ : b \u2264 a\u271d), Ico a b) ** refine' MeasurableSet.iUnion fun hab => MeasurableSet.iUnion fun _ => _ ** case pos \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d\u00b9 b\u271d x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a\u271d : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha\u271d : \u2200 (b : \u03b1), b < a\u271d \u2192 Set.Nonempty (Ioo b a\u271d) a : \u03b1 ha : a \u2208 t b : \u03b1 hb : b \u2208 t hab : a < b x\u271d : b \u2264 a\u271d \u22a2 MeasurableSet (Ico a b) ** exact .basic _ \u27e8a, hts ha, b, hts hb, hab, mem_singleton _\u27e9 ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u00ac\u2200 (b : \u03b1), b < a \u2192 Set.Nonempty (Ioo b a) \u22a2 MeasurableSet (Iio a) ** simp only [not_forall, not_nonempty_iff_eq_empty] at ha ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : \u2203 x h, Ioo x a = \u2205 \u22a2 MeasurableSet (Iio a) ** replace ha : a \u2208 s := hIoo ha.choose a ha.choose_spec.fst ha.choose_spec.snd ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : a \u2208 s \u22a2 MeasurableSet (Iio a) ** convert_to MeasurableSet (\u22c3 (l \u2208 t) (_ : l < a), Ico l a) ** case h.e'_3 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : a \u2208 s \u22a2 Iio a = \u22c3 l \u2208 t, \u22c3 (_ : l < a), Ico l a ** symm ** case h.e'_3 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : a \u2208 s \u22a2 \u22c3 l \u2208 t, \u22c3 (_ : l < a), Ico l a = Iio a ** simp only [\u2190 Ici_inter_Iio, \u2190 iUnion_inter, inter_eq_right, subset_def, mem_iUnion,\n  mem_Ici, mem_Iio] ** case h.e'_3 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : a \u2208 s \u22a2 \u2200 (x : \u03b1), x < a \u2192 \u2203 i h h, i \u2264 x ** intro x hx ** case h.e'_3 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x\u271d : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : a \u2208 s x : \u03b1 hx : x < a \u22a2 \u2203 i h h, i \u2264 x ** rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) x with \u27e8z, hzt, hzx\u27e9 ** case h.e'_3.intro.intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x\u271d : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : a \u2208 s x : \u03b1 hx : x < a z : \u03b1 hzt : z \u2208 t hzx : z \u2264 x \u22a2 \u2203 i h h, i \u2264 x ** exact \u27e8z, hzt, hzx.trans_lt hx, hzx\u27e9 ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : a \u2208 s \u22a2 MeasurableSet (\u22c3 l \u2208 t, \u22c3 (_ : l < a), Ico l a) ** refine' .biUnion hc fun x hx => MeasurableSet.iUnion fun hlt => _ ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1\u271d inst\u271d\u00b2\u2070 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2079 : MeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2078 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2077 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2076 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2075 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2074 : TopologicalSpace \u03b3 inst\u271d\u00b9\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b9\u00b2 : BorelSpace \u03b3 inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b3\u2082 inst\u271d\u2079 : BorelSpace \u03b3\u2082 inst\u271d\u2078 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2077 : TopologicalSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : LinearOrder \u03b1\u271d inst\u271d\u2074 : OrderClosedTopology \u03b1\u271d a\u271d b x\u271d : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 s : Set \u03b1 hd : Dense s hbot : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s hIoo : \u2200 (x y : \u03b1), x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s S : Set (Set \u03b1) := {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 s \u2227 l < u \u2227 Ico l u = S} this : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom S a : \u03b1 t : Set \u03b1 hts : t \u2286 s hc : Set.Countable t htd : Dense t htb : \u2200 (x : \u03b1), IsBot x \u2192 x \u2208 s \u2192 x \u2208 t ha : a \u2208 s x : \u03b1 hx : x \u2208 t hlt : x < a \u22a2 MeasurableSet (Ico x a) ** exact .basic _ \u27e8x, hts hx, a, ha, hlt, mem_singleton _\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.isometry_comap_mkMetric ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u22a2 \u2191(comap f) (mkMetric m) = mkMetric m ** simp only [mkMetric, mkMetric', mkMetric'.pre, inducedOuterMeasure, comap_iSup] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u22a2 \u2a06 i, \u2a06 (_ : i > 0), \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) =     \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => m (diam s)) ** refine' surjective_id.iSup_congr id fun \u03b5 => surjective_id.iSup_congr id fun h\u03b5 => _ ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 \u22a2 \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) = boundedBy (extend fun s x => m (diam s)) ** rw [comap_boundedBy _ (H.imp _ id)] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 \u22a2 (boundedBy fun s => extend (fun s x => m (diam s)) (f '' s)) = boundedBy (extend fun s x => m (diam s)) ** congr with s : 1 ** case e_m.h \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 s : Set X \u22a2 extend (fun s x => m (diam s)) (f '' s) = extend (fun s x => m (diam s)) s ** apply extend_congr ** case e_m.h.hP \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 s : Set X \u22a2 diam (f '' s) \u2264 \u03b5 \u2194 diam s \u2264 id \u03b5 ** simp [hf.ediam_image] ** case e_m.h.hm \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 s : Set X \u22a2 diam (f '' s) \u2264 \u03b5 \u2192 diam s \u2264 id \u03b5 \u2192 m (diam (f '' s)) = m (diam s) ** intros ** case e_m.h.hm \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 s : Set X ha\u271d : diam (f '' s) \u2264 \u03b5 hb\u271d : diam s \u2264 id \u03b5 \u22a2 m (diam (f '' s)) = m (diam s) ** simp [hf.injective.subsingleton_image_iff, hf.ediam_image] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 \u22a2 Monotone m \u2192 Monotone fun s => extend (fun s x => m (diam s)) \u2191s ** intro h_mono s t hst ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 h_mono : Monotone m s t : { s // Set.Nonempty s } hst : s \u2264 t \u22a2 (fun s => extend (fun s x => m (diam s)) \u2191s) s \u2264 (fun s => extend (fun s x => m (diam s)) \u2191s) t ** simp only [extend, le_iInf_iff] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 h_mono : Monotone m s t : { s // Set.Nonempty s } hst : s \u2264 t \u22a2 diam \u2191t \u2264 \u03b5 \u2192 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t) ** intro ht ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 h_mono : Monotone m s t : { s // Set.Nonempty s } hst : s \u2264 t ht : diam \u2191t \u2264 \u03b5 \u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t) ** apply le_trans _ (h_mono (diam_mono hst)) ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 h_mono : Monotone m s t : { s // Set.Nonempty s } hst : s \u2264 t ht : diam \u2191t \u2264 \u03b5 \u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam ((fun a => \u2191a) s)) ** simp only [(diam_mono hst).trans ht, le_refl, ciInf_pos] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ordConnectedComponent_eq_empty ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x y z : \u03b1 \u22a2 ordConnectedComponent s x = \u2205 \u2194 \u00acx \u2208 s ** rw [\u2190 not_nonempty_iff_eq_empty, nonempty_ordConnectedComponent] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.xgcdAux_val ** x y : \u2115 \u22a2 xgcdAux x 1 0 y 0 1 = (gcd x y, xgcd x y) ** rw [xgcd, \u2190 xgcdAux_fst x y 1 0 0 1] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.val_add_val_of_le ** n : \u2115 inst\u271d : NeZero n a b : ZMod n h : n \u2264 val a + val b \u22a2 val a + val b = val (a + b) + n ** rw [val_add, Nat.add_mod_add_of_le_add_mod, Nat.mod_eq_of_lt (val_lt _),\n  Nat.mod_eq_of_lt (val_lt _)] ** n : \u2115 inst\u271d : NeZero n a b : ZMod n h : n \u2264 val a + val b \u22a2 n \u2264 val a % n + val b % n ** rwa [Nat.mod_eq_of_lt (val_lt _), Nat.mod_eq_of_lt (val_lt _)] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.pred_true_of_condCount_eq_one ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 h : \u2191\u2191(condCount s) t = 1 \u22a2 s \u2286 t ** have hsf := finite_of_condCount_ne_zero (by rw [h]; exact one_ne_zero) ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 h : \u2191\u2191(condCount s) t = 1 hsf : Set.Finite s \u22a2 s \u2286 t ** rw [condCount, cond_apply _ hsf.measurableSet, mul_comm] at h ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 h : \u2191\u2191Measure.count (s \u2229 t) * (\u2191\u2191Measure.count s)\u207b\u00b9 = 1 hsf : Set.Finite s \u22a2 s \u2286 t ** replace h := ENNReal.eq_inv_of_mul_eq_one_left h ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hsf : Set.Finite s h : \u2191\u2191Measure.count (s \u2229 t) = (\u2191\u2191Measure.count s)\u207b\u00b9\u207b\u00b9 \u22a2 s \u2286 t ** rw [inv_inv, Measure.count_apply_finite _ hsf, Measure.count_apply_finite _ (hsf.inter_of_left _),\n  Nat.cast_inj] at h ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hsf : Set.Finite s h : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf) \u22a2 s \u2286 t ** suffices s \u2229 t = s by exact this \u25b8 fun x hx => hx.2 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hsf : Set.Finite s h : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf) \u22a2 s \u2229 t = s ** rw [\u2190 @Set.Finite.toFinset_inj _ _ _ (hsf.inter_of_left _) hsf] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hsf : Set.Finite s h : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf) \u22a2 Set.Finite.toFinset (_ : Set.Finite (s \u2229 t)) = Set.Finite.toFinset hsf ** exact Finset.eq_of_subset_of_card_le (Set.Finite.toFinset_mono <| s.inter_subset_left t) h.ge ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 h : \u2191\u2191(condCount s) t = 1 \u22a2 \u2191\u2191(condCount ?m.9264) ?m.9265 \u2260 0 ** rw [h] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 h : \u2191\u2191(condCount s) t = 1 \u22a2 1 \u2260 0 ** exact one_ne_zero ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hsf : Set.Finite s h : Finset.card (Set.Finite.toFinset (_ : Set.Finite (s \u2229 t))) = Finset.card (Set.Finite.toFinset hsf) this : s \u2229 t = s \u22a2 s \u2286 t ** exact this \u25b8 fun x hx => hx.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.mem_prod_list_ofFn ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s\u271d t : Finset \u03b1 a\u271d : \u03b1 m n : \u2115 a : \u03b1 s : Fin n \u2192 Finset \u03b1 \u22a2 a \u2208 List.prod (List.ofFn s) \u2194 \u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = a ** rw [\u2190 mem_coe, coe_list_prod, List.map_ofFn, Set.mem_prod_list_ofFn] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s\u271d t : Finset \u03b1 a\u271d : \u03b1 m n : \u2115 a : \u03b1 s : Fin n \u2192 Finset \u03b1 \u22a2 (\u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = a) \u2194 \u2203 f, List.prod (List.ofFn fun i => \u2191(f i)) = a ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.ppred ** this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1 n : \u2115 \u22a2 (Nat.rfind fun n_1 =>       (fun m => decide (m = 0)) <$> \u2191(unpaired fun n m => if n = Nat.succ m then 0 else 1) (Nat.pair n n_1)) =     \u2191(Nat.ppred n) ** cases n <;> simp ** case zero this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1 \u22a2 (Nat.rfind fun n => Part.some false) = Part.none ** exact\n  eq_none_iff.2 fun a \u27e8\u27e8m, h, _\u27e9, _\u27e9 => by\n    simp [show 0 \u2260 m.succ by intro h; injection h] at h ** this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1 a : \u2115 x\u271d : a \u2208 Nat.rfind fun n => Part.some false m : \u2115 h : true \u2208 (fun n => Part.some false) m right\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom h\u271d :   Part.get (Nat.rfind fun n => Part.some false)       (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =     a \u22a2 False ** simp [show 0 \u2260 m.succ by intro h; injection h] at h ** this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1 a : \u2115 x\u271d : a \u2208 Nat.rfind fun n => Part.some false m : \u2115 h : true \u2208 (fun n => Part.some false) m right\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom h\u271d :   Part.get (Nat.rfind fun n => Part.some false)       (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =     a \u22a2 0 \u2260 Nat.succ m ** intro h ** this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1 a : \u2115 x\u271d : a \u2208 Nat.rfind fun n => Part.some false m : \u2115 h\u271d\u00b9 : true \u2208 (fun n => Part.some false) m right\u271d : \u2200 (k : \u2115), k < m \u2192 ((fun n => Part.some false) k).Dom h\u271d :   Part.get (Nat.rfind fun n => Part.some false)       (_ : \u2203 n, true \u2208 (fun n => Part.some false) n \u2227 \u2200 (k : \u2115), k < n \u2192 ((fun n => Part.some false) k).Dom) =     a h : 0 = Nat.succ m \u22a2 False ** injection h ** case succ this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1 n\u271d : \u2115 \u22a2 (Nat.rfind fun n => Part.some (decide (\u00acn\u271d = n \u2192 False))) = Part.some n\u271d ** refine' eq_some_iff.2 _ ** case succ this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1 n\u271d : \u2115 \u22a2 n\u271d \u2208 Nat.rfind fun n => Part.some (decide (\u00acn\u271d = n \u2192 False)) ** simp only [mem_rfind, not_true, IsEmpty.forall_iff, decide_True, mem_some_iff,\n  Bool.false_eq_decide_iff, true_and] ** case succ this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1 n\u271d : \u2115 \u22a2 \u2200 {m : \u2115}, m < n\u271d \u2192 \u00ac(\u00acn\u271d = m \u2192 False) ** intro m h ** case succ this : Primrec\u2082 fun n m => if n = Nat.succ m then 0 else 1 n\u271d m : \u2115 h : m < n\u271d \u22a2 \u00ac(\u00acn\u271d = m \u2192 False) ** simp [ne_of_gt h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Acc.rec_eq_recC ** \u22a2 @rec = @Acc.recC ** funext \u03b1 r motive intro a t ** case h.h.h.h.h.h \u03b1 : Sort u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop motive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2 intro :   (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x) a : \u03b1 t : Acc r a \u22a2 rec intro t = Acc.recC intro t ** induction t with\n| intro x h ih =>\n  dsimp only [recC_intro intro h]\n  congr; funext y hr; exact ih _ hr ** case h.h.h.h.h.h.intro \u03b1 : Sort u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop motive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2 intro :   (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x) a x : \u03b1 h : \u2200 (y : \u03b1), r y x \u2192 Acc r y ih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y) \u22a2 rec intro (_ : Acc r x) = Acc.recC intro (_ : Acc r x) ** dsimp only [recC_intro intro h] ** case h.h.h.h.h.h.intro \u03b1 : Sort u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop motive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2 intro :   (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x) a x : \u03b1 h : \u2200 (y : \u03b1), r y x \u2192 Acc r y ih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y) \u22a2 (intro x h fun y a => rec intro (_ : Acc ?m.2399 y)) = intro x h fun y hr => Acc.recC intro (_ : Acc ?m.2399 y) ** congr ** case h.h.h.h.h.h.intro.e_h_ih \u03b1 : Sort u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop motive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2 intro :   (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x) a x : \u03b1 h : \u2200 (y : \u03b1), r y x \u2192 Acc r y ih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y) \u22a2 (fun y a => rec intro (_ : Acc ?m.2399 y)) = fun y hr => Acc.recC intro (_ : Acc ?m.2399 y) ** funext y hr ** case h.h.h.h.h.h.intro.e_h_ih.h.h \u03b1 : Sort u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop motive : (a : \u03b1) \u2192 Acc r a \u2192 Sort u_2 intro :   (x : \u03b1) \u2192 (h : \u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 (a : r y x) \u2192 motive y (_ : Acc r y)) \u2192 motive x (_ : Acc r x) a x : \u03b1 h : \u2200 (y : \u03b1), r y x \u2192 Acc r y ih : \u2200 (y : \u03b1) (a : r y x), rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y) y : \u03b1 hr : r y x \u22a2 rec intro (_ : Acc ?m.2399 y) = Acc.recC intro (_ : Acc ?m.2399 y) ** exact ih _ hr ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_map_of_stronglyMeasurable ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc ** by_cases hG : CompleteSpace G ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc  case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : \u00acCompleteSpace G \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc ** by_cases hfi : Integrable f (Measure.map \u03c6 \u03bc) ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc  case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : \u00acIntegrable f \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc ** borelize G ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc ** have : SeparableSpace (range f \u222a {0} : Set G) := hfm.separableSpace_range_union_singleton ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(range f \u222a {0}) \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc ** refine' tendsto_nhds_unique\n  (tendsto_integral_approxOn_of_measurable_of_range_subset hfm.measurable hfi _ Subset.rfl) _ ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(range f \u222a {0}) \u22a2 Tendsto     (fun n =>       SimpleFunc.integral (Measure.map \u03c6 \u03bc) (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n))     atTop (\ud835\udcdd (\u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc)) ** convert tendsto_integral_approxOn_of_measurable_of_range_subset (hfm.measurable.comp h\u03c6)\n  ((integrable_map_measure hfm.aestronglyMeasurable h\u03c6.aemeasurable).1 hfi) (range f \u222a {0})\n  (by simp [insert_subset_insert, Set.range_comp_subset_range]) using 1 ** case h.e'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(range f \u222a {0}) \u22a2 (fun n =>       SimpleFunc.integral (Measure.map \u03c6 \u03bc)         (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n)) =     fun n =>     SimpleFunc.integral \u03bc (approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) n) ** ext1 i ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(range f \u222a {0}) i : \u2115 \u22a2 SimpleFunc.integral (Measure.map \u03c6 \u03bc) (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) =     SimpleFunc.integral \u03bc (approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) ** simp only [SimpleFunc.approxOn_comp, SimpleFunc.integral_eq, Measure.map_apply, h\u03c6,\n  SimpleFunc.measurableSet_preimage, \u2190 preimage_comp, SimpleFunc.coe_comp] ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(range f \u222a {0}) i : \u2115 \u22a2 \u2211 x in SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i),       ENNReal.toReal (\u2191\u2191\u03bc (\u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {x})) \u2022         x =     \u2211 x in SimpleFunc.range (approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i),       ENNReal.toReal           (\u2191\u2191\u03bc (\u2191(approxOn (f \u2218 \u03c6) (_ : Measurable (f \u2218 \u03c6)) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u207b\u00b9' {x})) \u2022         x ** refine' (Finset.sum_subset (SimpleFunc.range_comp_subset_range _ h\u03c6) fun y _ hy => _).symm ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(range f \u222a {0}) i : \u2115 y : G x\u271d : y \u2208 SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) hy :   \u00acy \u2208       SimpleFunc.range         (SimpleFunc.comp (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u03c6 h\u03c6) \u22a2 ENNReal.toReal (\u2191\u2191\u03bc (\u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y})) \u2022 y =     0 ** rw [SimpleFunc.mem_range, \u2190 Set.preimage_singleton_eq_empty, SimpleFunc.coe_comp] at hy ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(range f \u222a {0}) i : \u2115 y : G x\u271d : y \u2208 SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) hy : \u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y} = \u2205 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc (\u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y})) \u2022 y =     0 ** rw [hy] ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(range f \u222a {0}) i : \u2115 y : G x\u271d : y \u2208 SimpleFunc.range (approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) hy : \u2191(approxOn f (_ : Measurable f) (range f \u222a {0}) 0 (_ : 0 \u2208 range f \u222a {0}) i) \u2218 \u03c6 \u207b\u00b9' {y} = \u2205 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc \u2205) \u2022 y = 0 ** simp ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : \u00acCompleteSpace G \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc ** simp [integral, hG] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : \u00acIntegrable f \u22a2 \u222b (y : \u03b2), f y \u2202Measure.map \u03c6 \u03bc = \u222b (x : \u03b1), f (\u03c6 x) \u2202\u03bc ** rw [integral_undef hfi, integral_undef] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : \u00acIntegrable f \u22a2 \u00acIntegrable fun x => f (\u03c6 x) ** exact fun hf\u03c6 => hfi ((integrable_map_measure hfm.aestronglyMeasurable h\u03c6.aemeasurable).2 hf\u03c6) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : Measurable \u03c6 f : \u03b2 \u2192 G hfm : StronglyMeasurable f hG : CompleteSpace G hfi : Integrable f this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(range f \u222a {0}) \u22a2 range (f \u2218 \u03c6) \u222a {0} \u2286 range f \u222a {0} ** simp [insert_subset_insert, Set.range_comp_subset_range] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.ker_int_castAddHom ** n : \u2115 \u22a2 AddMonoidHom.ker (Int.castAddHom (ZMod n)) = AddSubgroup.zmultiples \u2191n ** ext ** case h n : \u2115 x\u271d : \u2124 \u22a2 x\u271d \u2208 AddMonoidHom.ker (Int.castAddHom (ZMod n)) \u2194 x\u271d \u2208 AddSubgroup.zmultiples \u2191n ** rw [Int.mem_zmultiples_iff, AddMonoidHom.mem_ker, Int.coe_castAddHom,\n  int_cast_zmod_eq_zero_iff_dvd] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.smul_univ ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b2 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : MulAction \u03b1 \u03b2 s\u271d t : Finset \u03b2 a : \u03b1 b : \u03b2 inst\u271d : Fintype \u03b2 s : Finset \u03b1 hs : Finset.Nonempty s \u22a2 \u2191(s \u2022 univ) = \u2191univ ** push_cast ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b2 inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : MulAction \u03b1 \u03b2 s\u271d t : Finset \u03b2 a : \u03b1 b : \u03b2 inst\u271d : Fintype \u03b2 s : Finset \u03b1 hs : Finset.Nonempty s \u22a2 \u2191s \u2022 Set.univ = Set.univ ** exact Set.smul_univ hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smul ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 F \u22a2 \u222b (x : E) in f '' s, g x \u2202\u03bc = \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc ** rw [\u2190 restrict_map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs hf' hf,\n  (measurableEmbedding_of_fderivWithin hs hf' hf).integral_map] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 F \u22a2 \u222b (x : \u2191s),       g         (Set.restrict s f           x) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =     \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc ** have : \u2200 x : s, g (s.restrict f x) = (g \u2218 f) x := fun x => rfl ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 F this : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x \u22a2 \u222b (x : \u2191s),       g         (Set.restrict s f           x) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =     \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc ** simp only [this, ENNReal.ofReal] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 F this : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x \u22a2 \u222b (x : \u2191s),       (g \u2218 f) \u2191x \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => \u2191(Real.toNNReal |ContinuousLinearMap.det (f' x)|)) =     \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc ** rw [\u2190 (MeasurableEmbedding.subtype_coe hs).integral_map, map_comap_subtype_coe hs,\n  set_integral_withDensity_eq_set_integral_smul\u2080\n    (aemeasurable_toNNReal_abs_det_fderivWithin \u03bc hs hf') _ hs] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 F this : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x \u22a2 \u222b (a : E) in s, Real.toNNReal |ContinuousLinearMap.det (f' a)| \u2022 (g \u2218 f) a \u2202\u03bc =     \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc ** congr with x ** case e_f.h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 F this : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x x : E \u22a2 Real.toNNReal |ContinuousLinearMap.det (f' x)| \u2022 (g \u2218 f) x = |ContinuousLinearMap.det (f' x)| \u2022 g (f x) ** conv_rhs => rw [\u2190 Real.coe_toNNReal _ (abs_nonneg (f' x).det)] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.valMinAbs_zero ** \u22a2 valMinAbs 0 = 0 ** simp only [valMinAbs_def_zero] ** n : \u2115 \u22a2 valMinAbs 0 = 0 ** simp only [valMinAbs_def_pos, if_true, Int.ofNat_zero, zero_le, val_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.tendsto_preCdf_atBot_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0) ** suffices \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) by\n  filter_upwards [this] with a ha\n  have h_eq_neg : (fun r : \u211a => preCdf \u03c1 r a) = fun r : \u211a => preCdf \u03c1 (- -r) a := by\n    simp_rw [neg_neg]\n  rw [h_eq_neg]\n  exact ha.comp tendsto_neg_atBot_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 this : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0) ** filter_upwards [this] with a ha ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 this : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) a : \u03b1 ha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0) ** have h_eq_neg : (fun r : \u211a => preCdf \u03c1 r a) = fun r : \u211a => preCdf \u03c1 (- -r) a := by\n  simp_rw [neg_neg] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 this : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) a : \u03b1 ha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) h_eq_neg : (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a \u22a2 Tendsto (fun r => preCdf \u03c1 r a) atBot (\ud835\udcdd 0) ** rw [h_eq_neg] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 this : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) a : \u03b1 ha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) h_eq_neg : (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a \u22a2 Tendsto (fun r => preCdf \u03c1 (- -r) a) atBot (\ud835\udcdd 0) ** exact ha.comp tendsto_neg_atBot_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 this : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) a : \u03b1 ha : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) \u22a2 (fun r => preCdf \u03c1 r a) = fun r => preCdf \u03c1 (- -r) a ** simp_rw [neg_neg] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) ** filter_upwards [monotone_preCdf \u03c1] with a ha ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 a : \u03b1 ha : Monotone fun r => preCdf \u03c1 r a \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) ** have h_anti : Antitone fun r => preCdf \u03c1 (-r) a := fun p q hpq => ha (neg_le_neg hpq) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 a : \u03b1 ha : Monotone fun r => preCdf \u03c1 r a h_anti : Antitone fun r => preCdf \u03c1 (-r) a \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) ** have h_tendsto :\n  Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot \u2228\n    \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) :=\n  tendsto_of_antitone h_anti ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 a : \u03b1 ha : Monotone fun r => preCdf \u03c1 r a h_anti : Antitone fun r => preCdf \u03c1 (-r) a h_tendsto : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) ** cases' h_tendsto with h_bot h_tendsto ** case h.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 a : \u03b1 ha : Monotone fun r => preCdf \u03c1 r a h_anti : Antitone fun r => preCdf \u03c1 (-r) a h_bot : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop atBot \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) ** exact \u27e80, Tendsto.mono_right h_bot atBot_le_nhds_bot\u27e9 ** case h.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 a : \u03b1 ha : Monotone fun r => preCdf \u03c1 r a h_anti : Antitone fun r => preCdf \u03c1 (-r) a h_tendsto : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) ** exact h_tendsto ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) ** let F : \u03b1 \u2192 \u211d\u22650\u221e := fun a =>\n  if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then h.choose else 0 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) ** have h_tendsto : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_exists] with a ha\n  simp_rw [dif_pos ha]\n  exact ha.choose_spec ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) ** suffices h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = 0 ** case h_lintegral_eq \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) \u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0 ** have h_lintegral' : Tendsto (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) atTop (\ud835\udcdd 0) := by\n  have h_lintegral_eq :\n    (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) = fun r : \u211a => \u03c1 (univ \u00d7\u02e2 Iic (-r : \u211d)) := by\n    ext1 n\n    rw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ,\n      Measure.IicSnd_univ]\n    norm_cast\n  rw [h_lintegral_eq]\n  have h_zero_eq_measure_iInter : (0 : \u211d\u22650\u221e) = \u03c1 (\u22c2 r : \u211a, univ \u00d7\u02e2 Iic (-r : \u211d)) := by\n    suffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty]\n    ext1 x\n    simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]\n    simp_rw [neg_lt]\n    exact exists_rat_gt _\n  rw [h_zero_eq_measure_iInter]\n  refine'\n    tendsto_measure_iInter (fun n => MeasurableSet.univ.prod measurableSet_Iic)\n      (fun i j hij x => _) \u27e80, measure_ne_top \u03c1 _\u27e9\n  simp only [mem_prod, mem_univ, mem_Iic, true_and_iff]\n  refine' fun hxj => hxj.trans (neg_le_neg _)\n  exact_mod_cast hij ** case h_lintegral_eq \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral' : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0) \u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0 ** exact tendsto_nhds_unique h_lintegral h_lintegral' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) ** filter_upwards [h_exists] with a ha ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 a : \u03b1 ha : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) \u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) ** simp_rw [dif_pos ha] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 a : \u03b1 ha : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) \u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (Exists.choose ha)) ** exact ha.choose_spec ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) ** have hF_ae_meas : AEMeasurable F \u03c1.fst := by\n  refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto\n  exact measurable_preCdf.aemeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0 hF_ae_meas : AEMeasurable F \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) ** rw [lintegral_eq_zero_iff' hF_ae_meas] at h_lintegral_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral_eq : F =\u1d50[Measure.fst \u03c1] 0 hF_ae_meas : AEMeasurable F \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) ** filter_upwards [h_tendsto, h_lintegral_eq] with a ha_tendsto ha_eq ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral_eq : F =\u1d50[Measure.fst \u03c1] 0 hF_ae_meas : AEMeasurable F a : \u03b1 ha_tendsto : Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) ha_eq : F a = OfNat.ofNat 0 a \u22a2 Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd 0) ** rwa [ha_eq] at ha_tendsto ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0 \u22a2 AEMeasurable F ** refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = 0 n : \u211a \u22a2 AEMeasurable fun x => preCdf \u03c1 (-n) x ** exact measurable_preCdf.aemeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) \u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) ** refine'\n  tendsto_lintegral_filter_of_dominated_convergence (fun _ => 1)\n    (eventually_of_forall fun _ => measurable_preCdf) (eventually_of_forall fun _ => _) _\n    h_tendsto ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) x\u271d : \u211a \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, preCdf \u03c1 (-x\u271d) a \u2264 (fun x => 1) a ** filter_upwards [preCdf_le_one \u03c1] with a ha using ha _ ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) \u22a2 \u222b\u207b (a : \u03b1), (fun x => 1) a \u2202Measure.fst \u03c1 \u2260 \u22a4 ** rw [lintegral_one] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) \u22a2 \u2191\u2191(Measure.fst \u03c1) univ \u2260 \u22a4 ** exact measure_ne_top _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) \u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0) ** have h_lintegral_eq :\n  (fun r => \u222b\u207b a, preCdf \u03c1 (-r) a \u2202\u03c1.fst) = fun r : \u211a => \u03c1 (univ \u00d7\u02e2 Iic (-r : \u211d)) := by\n  ext1 n\n  rw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ,\n    Measure.IicSnd_univ]\n  norm_cast ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) \u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd 0) ** rw [h_lintegral_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd 0) ** have h_zero_eq_measure_iInter : (0 : \u211d\u22650\u221e) = \u03c1 (\u22c2 r : \u211a, univ \u00d7\u02e2 Iic (-r : \u211d)) := by\n  suffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty]\n  ext1 x\n  simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]\n  simp_rw [neg_lt]\n  exact exists_rat_gt _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) h_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r)) \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd 0) ** rw [h_zero_eq_measure_iInter] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) h_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r)) \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r)))) ** refine'\n  tendsto_measure_iInter (fun n => MeasurableSet.univ.prod measurableSet_Iic)\n    (fun i j hij x => _) \u27e80, measure_ne_top \u03c1 _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) h_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r)) i j : \u211a hij : i \u2264 j x : \u03b1 \u00d7 \u211d \u22a2 x \u2208 univ \u00d7\u02e2 Iic (-\u2191j) \u2192 x \u2208 univ \u00d7\u02e2 Iic (-\u2191i) ** simp only [mem_prod, mem_univ, mem_Iic, true_and_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) h_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r)) i j : \u211a hij : i \u2264 j x : \u03b1 \u00d7 \u211d \u22a2 x.2 \u2264 -\u2191j \u2192 x.2 \u2264 -\u2191i ** refine' fun hxj => hxj.trans (neg_le_neg _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) h_zero_eq_measure_iInter : 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r)) i j : \u211a hij : i \u2264 j x : \u03b1 \u00d7 \u211d hxj : x.2 \u2264 -\u2191j \u22a2 \u2191i \u2264 \u2191j ** exact_mod_cast hij ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) \u22a2 (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) ** ext1 n ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) n : \u211a \u22a2 \u222b\u207b (a : \u03b1), preCdf \u03c1 (-n) a \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191n)) ** rw [\u2190 set_lintegral_univ, set_lintegral_preCdf_fst \u03c1 _ MeasurableSet.univ,\n  Measure.IicSnd_univ] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) n : \u211a \u22a2 \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic \u2191(-n)) = \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191n)) ** norm_cast ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) \u22a2 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r)) ** suffices \u22c2 r : \u211a, Iic (-(r : \u211d)) = \u2205 by rw [\u2190 prod_iInter, this, prod_empty, measure_empty] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) \u22a2 \u22c2 r, Iic (-\u2191r) = \u2205 ** ext1 x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) x : \u211d \u22a2 x \u2208 \u22c2 r, Iic (-\u2191r) \u2194 x \u2208 \u2205 ** simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) x : \u211d \u22a2 \u2203 x_1, -\u2191x_1 < x ** simp_rw [neg_lt] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) x : \u211d \u22a2 \u2203 x_1, -x < \u2191x_1 ** exact exists_rat_gt _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 (-r) a) atTop (\ud835\udcdd (F a)) h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral_eq : (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (-r) a \u2202Measure.fst \u03c1) = fun r => \u2191\u2191\u03c1 (univ \u00d7\u02e2 Iic (-\u2191r)) this : \u22c2 r, Iic (-\u2191r) = \u2205 \u22a2 0 = \u2191\u2191\u03c1 (\u22c2 r, univ \u00d7\u02e2 Iic (-\u2191r)) ** rw [\u2190 prod_iInter, this, prod_empty, measure_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "Measurable.lintegral_kernel_prod_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a ** let F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a ** have h : \u2200 a, \u2a06 n, F n a = uncurry f a := SimpleFunc.iSup_eapprox_apply (uncurry f) hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1 \u00d7 \u03b2), \u2a06 n, \u2191(F n) a = uncurry f a \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a ** simp only [Prod.forall, uncurry_apply_pair] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), f a b \u2202\u2191\u03ba a ** simp_rw [\u2190 h] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) \u2202\u2191\u03ba a ** simp_rw [this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a \u22a2 Measurable fun a => \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) \u2202\u2191\u03ba a ** refine' measurable_iSup fun n => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) \u2202\u2191\u03ba a ** refine' SimpleFunc.induction\n  (P := fun f => Measurable (fun (a : \u03b1) => \u222b\u207b (b : \u03b2), f (a, b) \u2202\u03ba a)) _ _ (F n) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b \u22a2 \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a ** intro a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a\u271d : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b a : \u03b1 \u22a2 \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a ** rw [lintegral_iSup] ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a\u271d : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b a : \u03b1 \u22a2 \u2200 (n : \u2115), Measurable fun b => \u2191(F n) (a, b) ** exact fun n => (F n).measurable.comp measurable_prod_mk_left ** case h_mono \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a\u271d : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b a : \u03b1 \u22a2 Monotone fun n b => \u2191(F n) (a, b) ** exact fun i j hij b => SimpleFunc.monotone_eapprox (uncurry f) hij _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 \u22a2 \u2200 (c : \u211d\u22650\u221e) {s : Set (\u03b1 \u00d7 \u03b2)} (hs : MeasurableSet s),     (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a)       (SimpleFunc.piecewise s hs (SimpleFunc.const (\u03b1 \u00d7 \u03b2) c) (SimpleFunc.const (\u03b1 \u00d7 \u03b2) 0)) ** intro c t ht ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 c : \u211d\u22650\u221e t : Set (\u03b1 \u00d7 \u03b2) ht : MeasurableSet t \u22a2 Measurable fun a =>     \u222b\u207b (b : \u03b2), \u2191(SimpleFunc.piecewise t ht (SimpleFunc.const (\u03b1 \u00d7 \u03b2) c) (SimpleFunc.const (\u03b1 \u00d7 \u03b2) 0)) (a, b) \u2202\u2191\u03ba a ** simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,\n  SimpleFunc.coe_zero, Set.piecewise_eq_indicator] ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 c : \u211d\u22650\u221e t : Set (\u03b1 \u00d7 \u03b2) ht : MeasurableSet t \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), Set.piecewise t (Function.const (\u03b1 \u00d7 \u03b2) c) 0 (a, b) \u2202\u2191\u03ba a ** exact kernel.measurable_lintegral_indicator_const (\u03ba := \u03ba) ht c ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 \u22a2 \u2200 \u2983f g : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e\u2984,     Disjoint (support \u2191f) (support \u2191g) \u2192       (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a) f \u2192         (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a) g \u2192           (fun f => Measurable fun a => \u222b\u207b (b : \u03b2), \u2191f (a, b) \u2202\u2191\u03ba a) (f + g) ** intro g\u2081 g\u2082 _ hm\u2081 hm\u2082 ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 g\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e a\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082) hm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a hm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191(g\u2081 + g\u2082) (a, b) \u2202\u2191\u03ba a ** simp only [SimpleFunc.coe_add, Pi.add_apply] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 g\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e a\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082) hm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a hm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a ** have h_add :\n  (fun a => \u222b\u207b b, g\u2081 (a, b) + g\u2082 (a, b) \u2202\u03ba a) =\n    (fun a => \u222b\u207b b, g\u2081 (a, b) \u2202\u03ba a) + fun a => \u222b\u207b b, g\u2082 (a, b) \u2202\u03ba a := by\n  ext1 a\n  rw [Pi.add_apply]\n  erw [lintegral_add_left (g\u2081.measurable.comp measurable_prod_mk_left)]\n  simp_rw [Function.comp_apply] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 g\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e a\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082) hm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a hm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a h_add :   (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) =     (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a \u22a2 Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a ** rw [h_add] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 g\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e a\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082) hm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a hm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a h_add :   (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) =     (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a \u22a2 Measurable ((fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) ** exact Measurable.add hm\u2081 hm\u2082 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 g\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e a\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082) hm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a hm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a \u22a2 (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) =     (fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a ** ext1 a ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a\u271d\u00b9 : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 g\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e a\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082) hm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a hm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a a : \u03b1 \u22a2 \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a =     ((fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a) + fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a) a ** rw [Pi.add_apply] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a\u271d\u00b9 : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 g\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e a\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082) hm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a hm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a a : \u03b1 \u22a2 \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) + \u2191g\u2082 (a, b) \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a + \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a ** erw [lintegral_add_left (g\u2081.measurable.comp measurable_prod_mk_left)] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } a\u271d\u00b9 : \u03b1 inst\u271d\u00b9 : IsSFiniteKernel \u03ba inst\u271d : IsSFiniteKernel \u03b7 f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (uncurry f) F : \u2115 \u2192 SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e := SimpleFunc.eapprox (uncurry f) h : \u2200 (a : \u03b1) (b : \u03b2), \u2a06 n, \u2191(SimpleFunc.eapprox (uncurry f) n) (a, b) = f a b this : \u2200 (a : \u03b1), \u222b\u207b (b : \u03b2), \u2a06 n, \u2191(F n) (a, b) \u2202\u2191\u03ba a = \u2a06 n, \u222b\u207b (b : \u03b2), \u2191(F n) (a, b) \u2202\u2191\u03ba a n : \u2115 g\u2081 g\u2082 : SimpleFunc (\u03b1 \u00d7 \u03b2) \u211d\u22650\u221e a\u271d : Disjoint (support \u2191g\u2081) (support \u2191g\u2082) hm\u2081 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a hm\u2082 : Measurable fun a => \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a a : \u03b1 \u22a2 \u222b\u207b (a_1 : \u03b2), (\u2191g\u2081 \u2218 Prod.mk a) a_1 \u2202\u2191\u03ba a + \u222b\u207b (a_1 : \u03b2), \u2191g\u2082 (a, a_1) \u2202\u2191\u03ba a =     \u222b\u207b (b : \u03b2), \u2191g\u2081 (a, b) \u2202\u2191\u03ba a + \u222b\u207b (b : \u03b2), \u2191g\u2082 (a, b) \u2202\u2191\u03ba a ** simp_rw [Function.comp_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.norm_set_integral_le_of_norm_le_const_ae ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E C : \u211d hs : \u2191\u2191\u03bc s < \u22a4 hC : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016f x\u2016 \u2264 C \u22a2 \u2016\u222b (x : \u03b1) in s, f x \u2202\u03bc\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s) ** rw [\u2190 Measure.restrict_apply_univ] at * ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E C : \u211d hs : \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4 hC : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016f x\u2016 \u2264 C \u22a2 \u2016\u222b (x : \u03b1) in s, f x \u2202\u03bc\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc s) univ) ** haveI : IsFiniteMeasure (\u03bc.restrict s) := \u27e8\u2039_\u203a\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E C : \u211d hs : \u2191\u2191(Measure.restrict \u03bc s) univ < \u22a4 hC : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016f x\u2016 \u2264 C this : IsFiniteMeasure (Measure.restrict \u03bc s) \u22a2 \u2016\u222b (x : \u03b1) in s, f x \u2202\u03bc\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191(Measure.restrict \u03bc s) univ) ** exact norm_integral_le_of_norm_le_const hC ** Qed",
        "informal": ""
    },
    {
        "formal": "IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul' ** \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d hK : 0 < K \u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : \u03b1), \u2191\u2191\u03bc (closedBall x r) \u2264 \u2191(scalingConstantOf \u03bc K\u207b\u00b9) * \u2191\u2191\u03bc (closedBall x (K * r)) ** convert eventually_nhdsWithin_pos_mul_left hK (eventually_measure_le_scaling_constant_mul \u03bc K\u207b\u00b9) ** case h.e'_2.h.h.h.e'_3.h.e'_3.h.e'_4 \u03b1 : Type u_1 inst\u271d\u00b2 : MetricSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsUnifLocDoublingMeasure \u03bc K : \u211d hK : 0 < K x\u271d : \u211d a\u271d : \u03b1 \u22a2 x\u271d = K\u207b\u00b9 * (K * x\u271d) ** simp [inv_mul_cancel_left\u2080 hK.ne'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.restrict_finset_biUnion_congr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 s : Finset \u03b9 t : \u03b9 \u2192 Set \u03b1 \u22a2 restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i) ** induction' s using Finset.induction_on with i s _ hs ** case insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 t : \u03b9 \u2192 Set \u03b1 i : \u03b9 s : Finset \u03b9 a\u271d : \u00aci \u2208 s hs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i) \u22a2 restrict \u03bc (\u22c3 i_1 \u2208 insert i s, t i_1) = restrict \u03bd (\u22c3 i_1 \u2208 insert i s, t i_1) \u2194     \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 restrict \u03bc (t i_1) = restrict \u03bd (t i_1) ** simp only [forall_eq_or_imp, iUnion_iUnion_eq_or_left, Finset.mem_insert] ** case insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 t : \u03b9 \u2192 Set \u03b1 i : \u03b9 s : Finset \u03b9 a\u271d : \u00aci \u2208 s hs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i) \u22a2 restrict \u03bc (t i \u222a \u22c3 x \u2208 s, t x) = restrict \u03bd (t i \u222a \u22c3 x \u2208 s, t x) \u2194     restrict \u03bc (t i) = restrict \u03bd (t i) \u2227 \u2200 (a : \u03b9), a \u2208 s \u2192 restrict \u03bc (t a) = restrict \u03bd (t a) ** rw [restrict_union_congr, \u2190 hs] ** case empty \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 t : \u03b9 \u2192 Set \u03b1 \u22a2 restrict \u03bc (\u22c3 i \u2208 \u2205, t i) = restrict \u03bd (\u22c3 i \u2208 \u2205, t i) \u2194 \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 restrict \u03bc (t i) = restrict \u03bd (t i) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "List.range'_eq_range'TR ** \u22a2 range' = range'TR ** funext s n step ** case h.h.h s n : Nat step : optParam Nat 1 \u22a2 range' s n step = range'TR s n step ** exact (go s n 0).symm ** s\u271d n : Nat step : optParam Nat 1 s m : Nat \u22a2 range'TR.go step 0 (s + step * 0) (range' (s + step * 0) m step) = range' s (0 + m) step ** simp [range'TR.go] ** s\u271d n\u271d : Nat step : optParam Nat 1 s n m : Nat \u22a2 range'TR.go step (n + 1) (s + step * (n + 1)) (range' (s + step * (n + 1)) m step) = range' s (n + 1 + m) step ** simp [range'TR.go] ** s\u271d n\u271d : Nat step : optParam Nat 1 s n m : Nat \u22a2 range'TR.go step n (s + step * (n + 1) - step) ((s + step * (n + 1) - step) :: range' (s + step * (n + 1)) m step) =     range' s (n + 1 + m) step ** rw [Nat.mul_succ, \u2190 Nat.add_assoc, Nat.add_sub_cancel, Nat.add_right_comm n] ** s\u271d n\u271d : Nat step : optParam Nat 1 s n m : Nat \u22a2 range'TR.go step n (s + step * n) ((s + step * n) :: range' (s + step * n + step) m step) = range' s (n + m + 1) step ** exact go s n (m + 1) ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.get?_push_eq ** \u03b1 : Type u_1 a : Array \u03b1 x : \u03b1 \u22a2 (push a x)[size a]? = some x ** rw [getElem?_pos, get_push_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.get?_set_eq ** \u03b1 : Type u_1 a : \u03b1 n : Nat l : List \u03b1 \u22a2 get? (set l n a) n = (fun x => a) <$> get? l n ** simp only [set_eq_modifyNth, get?_modifyNth_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpIndSMul_nonneg ** \u03b1 : Type u_1 E\u271d : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d inst\u271d\u00b2\u2070 : CompleteSpace E\u271d inst\u271d\u00b9\u2079 : NormedAddCommGroup E' inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2077 : CompleteSpace E' inst\u271d\u00b9\u2076 : NormedSpace \u211d E' inst\u271d\u00b9\u2075 : NormedAddCommGroup F inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b3 : NormedAddCommGroup G inst\u271d\u00b9\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9\u00b9 : NormedSpace \u211d G' inst\u271d\u00b9\u2070 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2079 : IsROrC \ud835\udd5c' inst\u271d\u2078 : NormedAddCommGroup E'' inst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u2076 : CompleteSpace E'' inst\u271d\u2075 : NormedSpace \u211d E'' inst\u271d\u2074 : NormedSpace \u211d G hm : m \u2264 m0 E : Type u_10 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OrderedSMul \u211d E inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E hx : 0 \u2264 x \u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ** refine' EventuallyLE.trans_eq _ (condexpIndSMul_ae_eq_smul hm hs h\u03bcs x).symm ** \u03b1 : Type u_1 E\u271d : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d inst\u271d\u00b2\u2070 : CompleteSpace E\u271d inst\u271d\u00b9\u2079 : NormedAddCommGroup E' inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2077 : CompleteSpace E' inst\u271d\u00b9\u2076 : NormedSpace \u211d E' inst\u271d\u00b9\u2075 : NormedAddCommGroup F inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b3 : NormedAddCommGroup G inst\u271d\u00b9\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9\u00b9 : NormedSpace \u211d G' inst\u271d\u00b9\u2070 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2079 : IsROrC \ud835\udd5c' inst\u271d\u2078 : NormedAddCommGroup E'' inst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u2076 : CompleteSpace E'' inst\u271d\u2075 : NormedSpace \u211d E'' inst\u271d\u2074 : NormedSpace \u211d G hm : m \u2264 m0 E : Type u_10 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OrderedSMul \u211d E inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E hx : 0 \u2264 x \u22a2 0 \u2264\u1d50[\u03bc] fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x ** filter_upwards [condexpL2_indicator_nonneg hm hs h\u03bcs] with a ha ** case h \u03b1 : Type u_1 E\u271d : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d inst\u271d\u00b2\u2070 : CompleteSpace E\u271d inst\u271d\u00b9\u2079 : NormedAddCommGroup E' inst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u2077 : CompleteSpace E' inst\u271d\u00b9\u2076 : NormedSpace \u211d E' inst\u271d\u00b9\u2075 : NormedAddCommGroup F inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b3 : NormedAddCommGroup G inst\u271d\u00b9\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9\u00b9 : NormedSpace \u211d G' inst\u271d\u00b9\u2070 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2079 : IsROrC \ud835\udd5c' inst\u271d\u2078 : NormedAddCommGroup E'' inst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u2076 : CompleteSpace E'' inst\u271d\u2075 : NormedSpace \u211d E'' inst\u271d\u2074 : NormedSpace \u211d G hm : m \u2264 m0 E : Type u_10 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OrderedSMul \u211d E inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E hx : 0 \u2264 x a : \u03b1 ha : OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u22a2 OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x ** exact smul_nonneg ha hx ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.mul_inv_eq_gcd ** n : \u2115 a : ZMod n \u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) n) ** cases' n with n ** case zero a : ZMod Nat.zero \u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) Nat.zero) ** dsimp [ZMod] at a \u22a2 ** case zero a : \u2124 \u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) 0) ** calc\n  _ = a * Int.sign a := rfl\n  _ = a.natAbs := by rw [Int.mul_sign]\n  _ = a.natAbs.gcd 0 := by rw [Nat.gcd_zero_right] ** a : \u2124 \u22a2 a * Int.sign a = \u2191(Int.natAbs a) ** rw [Int.mul_sign] ** a : \u2124 \u22a2 \u2191(Int.natAbs a) = \u2191(Nat.gcd (Int.natAbs a) 0) ** rw [Nat.gcd_zero_right] ** case succ n : \u2115 a : ZMod (Nat.succ n) \u22a2 a * a\u207b\u00b9 = \u2191(Nat.gcd (val a) (Nat.succ n)) ** calc\n  a * a\u207b\u00b9 = a * a\u207b\u00b9 + n.succ * Nat.gcdB (val a) n.succ := by\n    rw [nat_cast_self, zero_mul, add_zero]\n  _ = \u2191(\u2191a.val * Nat.gcdA (val a) n.succ + n.succ * Nat.gcdB (val a) n.succ) := by\n    push_cast\n    rw [nat_cast_zmod_val]\n    rfl\n  _ = Nat.gcd a.val n.succ := by rw [\u2190 Nat.gcd_eq_gcd_ab a.val n.succ]; rfl ** n : \u2115 a : ZMod (Nat.succ n) \u22a2 a * a\u207b\u00b9 = a * a\u207b\u00b9 + \u2191(Nat.succ n) * \u2191(Nat.gcdB (val a) (Nat.succ n)) ** rw [nat_cast_self, zero_mul, add_zero] ** n : \u2115 a : ZMod (Nat.succ n) \u22a2 a * a\u207b\u00b9 + \u2191(Nat.succ n) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =     \u2191(\u2191(val a) * Nat.gcdA (val a) (Nat.succ n) + \u2191(Nat.succ n) * Nat.gcdB (val a) (Nat.succ n)) ** push_cast ** n : \u2115 a : ZMod (Nat.succ n) \u22a2 a * a\u207b\u00b9 + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =     \u2191(val a) * \u2191(Nat.gcdA (val a) (Nat.succ n)) + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) ** rw [nat_cast_zmod_val] ** n : \u2115 a : ZMod (Nat.succ n) \u22a2 a * a\u207b\u00b9 + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) =     a * \u2191(Nat.gcdA (val a) (Nat.succ n)) + (\u2191n + 1) * \u2191(Nat.gcdB (val a) (Nat.succ n)) ** rfl ** n : \u2115 a : ZMod (Nat.succ n) \u22a2 \u2191(\u2191(val a) * Nat.gcdA (val a) (Nat.succ n) + \u2191(Nat.succ n) * Nat.gcdB (val a) (Nat.succ n)) =     \u2191(Nat.gcd (val a) (Nat.succ n)) ** rw [\u2190 Nat.gcd_eq_gcd_ab a.val n.succ] ** n : \u2115 a : ZMod (Nat.succ n) \u22a2 \u2191\u2191(Nat.gcd (val a) (Nat.succ n)) = \u2191(Nat.gcd (val a) (Nat.succ n)) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.range_list_map_coe ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : List \u03b1 s : Set \u03b1 \u22a2 range (map Subtype.val) = {l | \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 s} ** rw [range_list_map, Subtype.range_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.get?_swap ** \u03b1 : Type u_1 a : Array \u03b1 i j : Fin (size a) k : Nat \u22a2 (swap a i j)[k]? = if j.val = k then some a[i.val] else if i.val = k then some a[j.val] else a[k]? ** simp [swap_def, get?_set, \u2190 getElem_fin_eq_data_get] ** Qed",
        "informal": ""
    },
    {
        "formal": "integral_comp_neg_Ioi ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E c : \u211d f : \u211d \u2192 E \u22a2 \u222b (x : \u211d) in Ioi c, f (-x) = \u222b (x : \u211d) in Iic (-c), f x ** rw [\u2190 neg_neg c, \u2190 integral_comp_neg_Iic] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E c : \u211d f : \u211d \u2192 E \u22a2 \u222b (x : \u211d) in Iic (-c), f (- -x) = \u222b (x : \u211d) in Iic (- - -c), f x ** simp only [neg_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.WeaklyRegular.restrict_of_measurableSet ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : WeaklyRegular \u03bc A : Set \u03b1 hA : MeasurableSet A h'A : \u2191\u2191\u03bc A \u2260 \u22a4 \u22a2 WeaklyRegular (restrict \u03bc A) ** haveI : Fact (\u03bc A < \u221e) := \u27e8h'A.lt_top\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : WeaklyRegular \u03bc A : Set \u03b1 hA : MeasurableSet A h'A : \u2191\u2191\u03bc A \u2260 \u22a4 this : Fact (\u2191\u2191\u03bc A < \u22a4) \u22a2 WeaklyRegular (restrict \u03bc A) ** refine' InnerRegular.weaklyRegular_of_finite (\u03bc.restrict A) fun V V_open => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : WeaklyRegular \u03bc A : Set \u03b1 hA : MeasurableSet A h'A : \u2191\u2191\u03bc A \u2260 \u22a4 this : Fact (\u2191\u2191\u03bc A < \u22a4) V : Set \u03b1 V_open : IsOpen V \u22a2 \u2200 (r : \u211d\u22650\u221e), r < \u2191\u2191(restrict \u03bc A) V \u2192 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191(restrict \u03bc A) K ** simp only [restrict_apply' hA] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : WeaklyRegular \u03bc A : Set \u03b1 hA : MeasurableSet A h'A : \u2191\u2191\u03bc A \u2260 \u22a4 this : Fact (\u2191\u2191\u03bc A < \u22a4) V : Set \u03b1 V_open : IsOpen V \u22a2 \u2200 (r : \u211d\u22650\u221e), r < \u2191\u2191\u03bc (V \u2229 A) \u2192 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A) ** intro r hr ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : WeaklyRegular \u03bc A : Set \u03b1 hA : MeasurableSet A h'A : \u2191\u2191\u03bc A \u2260 \u22a4 this : Fact (\u2191\u2191\u03bc A < \u22a4) V : Set \u03b1 V_open : IsOpen V r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc (V \u2229 A) \u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A) ** have : \u03bc (V \u2229 A) \u2260 \u221e := ne_top_of_le_ne_top h'A (measure_mono <| inter_subset_right _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : WeaklyRegular \u03bc A : Set \u03b1 hA : MeasurableSet A h'A : \u2191\u2191\u03bc A \u2260 \u22a4 this\u271d : Fact (\u2191\u2191\u03bc A < \u22a4) V : Set \u03b1 V_open : IsOpen V r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc (V \u2229 A) this : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4 \u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A) ** rcases (V_open.measurableSet.inter hA).exists_lt_isClosed_of_ne_top this hr with\n  \u27e8F, hFVA, hFc, hF\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : WeaklyRegular \u03bc A : Set \u03b1 hA : MeasurableSet A h'A : \u2191\u2191\u03bc A \u2260 \u22a4 this\u271d : Fact (\u2191\u2191\u03bc A < \u22a4) V : Set \u03b1 V_open : IsOpen V r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc (V \u2229 A) this : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4 F : Set \u03b1 hFVA : F \u2286 V \u2229 A hFc : IsClosed F hF : r < \u2191\u2191\u03bc F \u22a2 \u2203 K, K \u2286 V \u2227 IsClosed K \u2227 r < \u2191\u2191\u03bc (K \u2229 A) ** refine' \u27e8F, hFVA.trans (inter_subset_left _ _), hFc, _\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : WeaklyRegular \u03bc A : Set \u03b1 hA : MeasurableSet A h'A : \u2191\u2191\u03bc A \u2260 \u22a4 this\u271d : Fact (\u2191\u2191\u03bc A < \u22a4) V : Set \u03b1 V_open : IsOpen V r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc (V \u2229 A) this : \u2191\u2191\u03bc (V \u2229 A) \u2260 \u22a4 F : Set \u03b1 hFVA : F \u2286 V \u2229 A hFc : IsClosed F hF : r < \u2191\u2191\u03bc F \u22a2 r < \u2191\u2191\u03bc (F \u2229 A) ** rwa [inter_eq_self_of_subset_left (hFVA.trans <| inter_subset_right _ _)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.map_iInf ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 f : \u03b1 \u2192 \u03b2 hf : Injective f m : \u03b9 \u2192 OuterMeasure \u03b1 \u22a2 \u2191(map f) (\u2a05 i, m i) = \u2191(restrict (range f)) (\u2a05 i, \u2191(map f) (m i)) ** refine' Eq.trans _ (map_comap _ _) ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 f : \u03b1 \u2192 \u03b2 hf : Injective f m : \u03b9 \u2192 OuterMeasure \u03b1 \u22a2 \u2191(map f) (\u2a05 i, m i) = \u2191(map f) (\u2191(comap f) (\u2a05 i, \u2191(map f) (m i))) ** simp only [comap_iInf, comap_map hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.core_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 f\u271d f : \u03b1 \u2192. \u03b2 s : Set \u03b2 \u22a2 core f s = preimage f s \u222a (Dom f)\u1d9c ** rw [preimage_eq, Set.union_distrib_right, Set.union_comm (Dom f), Set.compl_union_self,\n  Set.inter_univ, Set.union_eq_self_of_subset_right (f.compl_dom_subset_core s)] ** Qed",
        "informal": ""
    },
    {
        "formal": "AddAction.orbitZmultiplesEquiv_symm_apply' ** n : \u2115 A : Type u_1 R : Type u_2 inst\u271d\u2075 : AddGroup A inst\u271d\u2074 : Ring R \u03b1\u271d : Type u_3 \u03b2\u271d : Type u_4 inst\u271d\u00b3 : Group \u03b1\u271d a\u271d : \u03b1\u271d inst\u271d\u00b2 : MulAction \u03b1\u271d \u03b2\u271d b\u271d : \u03b2\u271d \u03b1 : Type u_5 \u03b2 : Type u_6 inst\u271d\u00b9 : AddGroup \u03b1 a : \u03b1 inst\u271d : AddAction \u03b1 \u03b2 b : \u03b2 k : \u2124 \u22a2 \u2191(orbitZmultiplesEquiv a b).symm \u2191k =     k \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65       { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) } ** rw [AddAction.orbit_zmultiples_equiv_symm_apply, ZMod.coe_int_cast] ** n : \u2115 A : Type u_1 R : Type u_2 inst\u271d\u2075 : AddGroup A inst\u271d\u2074 : Ring R \u03b1\u271d : Type u_3 \u03b2\u271d : Type u_4 inst\u271d\u00b3 : Group \u03b1\u271d a\u271d : \u03b1\u271d inst\u271d\u00b2 : MulAction \u03b1\u271d \u03b2\u271d b\u271d : \u03b2\u271d \u03b1 : Type u_5 \u03b2 : Type u_6 inst\u271d\u00b9 : AddGroup \u03b1 a : \u03b1 inst\u271d : AddAction \u03b1 \u03b2 b : \u03b2 k : \u2124 \u22a2 (k % \u2191(minimalPeriod ((fun x x_1 => x +\u1d65 x_1) a) b)) \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65       { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) } =     k \u2022 { val := a, property := (_ : a \u2208 zmultiples a) } +\u1d65       { val := b, property := (_ : b \u2208 AddAction.orbit { x // x \u2208 zmultiples a } b) } ** exact Subtype.ext (zsmul_vadd_mod_minimalPeriod a b k) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.iSup_eapprox_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 K : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f a : \u03b1 \u22a2 \u2a06 n, \u2191(eapprox f n) a = f a ** rw [eapprox, iSup_approx_apply ennrealRatEmbed f a hf rfl] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 K : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f a : \u03b1 \u22a2 \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k = f a ** refine' le_antisymm (iSup_le fun i => iSup_le fun hi => hi) (le_of_not_gt _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 K : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f a : \u03b1 \u22a2 \u00acf a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k ** intro h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 K : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f a : \u03b1 h : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k \u22a2 False ** rcases ENNReal.lt_iff_exists_rat_btwn.1 h with \u27e8q, _, lt_q, q_lt\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 K : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f a : \u03b1 h : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k q : \u211a left\u271d : 0 \u2264 q lt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q) q_lt : \u2191(Real.toNNReal \u2191q) < f a \u22a2 False ** have :\n  (Real.toNNReal q : \u211d\u22650\u221e) \u2264 \u2a06 (k : \u2115) (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k := by\n  refine' le_iSup_of_le (Encodable.encode q) _\n  rw [ennrealRatEmbed_encode q]\n  exact le_iSup_of_le (le_of_lt q_lt) le_rfl ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 K : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f a : \u03b1 h : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k q : \u211a left\u271d : 0 \u2264 q lt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q) q_lt : \u2191(Real.toNNReal \u2191q) < f a this : \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k \u22a2 False ** exact lt_irrefl _ (lt_of_le_of_lt this lt_q) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 K : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f a : \u03b1 h : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k q : \u211a left\u271d : 0 \u2264 q lt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q) q_lt : \u2191(Real.toNNReal \u2191q) < f a \u22a2 \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k ** refine' le_iSup_of_le (Encodable.encode q) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 K : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f a : \u03b1 h : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k q : \u211a left\u271d : 0 \u2264 q lt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q) q_lt : \u2191(Real.toNNReal \u2191q) < f a \u22a2 \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 (_ : ennrealRatEmbed (Encodable.encode q) \u2264 f a), ennrealRatEmbed (Encodable.encode q) ** rw [ennrealRatEmbed_encode q] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 K : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f a : \u03b1 h : f a > \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k q : \u211a left\u271d : 0 \u2264 q lt_q : \u2a06 k, \u2a06 (_ : ennrealRatEmbed k \u2264 f a), ennrealRatEmbed k < \u2191(Real.toNNReal \u2191q) q_lt : \u2191(Real.toNNReal \u2191q) < f a \u22a2 \u2191(Real.toNNReal \u2191q) \u2264 \u2a06 (_ : \u2191(Real.toNNReal \u2191q) \u2264 f a), \u2191(Real.toNNReal \u2191q) ** exact le_iSup_of_le (le_of_lt q_lt) le_rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.one_lt_card_iff_nontrivial_coe ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 \u22a2 1 < card s \u2194 Nontrivial { x // x \u2208 s } ** rw [\u2190 not_iff_not, not_lt, not_nontrivial_iff_subsingleton, card_le_one_iff_subsingleton_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.pairwise_disjoint_fundamentalInterior ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9 : Group G inst\u271d : MulAction G \u03b1 s : Set \u03b1 x : \u03b1 \u22a2 Pairwise (Disjoint on fun g => g \u2022 fundamentalInterior G s) ** refine' fun a b hab => disjoint_left.2 _ ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9 : Group G inst\u271d : MulAction G \u03b1 s : Set \u03b1 x : \u03b1 a b : G hab : a \u2260 b \u22a2 \u2200 \u2983a_1 : \u03b1\u2984, a_1 \u2208 (fun g => g \u2022 fundamentalInterior G s) a \u2192 \u00aca_1 \u2208 (fun g => g \u2022 fundamentalInterior G s) b ** rintro _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, hxy\u27e9 ** case intro.intro.intro.intro G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9 : Group G inst\u271d : MulAction G \u03b1 s : Set \u03b1 x\u271d : \u03b1 a b : G hab : a \u2260 b x : \u03b1 hx : x \u2208 fundamentalInterior G s y : \u03b1 hy : y \u2208 fundamentalInterior G s hxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x \u22a2 False ** rw [mem_fundamentalInterior] at hx hy ** case intro.intro.intro.intro G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9 : Group G inst\u271d : MulAction G \u03b1 s : Set \u03b1 x\u271d : \u03b1 a b : G hab : a \u2260 b x : \u03b1 hx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s y : \u03b1 hy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s hxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x \u22a2 False ** refine' hx.2 (a\u207b\u00b9 * b) _ _ ** case intro.intro.intro.intro.refine'_1 G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9 : Group G inst\u271d : MulAction G \u03b1 s : Set \u03b1 x\u271d : \u03b1 a b : G hab : a \u2260 b x : \u03b1 hx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s y : \u03b1 hy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s hxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x \u22a2 a\u207b\u00b9 * b \u2260 1  case intro.intro.intro.intro.refine'_2 G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9 : Group G inst\u271d : MulAction G \u03b1 s : Set \u03b1 x\u271d : \u03b1 a b : G hab : a \u2260 b x : \u03b1 hx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s y : \u03b1 hy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s hxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x \u22a2 x \u2208 (a\u207b\u00b9 * b) \u2022 s ** rwa [Ne.def, inv_mul_eq_iff_eq_mul, mul_one, eq_comm] ** case intro.intro.intro.intro.refine'_2 G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9 : Group G inst\u271d : MulAction G \u03b1 s : Set \u03b1 x\u271d : \u03b1 a b : G hab : a \u2260 b x : \u03b1 hx : x \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acx \u2208 g \u2022 s y : \u03b1 hy : y \u2208 s \u2227 \u2200 (g : G), g \u2260 1 \u2192 \u00acy \u2208 g \u2022 s hxy : (fun x => b \u2022 x) y = (fun x => a \u2022 x) x \u22a2 x \u2208 (a\u207b\u00b9 * b) \u2022 s ** simpa [mul_smul, \u2190 hxy, mem_inv_smul_set_iff] using hy.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.isUnit_ne_iff_eq_neg ** u u' : \u2124 hu : IsUnit u hu' : IsUnit u' \u22a2 u \u2260 u' \u2194 u = -u' ** simpa only [Ne, Units.ext_iff] using units_ne_iff_eq_neg (u := hu.unit) (u' := hu'.unit) ** Qed",
        "informal": ""
    },
    {
        "formal": "range_circleMap ** E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d \u22a2 (c +\u1d65 R \u2022 range fun \u03b8 => cexp (\u2191\u03b8 * I)) = sphere c |R| ** rw [Complex.range_exp_mul_I, smul_sphere R 0 zero_le_one] ** E : Type u_1 inst\u271d : NormedAddCommGroup E c : \u2102 R : \u211d \u22a2 c +\u1d65 sphere (R \u2022 0) (\u2016R\u2016 * 1) = sphere c |R| ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "List.aestronglyMeasurable_prod' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2075 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 f g : \u03b1 \u2192 \u03b2 M : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : ContinuousMul M l : List (\u03b1 \u2192 M) hl : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc \u22a2 AEStronglyMeasurable (List.prod l) \u03bc ** induction' l with f l ihl ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2075 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 f\u271d g : \u03b1 \u2192 \u03b2 M : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : ContinuousMul M l\u271d : List (\u03b1 \u2192 M) hl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 M l : List (\u03b1 \u2192 M) ihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc) \u2192 AEStronglyMeasurable (List.prod l) \u03bc hl : \u2200 (f_1 : \u03b1 \u2192 M), f_1 \u2208 f :: l \u2192 AEStronglyMeasurable f_1 \u03bc \u22a2 AEStronglyMeasurable (List.prod (f :: l)) \u03bc ** rw [List.forall_mem_cons] at hl ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2075 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 f\u271d g : \u03b1 \u2192 \u03b2 M : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : ContinuousMul M l\u271d : List (\u03b1 \u2192 M) hl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 M l : List (\u03b1 \u2192 M) ihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc) \u2192 AEStronglyMeasurable (List.prod l) \u03bc hl : AEStronglyMeasurable f \u03bc \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEStronglyMeasurable x \u03bc \u22a2 AEStronglyMeasurable (List.prod (f :: l)) \u03bc ** rw [List.prod_cons] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2075 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 f\u271d g : \u03b1 \u2192 \u03b2 M : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : ContinuousMul M l\u271d : List (\u03b1 \u2192 M) hl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l\u271d \u2192 AEStronglyMeasurable f \u03bc f : \u03b1 \u2192 M l : List (\u03b1 \u2192 M) ihl : (\u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc) \u2192 AEStronglyMeasurable (List.prod l) \u03bc hl : AEStronglyMeasurable f \u03bc \u2227 \u2200 (x : \u03b1 \u2192 M), x \u2208 l \u2192 AEStronglyMeasurable x \u03bc \u22a2 AEStronglyMeasurable (f * List.prod l) \u03bc ** exact hl.1.mul (ihl hl.2) ** case nil \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2075 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 f g : \u03b1 \u2192 \u03b2 M : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : ContinuousMul M l : List (\u03b1 \u2192 M) hl\u271d : \u2200 (f : \u03b1 \u2192 M), f \u2208 l \u2192 AEStronglyMeasurable f \u03bc hl : \u2200 (f : \u03b1 \u2192 M), f \u2208 [] \u2192 AEStronglyMeasurable f \u03bc \u22a2 AEStronglyMeasurable (List.prod []) \u03bc ** exact aestronglyMeasurable_one ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.toListRev_eq ** \u03b1 : Type u_1 arr : Array \u03b1 \u22a2 toListRev arr = List.reverse arr.data ** rw [toListRev, foldl_eq_foldl_data, \u2190 List.foldr_reverse, List.foldr_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "WType.cardinal_mk_le_max_aleph0_of_finite ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) \u22a2 IsEmpty \u03b1 \u2192 #(WType \u03b2) \u2264 max #\u03b1 \u2135\u2080 ** intro h ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) h : IsEmpty \u03b1 \u22a2 #(WType \u03b2) \u2264 max #\u03b1 \u2135\u2080 ** rw [Cardinal.mk_eq_zero (WType \u03b2)] ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) h : IsEmpty \u03b1 \u22a2 0 \u2264 max #\u03b1 \u2135\u2080 ** exact zero_le _ ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) hn : Nonempty \u03b1 m : Cardinal.{u} := max #\u03b1 \u2135\u2080 \u22a2 Order.succ 0 \u2264 \u2a06 a, m ^ #(\u03b2 a) ** rw [succ_zero] ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) hn : Nonempty \u03b1 m : Cardinal.{u} := max #\u03b1 \u2135\u2080 \u22a2 1 \u2264 \u2a06 a, m ^ #(\u03b2 a) ** obtain \u27e8a\u27e9 : Nonempty \u03b1 := hn ** case intro \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) m : Cardinal.{u} := max #\u03b1 \u2135\u2080 a : \u03b1 \u22a2 1 \u2264 \u2a06 a, m ^ #(\u03b2 a) ** refine' le_trans _ (le_ciSup (bddAbove_range.{u, u} _) a) ** case intro \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) m : Cardinal.{u} := max #\u03b1 \u2135\u2080 a : \u03b1 \u22a2 1 \u2264 m ^ #(\u03b2 a) ** rw [\u2190 power_zero] ** case intro \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) m : Cardinal.{u} := max #\u03b1 \u2135\u2080 a : \u03b1 \u22a2 ?m.2501 ^ 0 \u2264 m ^ #(\u03b2 a)  \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) m : Cardinal.{u} := max #\u03b1 \u2135\u2080 a : \u03b1 \u22a2 Cardinal.{u} ** exact\n  power_le_power_left\n    (pos_iff_ne_zero.1 (aleph0_pos.trans_le (le_max_right _ _))) (zero_le _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.add_one_le_iff ** n : Nat \u22a2 \u2200 {k : Fin (n + 1)}, k + 1 \u2264 k \u2194 k = last n ** match n with\n| 0 =>\n  intro (k : Fin 1)\n  exact iff_of_true (Subsingleton.elim (\u03b1 := Fin 1) (k+1) _ \u25b8 Nat.le_refl _) (fin_one_eq_zero ..)\n| n + 1 =>\n  intro (k : Fin (n+2))\n  rw [\u2190 add_one_lt_iff, lt_def, le_def, Nat.lt_iff_le_and_ne, and_iff_left]\n  rw [val_add_one]\n  split <;> simp [*, (Nat.succ_ne_zero _).symm, Nat.ne_of_gt (Nat.lt_succ_self _)] ** n : Nat \u22a2 \u2200 {k : Fin (0 + 1)}, k + 1 \u2264 k \u2194 k = last 0 ** intro (k : Fin 1) ** n : Nat k : Fin 1 \u22a2 k + 1 \u2264 k \u2194 k = last 0 ** exact iff_of_true (Subsingleton.elim (\u03b1 := Fin 1) (k+1) _ \u25b8 Nat.le_refl _) (fin_one_eq_zero ..) ** n\u271d n : Nat \u22a2 \u2200 {k : Fin (n + 1 + 1)}, k + 1 \u2264 k \u2194 k = last (n + 1) ** intro (k : Fin (n+2)) ** n\u271d n : Nat k : Fin (n + 2) \u22a2 k + 1 \u2264 k \u2194 k = last (n + 1) ** rw [\u2190 add_one_lt_iff, lt_def, le_def, Nat.lt_iff_le_and_ne, and_iff_left] ** n\u271d n : Nat k : Fin (n + 2) \u22a2 \u2191(k + 1) \u2260 \u2191k ** rw [val_add_one] ** n\u271d n : Nat k : Fin (n + 2) \u22a2 (if k = last (n + 1) then 0 else \u2191k + 1) \u2260 \u2191k ** split <;> simp [*, (Nat.succ_ne_zero _).symm, Nat.ne_of_gt (Nat.lt_succ_self _)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lpMeasSubgroupToLpTrim_neg ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 lpMeasSubgroupToLpTrim F p \u03bc hm (-f) = -lpMeasSubgroupToLpTrim F p \u03bc hm f ** ext1 ** case h \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[Measure.trim \u03bc hm] \u2191\u2191(-lpMeasSubgroupToLpTrim F p \u03bc hm f) ** refine' EventuallyEq.trans _ (Lp.coeFn_neg _).symm ** case h \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[Measure.trim \u03bc hm] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) ** refine' ae_eq_trim_of_stronglyMeasurable hm (Lp.stronglyMeasurable _) _ _ ** case h.refine'_2 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (-f)) =\u1d50[\u03bc] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) ** refine' (lpMeasSubgroupToLpTrim_ae_eq hm _).trans _ ** case h.refine'_2 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191\u2191(-f) =\u1d50[\u03bc] -\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) ** refine' EventuallyEq.trans _ (EventuallyEq.neg (lpMeasSubgroupToLpTrim_ae_eq hm f).symm) ** case h.refine'_2 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191\u2191(-f) =\u1d50[\u03bc] fun x => -\u2191\u2191\u2191f x ** refine' (Lp.coeFn_neg _).trans _ ** case h.refine'_2 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 -\u2191\u2191\u2191f =\u1d50[\u03bc] fun x => -\u2191\u2191\u2191f x ** exact eventually_of_forall fun x => by rfl ** case h.refine'_1 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 StronglyMeasurable (-\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f)) ** exact @StronglyMeasurable.neg _ _ _ m _ _ _ (Lp.stronglyMeasurable _) ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f : { x // x \u2208 lpMeasSubgroup F m p \u03bc } x : \u03b1 \u22a2 (-\u2191\u2191\u2191f) x = (fun x => -\u2191\u2191\u2191f x) x ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.chaar_mono ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 \u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082 ** let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 \u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082 ** have : Continuous eval := (continuous_apply K\u2082).sub (continuous_apply K\u2081) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval \u22a2 chaar K\u2080 K\u2081 \u2264 chaar K\u2080 K\u2082 ** rw [\u2190 sub_nonneg] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval \u22a2 0 \u2264 chaar K\u2080 K\u2082 - chaar K\u2080 K\u2081 ** show chaar K\u2080 \u2208 eval \u207b\u00b9' Ici (0 : \u211d) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval \u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' Ici 0 ** apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval \u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' Ici 0 ** unfold clPrehaar ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval \u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' Ici 0 ** rw [IsClosed.closure_subset_iff] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval \u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' Ici 0 ** rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9 ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' Ici 0 ** simp only [mem_preimage, mem_Ici, sub_nonneg] ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U K\u2081 \u2264 prehaar (\u2191K\u2080) U K\u2082 ** apply prehaar_mono _ h ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Set.Nonempty (interior U) ** rw [h2U.interior_eq] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Set.Nonempty U ** exact \u27e81, h3U\u27e9 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval \u22a2 IsClosed (eval \u207b\u00b9' Ici 0) ** apply continuous_iff_isClosed.mp this ** case a G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 \u2191K\u2082 eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2082 - f K\u2081 this : Continuous eval \u22a2 IsClosed (Ici 0) ** exact isClosed_Ici ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.get!_eq_get? ** \u03b1 : Type u_1 n : Nat inst\u271d : Inhabited \u03b1 a : Array \u03b1 \u22a2 get! a n = Option.getD (get? a n) default ** simp [get!_eq_getD] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condCount_inter ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hs : Set.Finite s \u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t ** by_cases hst : s \u2229 t = \u2205 ** case neg \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hs : Set.Finite s hst : \u00acs \u2229 t = \u2205 \u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t ** rw [condCount, condCount, cond_apply _ hs.measurableSet, cond_apply _ hs.measurableSet,\n  cond_apply _ (hs.inter_of_left _).measurableSet, mul_comm _ (Measure.count (s \u2229 t)),\n  \u2190 mul_assoc, mul_comm _ (Measure.count (s \u2229 t)), \u2190 mul_assoc, ENNReal.mul_inv_cancel, one_mul,\n  mul_comm, Set.inter_assoc] ** case pos \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hs : Set.Finite s hst : s \u2229 t = \u2205 \u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 t)) u * \u2191\u2191(condCount s) t ** rw [hst, condCount_empty_meas, Measure.coe_zero, Pi.zero_apply, zero_mul,\n  condCount_eq_zero_iff hs, \u2190 Set.inter_assoc, hst, Set.empty_inter] ** case neg.h0 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hs : Set.Finite s hst : \u00acs \u2229 t = \u2205 \u22a2 \u2191\u2191Measure.count (s \u2229 t) \u2260 0 ** rwa [\u2190 Measure.count_eq_zero_iff] at hst ** case neg.ht \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hs : Set.Finite s hst : \u00acs \u2229 t = \u2205 \u22a2 \u2191\u2191Measure.count (s \u2229 t) \u2260 \u22a4 ** exact (Measure.count_apply_lt_top.2 <| hs.inter_of_left _).ne ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.lookup_insert_of_ne ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a a' : \u03b1 b : \u03b2 a s\u271d : Finmap \u03b2 h : a' \u2260 a s : AList \u03b2 \u22a2 lookup a' (insert a b \u27e6s\u27e7) = lookup a' \u27e6s\u27e7 ** simp only [insert_toFinmap, lookup_toFinmap, lookup_insert_ne h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.boundedBy_zero ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e \u22a2 boundedBy 0 = 0 ** rw [\u2190 coe_bot, eq_bot_iff] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e \u22a2 boundedBy 0 \u2264 \u22a5 ** apply boundedBy_le ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.inter_filter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b2 : DecidablePred p inst\u271d\u00b9 : DecidablePred q s\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 \u22a2 s \u2229 filter p t = filter p (s \u2229 t) ** rw [inter_comm, filter_inter, inter_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.lcm_one_left ** i : \u2124 \u22a2 lcm 1 i = natAbs i ** rw [Int.lcm] ** i : \u2124 \u22a2 Nat.lcm (natAbs 1) (natAbs i) = natAbs i ** apply Nat.lcm_one_left ** Qed",
        "informal": ""
    },
    {
        "formal": "ContinuousLinearMap.smul_compLpL ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedAddCommGroup G g : E \u2192 F c\u271d : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedSpace \ud835\udd5c F inst\u271d\u2074 : Fact (1 \u2264 p) \ud835\udd5c' : Type u_6 inst\u271d\u00b3 : NormedRing \ud835\udd5c' inst\u271d\u00b2 : Module \ud835\udd5c' F inst\u271d\u00b9 : BoundedSMul \ud835\udd5c' F inst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' F c : \ud835\udd5c' L : E \u2192L[\ud835\udd5c] F \u22a2 compLpL p \u03bc (c \u2022 L) = c \u2022 compLpL p \u03bc L ** ext1 f ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedAddCommGroup G g : E \u2192 F c\u271d : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedSpace \ud835\udd5c F inst\u271d\u2074 : Fact (1 \u2264 p) \ud835\udd5c' : Type u_6 inst\u271d\u00b3 : NormedRing \ud835\udd5c' inst\u271d\u00b2 : Module \ud835\udd5c' F inst\u271d\u00b9 : BoundedSMul \ud835\udd5c' F inst\u271d : SMulCommClass \ud835\udd5c \ud835\udd5c' F c : \ud835\udd5c' L : E \u2192L[\ud835\udd5c] F f : { x // x \u2208 Lp E p } \u22a2 \u2191(compLpL p \u03bc (c \u2022 L)) f = \u2191(c \u2022 compLpL p \u03bc L) f ** exact smul_compLp c L f ** Qed",
        "informal": ""
    },
    {
        "formal": "Holor.slice_sum ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : AddCommMonoid \u03b1 \u03b2 : Type i : \u2115 hid : i < d s : Finset \u03b2 f : \u03b2 \u2192 Holor \u03b1 (d :: ds) \u22a2 \u2211 x in s, slice (f x) i hid = slice (\u2211 x in s, f x) i hid ** letI := Classical.decEq \u03b2 ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : AddCommMonoid \u03b1 \u03b2 : Type i : \u2115 hid : i < d s : Finset \u03b2 f : \u03b2 \u2192 Holor \u03b1 (d :: ds) this : DecidableEq \u03b2 := Classical.decEq \u03b2 \u22a2 \u2211 x in s, slice (f x) i hid = slice (\u2211 x in s, f x) i hid ** refine' Finset.induction_on s _ _ ** case refine'_1 \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : AddCommMonoid \u03b1 \u03b2 : Type i : \u2115 hid : i < d s : Finset \u03b2 f : \u03b2 \u2192 Holor \u03b1 (d :: ds) this : DecidableEq \u03b2 := Classical.decEq \u03b2 \u22a2 \u2211 x in \u2205, slice (f x) i hid = slice (\u2211 x in \u2205, f x) i hid ** simp [slice_zero] ** case refine'_2 \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : AddCommMonoid \u03b1 \u03b2 : Type i : \u2115 hid : i < d s : Finset \u03b2 f : \u03b2 \u2192 Holor \u03b1 (d :: ds) this : DecidableEq \u03b2 := Classical.decEq \u03b2 \u22a2 \u2200 \u2983a : \u03b2\u2984 {s : Finset \u03b2},     \u00aca \u2208 s \u2192       \u2211 x in s, slice (f x) i hid = slice (\u2211 x in s, f x) i hid \u2192         \u2211 x in insert a s, slice (f x) i hid = slice (\u2211 x in insert a s, f x) i hid ** intro _ _ h_not_in ih ** case refine'_2 \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : AddCommMonoid \u03b1 \u03b2 : Type i : \u2115 hid : i < d s : Finset \u03b2 f : \u03b2 \u2192 Holor \u03b1 (d :: ds) this : DecidableEq \u03b2 := Classical.decEq \u03b2 a\u271d : \u03b2 s\u271d : Finset \u03b2 h_not_in : \u00aca\u271d \u2208 s\u271d ih : \u2211 x in s\u271d, slice (f x) i hid = slice (\u2211 x in s\u271d, f x) i hid \u22a2 \u2211 x in insert a\u271d s\u271d, slice (f x) i hid = slice (\u2211 x in insert a\u271d s\u271d, f x) i hid ** rw [Finset.sum_insert h_not_in, ih, slice_add, Finset.sum_insert h_not_in] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.tprod_tprod ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i) \u03b4 : Type u_4 \u03c0 : \u03b4 \u2192 Type u_5 inst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x) l : List \u03b4 \u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i) inst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i) s : (i : \u03b4) \u2192 Set (\u03c0 i) \u22a2 \u2191\u2191(Measure.tprod l \u03bc) (Set.tprod l s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) l) ** induction' l with i l ih ** case cons \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i) \u03b4 : Type u_4 \u03c0 : \u03b4 \u2192 Type u_5 inst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x) \u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i) inst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i) s : (i : \u03b4) \u2192 Set (\u03c0 i) i : \u03b4 l : List \u03b4 ih : \u2191\u2191(Measure.tprod l \u03bc) (Set.tprod l s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) l) \u22a2 \u2191\u2191(Measure.tprod (i :: l) \u03bc) (Set.tprod (i :: l) s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) (i :: l)) ** rw [tprod_cons, Set.tprod, prod_prod, map_cons, prod_cons, ih] ** case nil \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i) \u03b4 : Type u_4 \u03c0 : \u03b4 \u2192 Type u_5 inst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x) \u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i) inst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i) s : (i : \u03b4) \u2192 Set (\u03c0 i) \u22a2 \u2191\u2191(Measure.tprod [] \u03bc) (Set.tprod [] s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) []) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.orderEmbOfFin_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : LinearOrder \u03b1 a : \u03b1 i : Fin 1 \u22a2 \u2191(orderEmbOfFin {a} (_ : card {a} = 1)) i = a ** rw [Subsingleton.elim i \u27e80, zero_lt_one\u27e9, orderEmbOfFin_zero _ zero_lt_one, min'_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_fract ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : LinearOrderedRing R inst\u271d\u2074 : FloorRing R inst\u271d\u00b3 : TopologicalSpace R inst\u271d\u00b2 : OrderTopology R inst\u271d\u00b9 : MeasurableSpace R inst\u271d : BorelSpace R \u22a2 Measurable Int.fract ** intro s hs ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : LinearOrderedRing R inst\u271d\u2074 : FloorRing R inst\u271d\u00b3 : TopologicalSpace R inst\u271d\u00b2 : OrderTopology R inst\u271d\u00b9 : MeasurableSpace R inst\u271d : BorelSpace R s : Set R hs : MeasurableSet s \u22a2 MeasurableSet (Int.fract \u207b\u00b9' s) ** rw [Int.preimage_fract] ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : LinearOrderedRing R inst\u271d\u2074 : FloorRing R inst\u271d\u00b3 : TopologicalSpace R inst\u271d\u00b2 : OrderTopology R inst\u271d\u00b9 : MeasurableSpace R inst\u271d : BorelSpace R s : Set R hs : MeasurableSet s \u22a2 MeasurableSet (\u22c3 m, (fun x => x - \u2191m) \u207b\u00b9' (s \u2229 Ico 0 1)) ** exact MeasurableSet.iUnion fun z => measurable_id.sub_const _ (hs.inter measurableSet_Ico) ** Qed",
        "informal": ""
    },
    {
        "formal": "Basis.parallelepiped_basisFun ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 \u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = \u2191(PositiveCompacts.piIcc01 \u03b9) ** refine' Eq.trans _ ((uIcc_of_le _).trans (Set.pi_univ_Icc _ _).symm) ** case refine'_1 \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 \u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = uIcc (fun i => 0) fun i => 1 ** classical convert parallelepiped_single (\u03b9 := \u03b9) 1 ** case refine'_1 \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 \u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = uIcc (fun i => 0) fun i => 1 ** convert parallelepiped_single (\u03b9 := \u03b9) 1 ** case refine'_2 \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 \u22a2 (fun i => 0) \u2264 fun i => 1 ** exact zero_le_one ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.exists_seq_iSup_eq_top_iff_countable ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x \u22a2 (\u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4) \u2194 \u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4 ** constructor ** case mp \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x \u22a2 (\u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4) \u2192 \u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4 ** rintro \u27e8s, hps, hs\u27e9 ** case mp.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x s : \u2115 \u2192 \u03b1 hps : \u2200 (n : \u2115), p (s n) hs : \u2a06 n, s n = \u22a4 \u22a2 \u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4 ** refine' \u27e8range s, countable_range s, forall_range_iff.2 hps, _\u27e9 ** case mp.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x s : \u2115 \u2192 \u03b1 hps : \u2200 (n : \u2115), p (s n) hs : \u2a06 n, s n = \u22a4 \u22a2 sSup (range s) = \u22a4 ** rwa [sSup_range] ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x \u22a2 (\u2203 S, Set.Countable S \u2227 (\u2200 (s : \u03b1), s \u2208 S \u2192 p s) \u2227 sSup S = \u22a4) \u2192 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4 ** rintro \u27e8S, hSc, hps, hS\u27e9 ** case mpr.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x S : Set \u03b1 hSc : Set.Countable S hps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s hS : sSup S = \u22a4 \u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4 ** rcases eq_empty_or_nonempty S with (rfl | hne) ** case mpr.intro.intro.intro.inl \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x hSc : Set.Countable \u2205 hps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s hS : sSup \u2205 = \u22a4 \u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4 ** rw [sSup_empty] at hS ** case mpr.intro.intro.intro.inl \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x hSc : Set.Countable \u2205 hps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s hS : \u22a5 = \u22a4 \u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4 ** haveI := subsingleton_of_bot_eq_top hS ** case mpr.intro.intro.intro.inl \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x hSc : Set.Countable \u2205 hps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s hS : \u22a5 = \u22a4 this : Subsingleton \u03b1 \u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4 ** rcases h with \u27e8x, hx\u27e9 ** case mpr.intro.intro.intro.inl.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop hSc : Set.Countable \u2205 hps : \u2200 (s : \u03b1), s \u2208 \u2205 \u2192 p s hS : \u22a5 = \u22a4 this : Subsingleton \u03b1 x : \u03b1 hx : p x \u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4 ** exact \u27e8fun _ => x, fun _ => hx, Subsingleton.elim _ _\u27e9 ** case mpr.intro.intro.intro.inr \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x S : Set \u03b1 hSc : Set.Countable S hps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s hS : sSup S = \u22a4 hne : Set.Nonempty S \u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4 ** rcases (Set.countable_iff_exists_surjective hne).1 hSc with \u27e8s, hs\u27e9 ** case mpr.intro.intro.intro.inr.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x S : Set \u03b1 hSc : Set.Countable S hps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s hS : sSup S = \u22a4 hne : Set.Nonempty S s : \u2115 \u2192 \u2191S hs : Surjective s \u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4 ** refine' \u27e8fun n => s n, fun n => hps _ (s n).coe_prop, _\u27e9 ** case mpr.intro.intro.intro.inr.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d : CompleteLattice \u03b1 p : \u03b1 \u2192 Prop h : \u2203 x, p x S : Set \u03b1 hSc : Set.Countable S hps : \u2200 (s : \u03b1), s \u2208 S \u2192 p s hS : sSup S = \u22a4 hne : Set.Nonempty S s : \u2115 \u2192 \u2191S hs : Surjective s \u22a2 \u2a06 n, (fun n => \u2191(s n)) n = \u22a4 ** rwa [hs.iSup_comp, \u2190 sSup_eq_iSup'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.exists_good\u03b4 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       \u03b4 < 1 \u2227         \u2200 (s : Finset E),           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2192             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2192 Finset.card s \u2264 multiplicity E ** by_contra' h ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s \u22a2 False ** set N := multiplicity E + 1 with hN ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 this : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016 \u22a2 False ** choose! F hF using this ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 this : \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 \u22a2 False ** rcases this with \u27e8f, hf, h'f\u27e9 ** case intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 \u22a2 False ** have finj : Function.Injective f := by\n  intro i j hij\n  by_contra h\n  have : 1 \u2264 \u2016f i - f j\u2016 := h'f i j h\n  simp only [hij, norm_zero, sub_self] at this\n  exact lt_irrefl _ (this.trans_lt zero_lt_one) ** case intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f \u22a2 False ** let s := Finset.image f Finset.univ ** case intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ \u22a2 False ** have s_card : s.card = N := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin N ** case intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N \u22a2 False ** have hs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2 := by\n  simp only [hf, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n    Finset.mem_image, true_and] ** case intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 \u22a2 False ** have h's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016 := by\n  simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n    Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]\n  intro i j hij\n  have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl\n  exact h'f i j this ** case intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016 \u22a2 False ** have : s.card \u2264 multiplicity E := card_le_multiplicity hs h's ** case intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016 this : Finset.card s \u2264 multiplicity E \u22a2 False ** rw [s_card, hN] at this ** case intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016 this : multiplicity E + 1 \u2264 multiplicity E \u22a2 False ** exact lt_irrefl _ ((Nat.lt_succ_self (multiplicity E)).trans_le this) ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u22a2 \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016 ** intro \u03b4 h\u03b4 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016 ** rcases lt_or_le \u03b4 1 with (h\u03b4' | h\u03b4') ** case inl E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : \u03b4 < 1 \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016 ** rcases h \u03b4 h\u03b4 h\u03b4' with \u27e8s, hs, h's, s_card\u27e9 ** case inl.intro.intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : \u03b4 < 1 s : Finset E hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016 s_card : multiplicity E < Finset.card s \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016 ** obtain \u27e8f, f_inj, hfs\u27e9 : \u2203 f : Fin N \u2192 E, Function.Injective f \u2227 range f \u2286 \u2191s := by\n  have : Fintype.card (Fin N) \u2264 s.card := by simp only [Fintype.card_fin]; exact s_card\n  rcases Function.Embedding.exists_of_card_le_finset this with \u27e8f, hf\u27e9\n  exact \u27e8f, f.injective, hf\u27e9 ** case inl.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : \u03b4 < 1 s : Finset E hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016 s_card : multiplicity E < Finset.card s f : Fin N \u2192 E f_inj : Function.Injective f hfs : range f \u2286 \u2191s \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016 ** simp only [range_subset_iff, Finset.mem_coe] at hfs ** case inl.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : \u03b4 < 1 s : Finset E hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016 s_card : multiplicity E < Finset.card s f : Fin N \u2192 E f_inj : Function.Injective f hfs : \u2200 (y : Fin (multiplicity E + 1)), f y \u2208 s \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016 ** refine' \u27e8f, fun i => hs _ (hfs i), fun i j hij => h's _ (hfs i) _ (hfs j) (f_inj.ne hij)\u27e9 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : \u03b4 < 1 s : Finset E hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016 s_card : multiplicity E < Finset.card s \u22a2 \u2203 f, Function.Injective f \u2227 range f \u2286 \u2191s ** have : Fintype.card (Fin N) \u2264 s.card := by simp only [Fintype.card_fin]; exact s_card ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : \u03b4 < 1 s : Finset E hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016 s_card : multiplicity E < Finset.card s this : Fintype.card (Fin N) \u2264 Finset.card s \u22a2 \u2203 f, Function.Injective f \u2227 range f \u2286 \u2191s ** rcases Function.Embedding.exists_of_card_le_finset this with \u27e8f, hf\u27e9 ** case intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : \u03b4 < 1 s : Finset E hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016 s_card : multiplicity E < Finset.card s this : Fintype.card (Fin N) \u2264 Finset.card s f : Fin N \u21aa E hf : range \u2191f \u2286 \u2191s \u22a2 \u2203 f, Function.Injective f \u2227 range f \u2286 \u2191s ** exact \u27e8f, f.injective, hf\u27e9 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : \u03b4 < 1 s : Finset E hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016 s_card : multiplicity E < Finset.card s \u22a2 Fintype.card (Fin N) \u2264 Finset.card s ** simp only [Fintype.card_fin] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : \u03b4 < 1 s : Finset E hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016 s_card : multiplicity E < Finset.card s \u22a2 multiplicity E + 1 \u2264 Finset.card s ** exact s_card ** case inr E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : 1 \u2264 \u03b4 \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016f i - f j\u2016 ** exact\n  \u27e8fun _ => 0, fun i => by simp; norm_num, fun i j _ => by\n    simpa only [norm_zero, sub_nonpos, sub_self]\u27e9 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : 1 \u2264 \u03b4 i : Fin N \u22a2 \u2016(fun x => 0) i\u2016 \u2264 2 ** simp ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : 1 \u2264 \u03b4 i : Fin N \u22a2 0 \u2264 2 ** norm_num ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h\u03b4' : 1 \u2264 \u03b4 i j : Fin N x\u271d : i \u2260 j \u22a2 1 - \u03b4 \u2264 \u2016(fun x => 0) i - (fun x => 0) j\u2016 ** simpa only [norm_zero, sub_nonpos, sub_self] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 ** obtain \u27e8u, _, zero_lt_u, hu\u27e9 :\n  \u2203 u : \u2115 \u2192 \u211d,\n    (\u2200 m n : \u2115, m < n \u2192 u n < u m) \u2227 (\u2200 n : \u2115, 0 < u n) \u2227 Filter.Tendsto u Filter.atTop (\ud835\udcdd 0) :=\n  exists_seq_strictAnti_tendsto (0 : \u211d) ** case intro.intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 ** have A : \u2200 n, F (u n) \u2208 closedBall (0 : Fin N \u2192 E) 2 := by\n  intro n\n  simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right,\n    (hF (u n) (zero_lt_u n)).left, forall_const] ** case intro.intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) A : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2 \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 ** obtain \u27e8f, fmem, \u03c6, \u03c6_mono, hf\u27e9 :\n  \u2203 f \u2208 closedBall (0 : Fin N \u2192 E) 2,\n    \u2203 \u03c6 : \u2115 \u2192 \u2115, StrictMono \u03c6 \u2227 Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f) :=\n  IsCompact.tendsto_subseq (isCompact_closedBall _ _) A ** case intro.intro.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) A : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2 f : Fin N \u2192 E fmem : f \u2208 closedBall 0 2 \u03c6 : \u2115 \u2192 \u2115 \u03c6_mono : StrictMono \u03c6 hf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f) \u22a2 \u2203 f, (\u2200 (i : Fin N), \u2016f i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 ** refine' \u27e8f, fun i => _, fun i j hij => _\u27e9 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) \u22a2 \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2 ** intro n ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) n : \u2115 \u22a2 F (u n) \u2208 closedBall 0 2 ** simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right,\n  (hF (u n) (zero_lt_u n)).left, forall_const] ** case intro.intro.intro.intro.intro.intro.intro.refine'_1 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) A : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2 f : Fin N \u2192 E fmem : f \u2208 closedBall 0 2 \u03c6 : \u2115 \u2192 \u2115 \u03c6_mono : StrictMono \u03c6 hf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f) i : Fin N \u22a2 \u2016f i\u2016 \u2264 2 ** simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right] at fmem ** case intro.intro.intro.intro.intro.intro.intro.refine'_1 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) A : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2 f : Fin N \u2192 E \u03c6 : \u2115 \u2192 \u2115 \u03c6_mono : StrictMono \u03c6 hf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f) i : Fin N fmem : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 \u22a2 \u2016f i\u2016 \u2264 2 ** exact fmem i ** case intro.intro.intro.intro.intro.intro.intro.refine'_2 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) A : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2 f : Fin N \u2192 E fmem : f \u2208 closedBall 0 2 \u03c6 : \u2115 \u2192 \u2115 \u03c6_mono : StrictMono \u03c6 hf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f) i j : Fin N hij : i \u2260 j \u22a2 1 \u2264 \u2016f i - f j\u2016 ** have A : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016) :=\n  ((hf.apply i).sub (hf.apply j)).norm ** case intro.intro.intro.intro.intro.intro.intro.refine'_2 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) A\u271d : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2 f : Fin N \u2192 E fmem : f \u2208 closedBall 0 2 \u03c6 : \u2115 \u2192 \u2115 \u03c6_mono : StrictMono \u03c6 hf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f) i j : Fin N hij : i \u2260 j A : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016) \u22a2 1 \u2264 \u2016f i - f j\u2016 ** have B : Tendsto (fun n => 1 - u (\u03c6 n)) atTop (\ud835\udcdd (1 - 0)) :=\n  tendsto_const_nhds.sub (hu.comp \u03c6_mono.tendsto_atTop) ** case intro.intro.intro.intro.intro.intro.intro.refine'_2 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) A\u271d : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2 f : Fin N \u2192 E fmem : f \u2208 closedBall 0 2 \u03c6 : \u2115 \u2192 \u2115 \u03c6_mono : StrictMono \u03c6 hf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f) i j : Fin N hij : i \u2260 j A : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016) B : Tendsto (fun n => 1 - u (\u03c6 n)) atTop (\ud835\udcdd (1 - 0)) \u22a2 1 \u2264 \u2016f i - f j\u2016 ** rw [sub_zero] at B ** case intro.intro.intro.intro.intro.intro.intro.refine'_2 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 u : \u2115 \u2192 \u211d left\u271d : \u2200 (m n : \u2115), m < n \u2192 u n < u m zero_lt_u : \u2200 (n : \u2115), 0 < u n hu : Tendsto u atTop (\ud835\udcdd 0) A\u271d : \u2200 (n : \u2115), F (u n) \u2208 closedBall 0 2 f : Fin N \u2192 E fmem : f \u2208 closedBall 0 2 \u03c6 : \u2115 \u2192 \u2115 \u03c6_mono : StrictMono \u03c6 hf : Tendsto ((F \u2218 u) \u2218 \u03c6) atTop (\ud835\udcdd f) i j : Fin N hij : i \u2260 j A : Tendsto (fun n => \u2016F (u (\u03c6 n)) i - F (u (\u03c6 n)) j\u2016) atTop (\ud835\udcdd \u2016f i - f j\u2016) B : Tendsto (fun n => 1 - u (\u03c6 n)) atTop (\ud835\udcdd 1) \u22a2 1 \u2264 \u2016f i - f j\u2016 ** exact le_of_tendsto_of_tendsto' B A fun n => (hF (u (\u03c6 n)) (zero_lt_u _)).2 i j hij ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 \u22a2 Function.Injective f ** intro i j hij ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 i j : Fin N hij : f i = f j \u22a2 i = j ** by_contra h ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h\u271d :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 i j : Fin N hij : f i = f j h : \u00aci = j \u22a2 False ** have : 1 \u2264 \u2016f i - f j\u2016 := h'f i j h ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h\u271d :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 i j : Fin N hij : f i = f j h : \u00aci = j this : 1 \u2264 \u2016f i - f j\u2016 \u22a2 False ** simp only [hij, norm_zero, sub_self] at this ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h\u271d :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 i j : Fin N hij : f i = f j h : \u00aci = j this : 1 \u2264 0 \u22a2 False ** exact lt_irrefl _ (this.trans_lt zero_lt_one) ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ \u22a2 Finset.card s = N ** rw [Finset.card_image_of_injective _ finj] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ \u22a2 Finset.card Finset.univ = N ** exact Finset.card_fin N ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N \u22a2 \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 ** simp only [hf, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n  Finset.mem_image, true_and] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 \u22a2 \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016 ** simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n  Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 \u22a2 \u2200 (a a_1 : Fin (multiplicity E + 1)), \u00acf a = f a_1 \u2192 1 \u2264 \u2016f a - f a_1\u2016 ** intro i j hij ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 i j : Fin (multiplicity E + 1) hij : \u00acf i = f j \u22a2 1 \u2264 \u2016f i - f j\u2016 ** have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 i j : Fin (multiplicity E + 1) hij : \u00acf i = f j this : i \u2260 j \u22a2 1 \u2264 \u2016f i - f j\u2016 ** exact h'f i j this ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h\u271d :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 i j : Fin (multiplicity E + 1) hij : \u00acf i = f j h : i = j \u22a2 False ** rw [h] at hij ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E h\u271d :   \u2200 (\u03b4 : \u211d),     0 < \u03b4 \u2192       \u03b4 < 1 \u2192         \u2203 s,           (\u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2) \u2227             (\u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - \u03b4 \u2264 \u2016c - d\u2016) \u2227 multiplicity E < Finset.card s N : \u2115 := multiplicity E + 1 hN : N = multiplicity E + 1 F : \u211d \u2192 Fin N \u2192 E hF : \u2200 (\u03b4 : \u211d), 0 < \u03b4 \u2192 (\u2200 (i : Fin N), \u2016F \u03b4 i\u2016 \u2264 2) \u2227 \u2200 (i j : Fin N), i \u2260 j \u2192 1 - \u03b4 \u2264 \u2016F \u03b4 i - F \u03b4 j\u2016 f : Fin N \u2192 E hf : \u2200 (i : Fin N), \u2016f i\u2016 \u2264 2 h'f : \u2200 (i j : Fin N), i \u2260 j \u2192 1 \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = N hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 i j : Fin (multiplicity E + 1) hij : \u00acf j = f j h : i = j \u22a2 False ** exact hij rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.Valid.validFor ** l mr lm r : List Char e :   lm ++ r =     { str := { data := l ++ mr }, startPos := { byteIdx := utf8ByteSize.go l },           stopPos := { byteIdx := utf8ByteSize.go lm } }.str.data h :   { str := { data := l ++ mr }, startPos := { byteIdx := utf8ByteSize.go l },         stopPos := { byteIdx := utf8ByteSize.go lm } }.startPos \u2264     { str := { data := l ++ mr }, startPos := { byteIdx := utf8ByteSize.go l },         stopPos := { byteIdx := utf8ByteSize.go lm } }.stopPos \u22a2 \u2203 l_1 m r,     ValidFor l_1 m r       { str := { data := l ++ mr }, startPos := { byteIdx := utf8ByteSize.go l },         stopPos := { byteIdx := utf8ByteSize.go lm } } ** simp at * ** l mr lm r : List Char e : lm ++ r = l ++ mr h : utf8Len l \u2264 utf8Len lm \u22a2 \u2203 l_1 m r,     ValidFor l_1 m r       { str := { data := l ++ mr }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len lm } } ** have := (or_iff_right_iff_imp.2 fun h => ?x).1 (List.append_eq_append_iff.1 e) ** l mr lm r : List Char e : lm ++ r = l ++ mr h : utf8Len l \u2264 utf8Len lm this : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r \u22a2 \u2203 l_1 m r,     ValidFor l_1 m r       { str := { data := l ++ mr }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len lm } }  case x l mr lm r : List Char e : lm ++ r = l ++ mr h\u271d : utf8Len l \u2264 utf8Len lm h : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr \u22a2 \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r ** case x =>\n  match l, r, h with | _, _, \u27e8m, rfl, rfl\u27e9 => ?_\n  simp at h\n  cases utf8Len_eq_zero.1 <| Nat.le_zero.1 (Nat.le_of_add_le_add_left (c := 0) h)\n  exact \u27e8[], by simp\u27e9 ** l mr lm r : List Char e : lm ++ r = l ++ mr h : utf8Len l \u2264 utf8Len lm this : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r \u22a2 \u2203 l_1 m r,     ValidFor l_1 m r       { str := { data := l ++ mr }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len lm } } ** match lm, mr, this with\n| _, _, \u27e8m, rfl, rfl\u27e9 => exact \u27e8l, m, r, by simpa using ValidFor.mk\u27e9 ** l mr lm r : List Char e : lm ++ r = l ++ mr h\u271d : utf8Len l \u2264 utf8Len lm h : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr \u22a2 \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r ** match l, r, h with | _, _, \u27e8m, rfl, rfl\u27e9 => ?_ ** l mr lm r : List Char h\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr m : List Char e : lm ++ (m ++ mr) = lm ++ m ++ mr h : utf8Len (lm ++ m) \u2264 utf8Len lm \u22a2 \u2203 c', lm = lm ++ m ++ c' \u2227 mr = c' ++ (m ++ mr) ** simp at h ** l mr lm r : List Char h\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr m : List Char e : lm ++ (m ++ mr) = lm ++ m ++ mr h : utf8Len lm + utf8Len m \u2264 utf8Len lm \u22a2 \u2203 c', lm = lm ++ m ++ c' \u2227 mr = c' ++ (m ++ mr) ** cases utf8Len_eq_zero.1 <| Nat.le_zero.1 (Nat.le_of_add_le_add_left (c := 0) h) ** case refl l mr lm r : List Char h\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr e : lm ++ ([] ++ mr) = lm ++ [] ++ mr h : utf8Len lm + utf8Len [] \u2264 utf8Len lm \u22a2 \u2203 c', lm = lm ++ [] ++ c' \u2227 mr = c' ++ ([] ++ mr) ** exact \u27e8[], by simp\u27e9 ** l mr lm r : List Char h\u271d : \u2203 a', l = lm ++ a' \u2227 r = a' ++ mr e : lm ++ ([] ++ mr) = lm ++ [] ++ mr h : utf8Len lm + utf8Len [] \u2264 utf8Len lm \u22a2 lm = lm ++ [] ++ [] \u2227 mr = [] ++ ([] ++ mr) ** simp ** l mr lm r : List Char this : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r m : List Char e : l ++ m ++ r = l ++ (m ++ r) h : utf8Len l \u2264 utf8Len (l ++ m) \u22a2 \u2203 l_1 m_1 r_1,     ValidFor l_1 m_1 r_1       { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },         stopPos := { byteIdx := utf8Len (l ++ m) } } ** exact \u27e8l, m, r, by simpa using ValidFor.mk\u27e9 ** l mr lm r : List Char this : \u2203 c', lm = l ++ c' \u2227 mr = c' ++ r m : List Char e : l ++ m ++ r = l ++ (m ++ r) h : utf8Len l \u2264 utf8Len (l ++ m) \u22a2 ValidFor l m r     { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l },       stopPos := { byteIdx := utf8Len (l ++ m) } } ** simpa using ValidFor.mk ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.cauchySeq_Lp_iff_cauchySeq_\u2112p ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : SemilatticeSup \u03b9 hp : Fact (1 \u2264 p) f : \u03b9 \u2192 { x // x \u2208 Lp E p } \u22a2 CauchySeq f \u2194 Tendsto (fun n => snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc) atTop (\ud835\udcdd 0) ** simp_rw [cauchySeq_iff_tendsto_dist_atTop_0, dist_def] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : SemilatticeSup \u03b9 hp : Fact (1 \u2264 p) f : \u03b9 \u2192 { x // x \u2208 Lp E p } \u22a2 Tendsto (fun n => ENNReal.toReal (snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc)) atTop (\ud835\udcdd 0) \u2194     Tendsto (fun n => snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc) atTop (\ud835\udcdd 0) ** rw [\u2190 ENNReal.zero_toReal, ENNReal.tendsto_toReal_iff (fun n => ?_) ENNReal.zero_ne_top] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : SemilatticeSup \u03b9 hp : Fact (1 \u2264 p) f : \u03b9 \u2192 { x // x \u2208 Lp E p } n : \u03b9 \u00d7 \u03b9 \u22a2 snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc \u2260 \u22a4 ** rw [snorm_congr_ae (Lp.coeFn_sub _ _).symm] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G \u03b9 : Type u_5 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : SemilatticeSup \u03b9 hp : Fact (1 \u2264 p) f : \u03b9 \u2192 { x // x \u2208 Lp E p } n : \u03b9 \u00d7 \u03b9 \u22a2 snorm (\u2191\u2191(f n.1 - f n.2)) p \u03bc \u2260 \u22a4 ** exact snorm_ne_top _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.InnerRegular.measurableSet_of_open ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) \u22a2 InnerRegular \u03bc p fun s => MeasurableSet s \u2227 \u2191\u2191\u03bc s \u2260 \u22a4 ** rintro s \u27e8hs, h\u03bcs\u27e9 r hr ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc s \u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 r < \u2191\u2191\u03bc K ** obtain \u27e8\u03b5, h\u03b5, h\u03b5s, rfl\u27e9 : \u2203 (\u03b5 : _) (_ : \u03b5 \u2260 0), \u03b5 + \u03b5 \u2264 \u03bc s \u2227 r = \u03bc s - (\u03b5 + \u03b5) := by\n  use (\u03bc s - r) / 2\n  simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right] ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s hr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s \u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K ** rcases hs.exists_isOpen_diff_lt h\u03bcs h\u03b5 with \u27e8U, hsU, hUo, hUt, h\u03bcU\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s hr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U hUt : \u2191\u2191\u03bc U < \u22a4 h\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5 \u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K ** rcases (U \\ s).exists_isOpen_lt_of_lt _ h\u03bcU with \u27e8U', hsU', hU'o, h\u03bcU'\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s hr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U hUt : \u2191\u2191\u03bc U < \u22a4 h\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5 U' : Set \u03b1 hsU' : U' \u2287 U \\ s hU'o : IsOpen U' h\u03bcU' : \u2191\u2191\u03bc U' < \u03b5 \u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K ** replace hsU' := diff_subset_comm.1 hsU' ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s hr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U hUt : \u2191\u2191\u03bc U < \u22a4 h\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5 U' : Set \u03b1 hU'o : IsOpen U' h\u03bcU' : \u2191\u2191\u03bc U' < \u03b5 hsU' : U \\ U' \u2286 s \u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K ** rcases H.exists_subset_lt_add h0 hUo hUt.ne h\u03b5 with \u27e8K, hKU, hKc, hKr\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s hr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U hUt : \u2191\u2191\u03bc U < \u22a4 h\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5 U' : Set \u03b1 hU'o : IsOpen U' h\u03bcU' : \u2191\u2191\u03bc U' < \u03b5 hsU' : U \\ U' \u2286 s K : Set \u03b1 hKU : K \u2286 U hKc : p K hKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5 \u22a2 \u2203 K, K \u2286 s \u2227 p K \u2227 \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc K ** refine' \u27e8K \\ U', fun x hx => hsU' \u27e8hKU hx.1, hx.2\u27e9, hd hKc hU'o, ENNReal.sub_lt_of_lt_add h\u03b5s _\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s hr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U hUt : \u2191\u2191\u03bc U < \u22a4 h\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5 U' : Set \u03b1 hU'o : IsOpen U' h\u03bcU' : \u2191\u2191\u03bc U' < \u03b5 hsU' : U \\ U' \u2286 s K : Set \u03b1 hKU : K \u2286 U hKc : p K hKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5 \u22a2 \u2191\u2191\u03bc s < \u2191\u2191\u03bc (K \\ U') + (\u03b5 + \u03b5) ** calc\n  \u03bc s \u2264 \u03bc U := \u03bc.mono hsU\n  _ < \u03bc K + \u03b5 := hKr\n  _ \u2264 \u03bc (K \\ U') + \u03bc U' + \u03b5 := (add_le_add_right (tsub_le_iff_right.1 le_measure_diff) _)\n  _ \u2264 \u03bc (K \\ U') + \u03b5 + \u03b5 := by\n    apply add_le_add_right; apply add_le_add_left\n    exact h\u03bcU'.le\n  _ = \u03bc (K \\ U') + (\u03b5 + \u03b5) := add_assoc _ _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc s \u22a2 \u2203 \u03b5 x, \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s \u2227 r = \u2191\u2191\u03bc s - (\u03b5 + \u03b5) ** use (\u03bc s - r) / 2 ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s\u271d : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc s \u22a2 \u2203 x, (\u2191\u2191\u03bc s - r) / 2 + (\u2191\u2191\u03bc s - r) / 2 \u2264 \u2191\u2191\u03bc s \u2227 r = \u2191\u2191\u03bc s - ((\u2191\u2191\u03bc s - r) / 2 + (\u2191\u2191\u03bc s - r) / 2) ** simp [*, hr.le, ENNReal.add_halves, ENNReal.sub_sub_cancel, le_add_right] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s hr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U hUt : \u2191\u2191\u03bc U < \u22a4 h\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5 U' : Set \u03b1 hU'o : IsOpen U' h\u03bcU' : \u2191\u2191\u03bc U' < \u03b5 hsU' : U \\ U' \u2286 s K : Set \u03b1 hKU : K \u2286 U hKc : p K hKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5 \u22a2 \u2191\u2191\u03bc (K \\ U') + \u2191\u2191\u03bc U' + \u03b5 \u2264 \u2191\u2191\u03bc (K \\ U') + \u03b5 + \u03b5 ** apply add_le_add_right ** case bc \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s hr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U hUt : \u2191\u2191\u03bc U < \u22a4 h\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5 U' : Set \u03b1 hU'o : IsOpen U' h\u03bcU' : \u2191\u2191\u03bc U' < \u03b5 hsU' : U \\ U' \u2286 s K : Set \u03b1 hKU : K \u2286 U hKc : p K hKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5 \u22a2 \u2191\u2191\u03bc (K \\ U') + \u2191\u2191\u03bc U' \u2264 \u2191\u2191\u03bc (K \\ U') + \u03b5 ** apply add_le_add_left ** case bc.bc \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U\u271d s\u271d : Set \u03b1 \u03b5\u271d r : \u211d\u22650\u221e inst\u271d : OuterRegular \u03bc H : InnerRegular \u03bc p IsOpen h0 : p \u2205 hd : \u2200 \u2983s U : Set \u03b1\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h\u03b5s : \u03b5 + \u03b5 \u2264 \u2191\u2191\u03bc s hr : \u2191\u2191\u03bc s - (\u03b5 + \u03b5) < \u2191\u2191\u03bc s U : Set \u03b1 hsU : U \u2287 s hUo : IsOpen U hUt : \u2191\u2191\u03bc U < \u22a4 h\u03bcU : \u2191\u2191\u03bc (U \\ s) < \u03b5 U' : Set \u03b1 hU'o : IsOpen U' h\u03bcU' : \u2191\u2191\u03bc U' < \u03b5 hsU' : U \\ U' \u2286 s K : Set \u03b1 hKU : K \u2286 U hKc : p K hKr : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u03b5 \u22a2 \u2191\u2191\u03bc U' \u2264 \u03b5 ** exact h\u03bcU'.le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.univ_pi_Ico_ae_eq_Icc ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u2074 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i) f g : (i : \u03b9) \u2192 \u03b1 i \u22a2 (Set.pi univ fun i => Ico (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Icc f g ** rw [\u2190 pi_univ_Icc] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u2074 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i) f g : (i : \u03b9) \u2192 \u03b1 i \u22a2 (Set.pi univ fun i => Ico (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Icc (f i) (g i) ** exact pi_Ico_ae_eq_pi_Icc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.simpleFunc.smul_toSimpleFunc ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedRing \ud835\udd5c inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : BoundedSMul \ud835\udd5c E k : \ud835\udd5c f : { x // x \u2208 simpleFunc E p \u03bc } \u22a2 \u2191(toSimpleFunc (k \u2022 f)) =\u1d50[\u03bc] k \u2022 \u2191(toSimpleFunc f) ** filter_upwards [toSimpleFunc_eq_toFun (k \u2022 f), toSimpleFunc_eq_toFun f,\n  Lp.coeFn_smul k (f : Lp E p \u03bc)] with _ ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedRing \ud835\udd5c inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : BoundedSMul \ud835\udd5c E k : \ud835\udd5c f : { x // x \u2208 simpleFunc E p \u03bc } a\u271d : \u03b1 \u22a2 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191\u2191(k \u2022 f) a\u271d \u2192     \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192       \u2191\u2191(k \u2022 \u2191f) a\u271d = (k \u2022 \u2191\u2191\u2191f) a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = (k \u2022 \u2191(toSimpleFunc f)) a\u271d ** simp only [Pi.smul_apply, coe_smul] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedRing \ud835\udd5c inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : BoundedSMul \ud835\udd5c E k : \ud835\udd5c f : { x // x \u2208 simpleFunc E p \u03bc } a\u271d : \u03b1 \u22a2 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d \u2192     \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = k \u2022 \u2191(toSimpleFunc f) a\u271d ** repeat intro h; rw [h] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedRing \ud835\udd5c inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : BoundedSMul \ud835\udd5c E k : \ud835\udd5c f : { x // x \u2208 simpleFunc E p \u03bc } a\u271d : \u03b1 h\u271d : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d h : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u22a2 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d ** intro h ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedRing \ud835\udd5c inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : BoundedSMul \ud835\udd5c E k : \ud835\udd5c f : { x // x \u2208 simpleFunc E p \u03bc } a\u271d : \u03b1 h\u271d\u00b9 : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d h\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d h : \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u22a2 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d ** rw [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.univ_pow ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b3 : DecidableEq \u03b1 inst\u271d\u00b2 : DecidableEq \u03b2 inst\u271d\u00b9 : Monoid \u03b1 s t : Finset \u03b1 a : \u03b1 m n : \u2115 inst\u271d : Fintype \u03b1 hn : n \u2260 0 \u22a2 \u2191(univ ^ n) = \u2191univ ** rw [coe_pow, coe_univ, Set.univ_pow hn] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PointedMap.headI_map ** \u0393 : Type u_1 \u0393' : Type u_2 inst\u271d\u00b9 : Inhabited \u0393 inst\u271d : Inhabited \u0393' f : PointedMap \u0393 \u0393' l : List \u0393 \u22a2 List.headI (List.map f.f l) = Turing.PointedMap.f f (List.headI l) ** cases l <;> [exact (PointedMap.map_pt f).symm; rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_ereal_toReal ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u22a2 Measurable fun p => EReal.toReal \u2191p ** simpa using measurable_id ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.filter_apply_ne_zero_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s : Set \u03b1 h : \u2203 a, a \u2208 s \u2227 a \u2208 support p a : \u03b1 \u22a2 \u2191(filter p s h) a \u2260 0 \u2194 a \u2208 s \u2227 a \u2208 support p ** rw [Ne.def, filter_apply_eq_zero_iff, not_or, Classical.not_not, Classical.not_not] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.ToPartrec.cont_eval_fix ** f : Code k : Cont v : List \u2115 fok : Code.Ok f \u22a2 eval step (stepNormal f (Cont.fix f k) v) = do     let v \u2190 Code.eval (Code.fix f) v     eval step (Cfg.ret k v) ** refine' Part.ext fun x => _ ** f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg \u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u2194     x \u2208 do       let v \u2190 Code.eval (Code.fix f) v       eval step (Cfg.ret k v) ** simp only [Part.bind_eq_bind, Part.mem_bind_iff] ** f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg \u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u2194 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a) ** constructor ** case mp f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg \u22a2 \u2200 (c : Cfg),     x \u2208 eval step c \u2192       \u2200 (v : List \u2115) (c' : Cfg),         c = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) ** refine' fun c he => evalInduction he fun y h IH => _ ** case mp f : Code k : Cont v : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c y : Cfg h : x \u2208 eval step y IH :   \u2200 (a' : Cfg),     step y = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) \u22a2 \u2200 (v : List \u2115) (c' : Cfg),     y = Cfg.then c' (Cont.fix f k) \u2192       Reaches step (stepNormal f Cont.halt v) c' \u2192         \u2203 v\u2081,           v\u2081 \u2208 Code.eval f v \u2227             \u2203 v\u2082,               (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                 x \u2208 eval step (Cfg.ret k v\u2082) ** rintro v (\u27e8v'\u27e9 | \u27e8k', v'\u27e9) rfl hr <;> rw [Cfg.then] at h IH <;> simp only [] at h IH ** f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg this :   \u2200 (c : Cfg),     x \u2208 eval step c \u2192       \u2200 (v : List \u2115) (c' : Cfg),         c = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) \u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u2192 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a) ** intro h ** f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg this :   \u2200 (c : Cfg),     x \u2208 eval step c \u2192       \u2200 (v : List \u2115) (c' : Cfg),         c = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) h : x \u2208 eval step (stepNormal f (Cont.fix f k) v) \u22a2 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a) ** obtain \u27e8v\u2081, hv\u2081, v\u2082, hv\u2082, h\u2083\u27e9 :=\n  this _ h _ _ (stepNormal_then _ Cont.halt _ _) ReflTransGen.refl ** case intro.intro.intro.intro f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg this :   \u2200 (c : Cfg),     x \u2208 eval step c \u2192       \u2200 (v : List \u2115) (c' : Cfg),         c = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) h : x \u2208 eval step (stepNormal f (Cont.fix f k) v) v\u2081 : List \u2115 hv\u2081 : v\u2081 \u2208 Code.eval f v v\u2082 : List \u2115 hv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081) h\u2083 : x \u2208 eval step (Cfg.ret k v\u2082) \u22a2 \u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a) ** refine' \u27e8v\u2082, PFun.mem_fix_iff.2 _, h\u2083\u27e9 ** case intro.intro.intro.intro f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg this :   \u2200 (c : Cfg),     x \u2208 eval step c \u2192       \u2200 (v : List \u2115) (c' : Cfg),         c = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) h : x \u2208 eval step (stepNormal f (Cont.fix f k) v) v\u2081 : List \u2115 hv\u2081 : v\u2081 \u2208 Code.eval f v v\u2082 : List \u2115 hv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081) h\u2083 : x \u2208 eval step (Cfg.ret k v\u2082) \u22a2 Sum.inl v\u2082 \u2208       Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2228     \u2203 a',       Sum.inr a' \u2208           Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))             (Code.eval f v) \u2227         v\u2082 \u2208           PFun.fix             (fun v =>               Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))                 (Code.eval f v))             a' ** simp only [Part.eq_some_iff.2 hv\u2081, Part.map_some] ** case intro.intro.intro.intro f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg this :   \u2200 (c : Cfg),     x \u2208 eval step c \u2192       \u2200 (v : List \u2115) (c' : Cfg),         c = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) h : x \u2208 eval step (stepNormal f (Cont.fix f k) v) v\u2081 : List \u2115 hv\u2081 : v\u2081 \u2208 Code.eval f v v\u2082 : List \u2115 hv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081) h\u2083 : x \u2208 eval step (Cfg.ret k v\u2082) \u22a2 Sum.inl v\u2082 \u2208 Part.some (if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2227         v\u2082 \u2208           PFun.fix             (fun v =>               Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))                 (Code.eval f v))             a' ** split_ifs at hv\u2082 \u22a2 ** case pos f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg this :   \u2200 (c : Cfg),     x \u2208 eval step c \u2192       \u2200 (v : List \u2115) (c' : Cfg),         c = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) h : x \u2208 eval step (stepNormal f (Cont.fix f k) v) v\u2081 : List \u2115 hv\u2081 : v\u2081 \u2208 Code.eval f v v\u2082 : List \u2115 h\u2083 : x \u2208 eval step (Cfg.ret k v\u2082) h\u271d : List.headI v\u2081 = 0 hv\u2082 : v\u2082 \u2208 pure (List.tail v\u2081) \u22a2 Sum.inl v\u2082 \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2227         v\u2082 \u2208           PFun.fix             (fun v =>               Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))                 (Code.eval f v))             a' ** rw [Part.mem_some_iff.1 hv\u2082] ** case pos f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg this :   \u2200 (c : Cfg),     x \u2208 eval step c \u2192       \u2200 (v : List \u2115) (c' : Cfg),         c = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) h : x \u2208 eval step (stepNormal f (Cont.fix f k) v) v\u2081 : List \u2115 hv\u2081 : v\u2081 \u2208 Code.eval f v v\u2082 : List \u2115 h\u2083 : x \u2208 eval step (Cfg.ret k v\u2082) h\u271d : List.headI v\u2081 = 0 hv\u2082 : v\u2082 \u2208 pure (List.tail v\u2081) \u22a2 Sum.inl (List.tail v\u2081) \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (Sum.inl (List.tail v\u2081)) \u2227         List.tail v\u2081 \u2208           PFun.fix             (fun v =>               Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))                 (Code.eval f v))             a' ** exact Or.inl (Part.mem_some _) ** case neg f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg this :   \u2200 (c : Cfg),     x \u2208 eval step c \u2192       \u2200 (v : List \u2115) (c' : Cfg),         c = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) h : x \u2208 eval step (stepNormal f (Cont.fix f k) v) v\u2081 : List \u2115 hv\u2081 : v\u2081 \u2208 Code.eval f v v\u2082 : List \u2115 h\u2083 : x \u2208 eval step (Cfg.ret k v\u2082) h\u271d : \u00acList.headI v\u2081 = 0 hv\u2082 : v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v\u2081) \u22a2 Sum.inl v\u2082 \u2208 Part.some (Sum.inr (List.tail v\u2081)) \u2228     \u2203 a',       Sum.inr a' \u2208 Part.some (Sum.inr (List.tail v\u2081)) \u2227         v\u2082 \u2208           PFun.fix             (fun v =>               Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))                 (Code.eval f v))             a' ** exact Or.inr \u27e8_, Part.mem_some _, hv\u2082\u27e9 ** case mp.halt f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') \u22a2 \u2203 v\u2081,     v\u2081 \u2208 Code.eval f v \u2227       \u2203 v\u2082,         (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227           x \u2208 eval step (Cfg.ret k v\u2082) ** have := mem_eval.2 \u27e8hr, rfl\u27e9 ** case mp.halt f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') this : Cfg.halt v' \u2208 eval step (stepNormal f Cont.halt v) \u22a2 \u2203 v\u2081,     v\u2081 \u2208 Code.eval f v \u2227       \u2203 v\u2082,         (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227           x \u2208 eval step (Cfg.ret k v\u2082) ** rw [fok, Part.bind_eq_bind, Part.mem_bind_iff] at this ** case mp.halt f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') this : \u2203 a, a \u2208 Code.eval f v \u2227 Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt a) \u22a2 \u2203 v\u2081,     v\u2081 \u2208 Code.eval f v \u2227       \u2203 v\u2082,         (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227           x \u2208 eval step (Cfg.ret k v\u2082) ** obtain \u27e8v'', h\u2081, h\u2082\u27e9 := this ** case mp.halt.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') v'' : List \u2115 h\u2081 : v'' \u2208 Code.eval f v h\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'') \u22a2 \u2203 v\u2081,     v\u2081 \u2208 Code.eval f v \u2227       \u2203 v\u2082,         (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227           x \u2208 eval step (Cfg.ret k v\u2082) ** rw [reaches_eval] at h\u2082 ** case mp.halt.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') v'' : List \u2115 h\u2081 : v'' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'') h\u2082 : Cfg.halt v' \u2208 eval step ?m.194376 \u22a2 \u2203 v\u2081,     v\u2081 \u2208 Code.eval f v \u2227       \u2203 v\u2082,         (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227           x \u2208 eval step (Cfg.ret k v\u2082)  case mp.halt.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') v'' : List \u2115 h\u2081 : v'' \u2208 Code.eval f v h\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'') \u22a2 Reaches step (Cfg.ret Cont.halt v'') ?m.194376  f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') v'' : List \u2115 h\u2081 : v'' \u2208 Code.eval f v h\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'') \u22a2 Cfg ** swap ** case mp.halt.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') v'' : List \u2115 h\u2081 : v'' \u2208 Code.eval f v h\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'') \u22a2 Reaches step (Cfg.ret Cont.halt v'') ?m.194376  case mp.halt.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') v'' : List \u2115 h\u2081 : v'' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'') h\u2082 : Cfg.halt v' \u2208 eval step ?m.194376 \u22a2 \u2203 v\u2081,     v\u2081 \u2208 Code.eval f v \u2227       \u2203 v\u2082,         (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227           x \u2208 eval step (Cfg.ret k v\u2082)  f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') v'' : List \u2115 h\u2081 : v'' \u2208 Code.eval f v h\u2082 : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'') \u22a2 Cfg ** exact ReflTransGen.single rfl ** case mp.halt.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') v'' : List \u2115 h\u2081 : v'' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v'') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v'') \u22a2 \u2203 v\u2081,     v\u2081 \u2208 Code.eval f v \u2227       \u2203 v\u2082,         (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227           x \u2208 eval step (Cfg.ret k v\u2082) ** cases Part.mem_unique h\u2082 (mem_eval.2 \u27e8ReflTransGen.refl, rfl\u27e9) ** case mp.halt.intro.intro.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') \u22a2 \u2203 v\u2081,     v\u2081 \u2208 Code.eval f v \u2227       \u2203 v\u2082,         (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227           x \u2208 eval step (Cfg.ret k v\u2082) ** refine' \u27e8v', h\u2081, _\u27e9 ** case mp.halt.intro.intro.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (stepRet (Cont.fix f k) v') IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') \u22a2 \u2203 v\u2082,     (v\u2082 \u2208 if List.headI v' = 0 then pure (List.tail v') else Code.eval (Code.fix f) (List.tail v')) \u2227       x \u2208 eval step (Cfg.ret k v\u2082) ** rw [stepRet] at h ** case mp.halt.intro.intro.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 h : x \u2208 eval step (if List.headI v' = 0 then stepRet k (List.tail v') else stepNormal f (Cont.fix f k) (List.tail v')) IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') \u22a2 \u2203 v\u2082,     (v\u2082 \u2208 if List.headI v' = 0 then pure (List.tail v') else Code.eval (Code.fix f) (List.tail v')) \u2227       x \u2208 eval step (Cfg.ret k v\u2082) ** revert h ** case mp.halt.intro.intro.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') \u22a2 x \u2208 eval step (if List.headI v' = 0 then stepRet k (List.tail v') else stepNormal f (Cont.fix f k) (List.tail v')) \u2192     \u2203 v\u2082,       (v\u2082 \u2208 if List.headI v' = 0 then pure (List.tail v') else Code.eval (Code.fix f) (List.tail v')) \u2227         x \u2208 eval step (Cfg.ret k v\u2082) ** by_cases he : v'.headI = 0 <;> simp only [exists_prop, if_pos, if_false, he] <;> intro h ** case pos f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : List.headI v' = 0 h : x \u2208 eval step (stepRet k (List.tail v')) \u22a2 \u2203 v\u2082, v\u2082 \u2208 pure (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082) ** refine' \u27e8_, Part.mem_some _, _\u27e9 ** case pos f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : List.headI v' = 0 h : x \u2208 eval step (stepRet k (List.tail v')) \u22a2 x \u2208 eval step (Cfg.ret k (List.tail v')) ** rw [reaches_eval] ** case pos f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : List.headI v' = 0 h : x \u2208 eval step (stepRet k (List.tail v')) \u22a2 x \u2208 eval step ?m.195368  case pos f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : List.headI v' = 0 h : x \u2208 eval step (stepRet k (List.tail v')) \u22a2 Reaches step (Cfg.ret k (List.tail v')) ?m.195368  f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : List.headI v' = 0 h : x \u2208 eval step (stepRet k (List.tail v')) \u22a2 Cfg ** exact h ** case pos f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : List.headI v' = 0 h : x \u2208 eval step (stepRet k (List.tail v')) \u22a2 Reaches step (Cfg.ret k (List.tail v')) (stepRet k (List.tail v')) ** exact ReflTransGen.single rfl ** case neg f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) \u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082) ** obtain \u27e8k\u2080, v\u2080, e\u2080\u27e9 := stepNormal.is_ret f Cont.halt v'.tail ** case neg.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 \u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082) ** have e\u2081 := stepNormal_then f Cont.halt (Cont.fix f k) v'.tail ** case neg.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 :   stepNormal f (Cont.then Cont.halt (Cont.fix f k)) (List.tail v') =     Cfg.then (stepNormal f Cont.halt (List.tail v')) (Cont.fix f k) \u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082) ** rw [e\u2080, Cont.then, Cfg.then] at e\u2081 ** case neg.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 :   stepNormal f ((fun k' => k') (Cont.fix f k)) (List.tail v') =     (match (motive := Cfg \u2192 Cont \u2192 Cfg) Cfg.ret k\u2080 v\u2080 with       | Cfg.halt v => fun k' => stepRet k' v       | Cfg.ret k v => fun k' => Cfg.ret (Cont.then k k') v)       (Cont.fix f k) \u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082) ** simp only [] at e\u2081 ** case neg.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080 \u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082) ** obtain \u27e8v\u2081, hv\u2081, v\u2082, hv\u2082, h\u2083\u27e9 :=\n  IH (stepRet (k\u2080.then (Cont.fix f k)) v\u2080) (by rw [stepRet, if_neg he, e\u2081]; rfl)\n    v'.tail _ stepRet_then (by apply ReflTransGen.single; rw [e\u2080]; rfl) ** f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080 \u22a2 step (stepRet (Cont.fix f k) v') = some (stepRet (Cont.then k\u2080 (Cont.fix f k)) v\u2080) ** rw [stepRet, if_neg he, e\u2081] ** f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080 \u22a2 step (Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080) = some (stepRet (Cont.then k\u2080 (Cont.fix f k)) v\u2080) ** rfl ** f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080 \u22a2 Reaches step (stepNormal f Cont.halt (List.tail v')) (stepRet k\u2080 v\u2080) ** apply ReflTransGen.single ** case hab f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080 \u22a2 stepRet k\u2080 v\u2080 \u2208 step (stepNormal f Cont.halt (List.tail v')) ** rw [e\u2080] ** case hab f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080 \u22a2 stepRet k\u2080 v\u2080 \u2208 step (Cfg.ret k\u2080 v\u2080) ** rfl ** case neg.intro.intro.intro.intro.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080 v\u2081 : List \u2115 hv\u2081 : v\u2081 \u2208 Code.eval f (List.tail v') v\u2082 : List \u2115 hv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081) h\u2083 : x \u2208 eval step (Cfg.ret k v\u2082) \u22a2 \u2203 v\u2082, v\u2082 \u2208 Code.eval (Code.fix f) (List.tail v') \u2227 x \u2208 eval step (Cfg.ret k v\u2082) ** refine' \u27e8_, PFun.mem_fix_iff.2 _, h\u2083\u27e9 ** case neg.intro.intro.intro.intro.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080 v\u2081 : List \u2115 hv\u2081 : v\u2081 \u2208 Code.eval f (List.tail v') v\u2082 : List \u2115 hv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081) h\u2083 : x \u2208 eval step (Cfg.ret k v\u2082) \u22a2 Sum.inl v\u2082 \u2208       Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))         (Code.eval f (List.tail v')) \u2228     \u2203 a',       Sum.inr a' \u2208           Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))             (Code.eval f (List.tail v')) \u2227         v\u2082 \u2208           PFun.fix             (fun v =>               Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))                 (Code.eval f v))             a' ** simp only [Part.eq_some_iff.2 hv\u2081, Part.map_some, Part.mem_some_iff] ** case neg.intro.intro.intro.intro.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he\u271d : x \u2208 eval step c v v' : List \u2115 IH :   \u2200 (a' : Cfg),     step (stepRet (Cont.fix f k) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v') h\u2081 : v' \u2208 Code.eval f v h\u2082\u271d : Cfg.halt v' \u2208 eval step (Cfg.ret Cont.halt v') h\u2082 : Cfg.halt v' \u2208 eval step (stepRet Cont.halt v') he : \u00acList.headI v' = 0 h : x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v')) k\u2080 : Cont v\u2080 : List \u2115 e\u2080 : stepNormal f Cont.halt (List.tail v') = Cfg.ret k\u2080 v\u2080 e\u2081 : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k\u2080 (Cont.fix f k)) v\u2080 v\u2081 : List \u2115 hv\u2081 : v\u2081 \u2208 Code.eval f (List.tail v') v\u2082 : List \u2115 hv\u2082 : v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081) h\u2083 : x \u2208 eval step (Cfg.ret k v\u2082) \u22a2 (Sum.inl v\u2082 = if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2228     \u2203 a',       (Sum.inr a' = if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) \u2227         v\u2082 \u2208           PFun.fix             (fun v =>               Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))                 (Code.eval f v))             a' ** split_ifs at hv\u2082 \u22a2 <;> [exact Or.inl (congr_arg Sum.inl (Part.mem_some_iff.1 hv\u2082));\n  exact Or.inr \u27e8_, rfl, hv\u2082\u27e9] ** case mp.ret f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x c : Cfg he : x \u2208 eval step c v : List \u2115 k' : Cont v' : List \u2115 h : x \u2208 eval step (Cfg.ret (Cont.then k' (Cont.fix f k)) v') IH :   \u2200 (a' : Cfg),     step (Cfg.ret (Cont.then k' (Cont.fix f k)) v') = some a' \u2192       \u2200 (v : List \u2115) (c' : Cfg),         a' = Cfg.then c' (Cont.fix f k) \u2192           Reaches step (stepNormal f Cont.halt v) c' \u2192             \u2203 v\u2081,               v\u2081 \u2208 Code.eval f v \u2227                 \u2203 v\u2082,                   (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227                     x \u2208 eval step (Cfg.ret k v\u2082) hr : Reaches step (stepNormal f Cont.halt v) (Cfg.ret k' v') \u22a2 \u2203 v\u2081,     v\u2081 \u2208 Code.eval f v \u2227       \u2203 v\u2082,         (v\u2082 \u2208 if List.headI v\u2081 = 0 then pure (List.tail v\u2081) else Code.eval (Code.fix f) (List.tail v\u2081)) \u2227           x \u2208 eval step (Cfg.ret k v\u2082) ** exact IH _ rfl _ _ stepRet_then (ReflTransGen.tail hr rfl) ** case mpr f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg \u22a2 (\u2203 a, a \u2208 Code.eval (Code.fix f) v \u2227 x \u2208 eval step (Cfg.ret k a)) \u2192 x \u2208 eval step (stepNormal f (Cont.fix f k) v) ** rintro \u27e8v', he, hr\u27e9 ** case mpr.intro.intro f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v hr : x \u2208 eval step (Cfg.ret k v') \u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) ** rw [reaches_eval] at hr ** case mpr.intro.intro f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step ?m.197346 \u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)  case mpr.intro.intro f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v hr : x \u2208 eval step (Cfg.ret k v') \u22a2 Reaches step (Cfg.ret k v') ?m.197346  f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v hr : x \u2208 eval step (Cfg.ret k v') \u22a2 Cfg ** swap ** case mpr.intro.intro f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v hr : x \u2208 eval step (Cfg.ret k v') \u22a2 Reaches step (Cfg.ret k v') ?m.197346  case mpr.intro.intro f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step ?m.197346 \u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v)  f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v hr : x \u2208 eval step (Cfg.ret k v') \u22a2 Cfg ** exact ReflTransGen.single rfl ** case mpr.intro.intro f : Code k : Cont v : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') \u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) ** refine' PFun.fixInduction he fun v (he : v' \u2208 f.fix.eval v) IH => _ ** case mpr.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') \u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) v) ** rw [fok, Part.bind_eq_bind, Part.mem_bind_iff] ** case mpr.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** obtain he | \u27e8v'', he\u2081', _\u27e9 := PFun.mem_fix_iff.1 he ** case mpr.intro.intro.inl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d\u00b9 : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') he :   Sum.inl v' \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** obtain \u27e8v', he\u2081, he\u2082\u27e9 := (Part.mem_map_iff _).1 he ** case mpr.intro.intro.inl.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v'\u271d : List \u2115 he\u271d\u00b9 : v'\u271d \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v'\u271d) hr : x \u2208 eval step (stepRet k v'\u271d) v : List \u2115 he\u271d : v'\u271d \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') he :   Sum.inl v'\u271d \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v he\u2082 : (if List.headI v' = 0 then Sum.inl (List.tail v') else Sum.inr (List.tail v')) = Sum.inl v'\u271d \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** split_ifs at he\u2082 with h ** case pos f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v'\u271d : List \u2115 he\u271d\u00b9 : v'\u271d \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v'\u271d) hr : x \u2208 eval step (stepRet k v'\u271d) v : List \u2115 he\u271d : v'\u271d \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') he :   Sum.inl v'\u271d \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u2082 : Sum.inl (List.tail v') = Sum.inl v'\u271d \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** cases he\u2082 ** case pos.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v : List \u2115 IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v')) hr : x \u2208 eval step (stepRet k (List.tail v')) he\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v he :   Sum.inl (List.tail v') \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** refine' \u27e8_, he\u2081, _\u27e9 ** case pos.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v : List \u2115 IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v')) hr : x \u2208 eval step (stepRet k (List.tail v')) he\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v he :   Sum.inl (List.tail v') \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 x \u2208 eval step (Cfg.ret (Cont.fix f k) v') ** rw [reaches_eval] ** case pos.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v : List \u2115 IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v')) hr : x \u2208 eval step (stepRet k (List.tail v')) he\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v he :   Sum.inl (List.tail v') \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 x \u2208 eval step ?m.198247  case pos.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v : List \u2115 IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v')) hr : x \u2208 eval step (stepRet k (List.tail v')) he\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v he :   Sum.inl (List.tail v') \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 Reaches step (Cfg.ret (Cont.fix f k) v') ?m.198247  f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v : List \u2115 IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v')) hr : x \u2208 eval step (stepRet k (List.tail v')) he\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v he :   Sum.inl (List.tail v') \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 Cfg ** swap ** case pos.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v : List \u2115 IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v')) hr : x \u2208 eval step (stepRet k (List.tail v')) he\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v he :   Sum.inl (List.tail v') \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 Reaches step (Cfg.ret (Cont.fix f k) v') ?m.198247  case pos.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v : List \u2115 IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v')) hr : x \u2208 eval step (stepRet k (List.tail v')) he\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v he :   Sum.inl (List.tail v') \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 x \u2208 eval step ?m.198247  f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v : List \u2115 IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v')) hr : x \u2208 eval step (stepRet k (List.tail v')) he\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v he :   Sum.inl (List.tail v') \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 Cfg ** exact ReflTransGen.single rfl ** case pos.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v : List \u2115 IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v' : List \u2115 he\u2081 : v' \u2208 Code.eval f v h : List.headI v' = 0 he\u271d\u00b9 : List.tail v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k (List.tail v')) hr : x \u2208 eval step (stepRet k (List.tail v')) he\u271d : List.tail v' \u2208 Code.eval (Code.fix f) v he :   Sum.inl (List.tail v') \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u22a2 x \u2208 eval step (stepRet (Cont.fix f k) v') ** rwa [stepRet, if_pos h] ** case mpr.intro.intro.inr.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v'' : List \u2115 he\u2081' :   Sum.inr v'' \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       v'' \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** obtain \u27e8v\u2081, he\u2081, he\u2082\u27e9 := (Part.mem_map_iff _).1 he\u2081' ** case mpr.intro.intro.inr.intro.intro.intro.intro f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v'' : List \u2115 he\u2081' :   Sum.inr v'' \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       v'' v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v he\u2082 : (if List.headI v\u2081 = 0 then Sum.inl (List.tail v\u2081) else Sum.inr (List.tail v\u2081)) = Sum.inr v'' \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** split_ifs at he\u2082 with h ** case neg f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v'' : List \u2115 he\u2081' :   Sum.inr v'' \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       v'' v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 he\u2082 : Sum.inr (List.tail v\u2081) = Sum.inr v'' \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** cases he\u2082 ** case neg.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 he\u2081' :   Sum.inr (List.tail v\u2081) \u2208     Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** clear he\u2081' ** case neg.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 \u2203 a, a \u2208 Code.eval f v \u2227 x \u2208 eval step (Cfg.ret (Cont.fix f k) a) ** refine' \u27e8_, he\u2081, _\u27e9 ** case neg.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 x \u2208 eval step (Cfg.ret (Cont.fix f k) v\u2081) ** rw [reaches_eval] ** case neg.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 x \u2208 eval step ?m.198933  case neg.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 Reaches step (Cfg.ret (Cont.fix f k) v\u2081) ?m.198933  f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 Cfg ** swap ** case neg.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 Reaches step (Cfg.ret (Cont.fix f k) v\u2081) ?m.198933  case neg.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 x \u2208 eval step ?m.198933  f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 Cfg ** exact ReflTransGen.single rfl ** case neg.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 x \u2208 eval step (stepRet (Cont.fix f k) v\u2081) ** rw [stepRet, if_neg h] ** case neg.refl f : Code k : Cont v\u271d : List \u2115 fok : Code.Ok f x : Cfg v' : List \u2115 he\u271d : v' \u2208 Code.eval (Code.fix f) v\u271d hr\u271d : x \u2208 eval step (Cfg.ret k v') hr : x \u2208 eval step (stepRet k v') v : List \u2115 he : v' \u2208 Code.eval (Code.fix f) v IH :   \u2200 (a'' : List \u2115),     Sum.inr a'' \u2208         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v) \u2192       x \u2208 eval step (stepNormal f (Cont.fix f k) a'') v\u2081 : List \u2115 he\u2081 : v\u2081 \u2208 Code.eval f v h : \u00acList.headI v\u2081 = 0 right\u271d :   v' \u2208     PFun.fix       (fun v =>         Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (Code.eval f v))       (List.tail v\u2081) \u22a2 x \u2208 eval step (stepNormal f (Cont.fix f k) (List.tail v\u2081)) ** exact IH v\u2081.tail ((Part.mem_map_iff _).2 \u27e8_, he\u2081, if_neg h\u27e9) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.mem_compl_image ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f : \u03b1 \u2192 \u03b2 s t\u271d : Set \u03b1 inst\u271d : BooleanAlgebra \u03b1 t : \u03b1 S : Set \u03b1 \u22a2 t \u2208 compl '' S \u2194 t\u1d9c \u2208 S ** simp [\u2190 preimage_compl_eq_image_compl] ** Qed",
        "informal": ""
    },
    {
        "formal": "Language.mem_pow ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d : List \u03b1 l : Language \u03b1 x : List \u03b1 n : \u2115 \u22a2 x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** induction' n with n ihn generalizing x ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d\u00b9 : List \u03b1 l : Language \u03b1 x\u271d x : List \u03b1 \u22a2 x \u2208 l ^ Nat.zero \u2194 \u2203 S, x = join S \u2227 length S = Nat.zero \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** simp only [mem_one, pow_zero, length_eq_zero] ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d\u00b9 : List \u03b1 l : Language \u03b1 x\u271d x : List \u03b1 \u22a2 x = [] \u2194 \u2203 S, x = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** constructor ** case zero.mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d\u00b9 : List \u03b1 l : Language \u03b1 x\u271d x : List \u03b1 \u22a2 x = [] \u2192 \u2203 S, x = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** rintro rfl ** case zero.mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d : List \u03b1 l : Language \u03b1 x : List \u03b1 \u22a2 \u2203 S, [] = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** exact \u27e8[], rfl, rfl, fun _ h \u21a6 by contradiction\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d\u00b9 : List \u03b1 l : Language \u03b1 x x\u271d : List \u03b1 h : x\u271d \u2208 [] \u22a2 x\u271d \u2208 l ** contradiction ** case zero.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d\u00b9 : List \u03b1 l : Language \u03b1 x\u271d x : List \u03b1 \u22a2 (\u2203 S, x = join S \u2227 S = [] \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2192 x = [] ** rintro \u27e8_, rfl, rfl, _\u27e9 ** case zero.mpr.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d : List \u03b1 l : Language \u03b1 x : List \u03b1 right\u271d : \u2200 (y : List \u03b1), y \u2208 [] \u2192 y \u2208 l \u22a2 join [] = [] ** rfl ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d\u00b9 : List \u03b1 l : Language \u03b1 x\u271d : List \u03b1 n : \u2115 ihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l x : List \u03b1 \u22a2 x \u2208 l ^ Nat.succ n \u2194 \u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** simp only [pow_succ, mem_mul, ihn] ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d\u00b9 : List \u03b1 l : Language \u03b1 x\u271d : List \u03b1 n : \u2115 ihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l x : List \u03b1 \u22a2 (\u2203 a b, a \u2208 l \u2227 (\u2203 S, b = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2227 a ++ b = x) \u2194     \u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** constructor ** case succ.mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d\u00b9 : List \u03b1 l : Language \u03b1 x\u271d : List \u03b1 n : \u2115 ihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l x : List \u03b1 \u22a2 (\u2203 a b, a \u2208 l \u2227 (\u2203 S, b = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2227 a ++ b = x) \u2192     \u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ** rintro \u27e8a, b, ha, \u27e8S, rfl, rfl, hS\u27e9, rfl\u27e9 ** case succ.mp.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a\u271d b x\u271d : List \u03b1 l : Language \u03b1 x a : List \u03b1 ha : a \u2208 l S : List (List \u03b1) hS : \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l ihn : \u2200 {x : List \u03b1}, x \u2208 l ^ length S \u2194 \u2203 S_1, x = join S_1 \u2227 length S_1 = length S \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l \u22a2 \u2203 S_1, a ++ join S = join S_1 \u2227 length S_1 = Nat.succ (length S) \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l ** exact \u27e8a :: S, rfl, rfl, forall_mem_cons.2 \u27e8ha, hS\u27e9\u27e9 ** case succ.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a b x\u271d\u00b9 : List \u03b1 l : Language \u03b1 x\u271d : List \u03b1 n : \u2115 ihn : \u2200 {x : List \u03b1}, x \u2208 l ^ n \u2194 \u2203 S, x = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l x : List \u03b1 \u22a2 (\u2203 S, x = join S \u2227 length S = Nat.succ n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2192     \u2203 a b, a \u2208 l \u2227 (\u2203 S, b = join S \u2227 length S = n \u2227 \u2200 (y : List \u03b1), y \u2208 S \u2192 y \u2208 l) \u2227 a ++ b = x ** rintro \u27e8_ | \u27e8a, S\u27e9, rfl, hn, hS\u27e9 <;> cases hn ** case succ.mpr.intro.cons.intro.intro.refl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a\u271d b x\u271d : List \u03b1 l : Language \u03b1 x a : List \u03b1 S : List (List \u03b1) hS : \u2200 (y : List \u03b1), y \u2208 a :: S \u2192 y \u2208 l ihn :   \u2200 {x : List \u03b1},     x \u2208 l ^ Nat.add (length S) 0 \u2194       \u2203 S_1, x = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l \u22a2 \u2203 a_1 b,     a_1 \u2208 l \u2227       (\u2203 S_1, b = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l) \u2227         a_1 ++ b = join (a :: S) ** rw [forall_mem_cons] at hS ** case succ.mpr.intro.cons.intro.intro.refl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 l\u271d m : Language \u03b1 a\u271d b x\u271d : List \u03b1 l : Language \u03b1 x a : List \u03b1 S : List (List \u03b1) hS : a \u2208 l \u2227 \u2200 (x : List \u03b1), x \u2208 S \u2192 x \u2208 l ihn :   \u2200 {x : List \u03b1},     x \u2208 l ^ Nat.add (length S) 0 \u2194       \u2203 S_1, x = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l \u22a2 \u2203 a_1 b,     a_1 \u2208 l \u2227       (\u2203 S_1, b = join S_1 \u2227 length S_1 = Nat.add (length S) 0 \u2227 \u2200 (y : List \u03b1), y \u2208 S_1 \u2192 y \u2208 l) \u2227         a_1 ++ b = join (a :: S) ** exact \u27e8a, _, hS.1, \u27e8S, rfl, rfl, hS.2\u27e9, rfl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** rcases ENNReal.exists_pos_sum_of_countable' \u03b5pos \u2115 with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 ** case intro.intro \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 \u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** have :\n  \u2200 n,\n    \u2203 g : \u03b1 \u2192 \u211d\u22650,\n      (\u2200 x, SimpleFunc.eapproxDiff f n x \u2264 g x) \u2227\n        LowerSemicontinuous g \u2227\n          (\u222b\u207b x, g x \u2202\u03bc) \u2264 (\u222b\u207b x, SimpleFunc.eapproxDiff f n x \u2202\u03bc) + \u03b4 n :=\n  fun n =>\n  SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge \u03bc (SimpleFunc.eapproxDiff f n)\n    (\u03b4pos n).ne' ** case intro.intro \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 this :   \u2200 (n : \u2115),     \u2203 g,       (\u2200 (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g x) \u2227         LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n \u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** choose g f_le_g gcont hg using this ** case intro.intro \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n \u22a2 \u2203 g, (\u2200 (x : \u03b1), f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** refine' \u27e8fun x => \u2211' n, g n x, fun x => _, _, _\u27e9 ** case intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n x : \u03b1 \u22a2 f x \u2264 (fun x => \u2211' (n : \u2115), \u2191(g n x)) x ** rw [\u2190 SimpleFunc.tsum_eapproxDiff f hf] ** case intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n x : \u03b1 \u22a2 \u2211' (n : \u2115), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2264 (fun x => \u2211' (n : \u2115), \u2191(g n x)) x ** exact ENNReal.tsum_le_tsum fun n => ENNReal.coe_le_coe.2 (f_le_g n x) ** case intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n \u22a2 LowerSemicontinuous fun x => \u2211' (n : \u2115), \u2191(g n x) ** refine' lowerSemicontinuous_tsum fun n => _ ** case intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n n : \u2115 \u22a2 LowerSemicontinuous fun x => \u2191(g n x) ** exact\n  ENNReal.continuous_coe.comp_lowerSemicontinuous (gcont n) fun x y hxy =>\n    ENNReal.coe_le_coe.2 hxy ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n \u22a2 \u222b\u207b (x : \u03b1), \u2211' (n : \u2115), \u2191(g n x) \u2202\u03bc = \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc ** rw [lintegral_tsum fun n => (gcont n).measurable.coe_nnreal_ennreal.aemeasurable] ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n \u22a2 \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u2211' (n : \u2115), \u03b4 n \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** refine' add_le_add _ h\u03b4.le ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n \u22a2 \u2211' (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc ** rw [\u2190 lintegral_tsum] ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n \u22a2 \u222b\u207b (a : \u03b1), \u2211' (i : \u2115), \u2191(\u2191(SimpleFunc.eapproxDiff f i) a) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc ** simp_rw [SimpleFunc.tsum_eapproxDiff f hf, le_refl] ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n \u22a2 \u2200 (i : \u2115), AEMeasurable fun x => \u2191(\u2191(SimpleFunc.eapproxDiff f i) x) ** intro n ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 \u2260 0 \u03b4 : \u2115 \u2192 \u211d\u22650\u221e \u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i h\u03b4 : \u2211' (i : \u2115), \u03b4 i < \u03b5 g : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650 f_le_g : \u2200 (n : \u2115) (x : \u03b1), \u2191(SimpleFunc.eapproxDiff f n) x \u2264 g n x gcont : \u2200 (n : \u2115), LowerSemicontinuous (g n) hg : \u2200 (n : \u2115), \u222b\u207b (x : \u03b1), \u2191(g n x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) \u2202\u03bc + \u03b4 n n : \u2115 \u22a2 AEMeasurable fun x => \u2191(\u2191(SimpleFunc.eapproxDiff f n) x) ** exact (SimpleFunc.measurable _).coe_nnreal_ennreal.aemeasurable ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.toMeasureOfZeroLE_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i hi\u2081 : MeasurableSet i hj\u2081 : MeasurableSet j \u22a2 \u2191\u2191(toMeasureOfZeroLE s i hi\u2081 hi) j = \u2191{ val := \u2191s (i \u2229 j), property := (_ : 0 \u2264 \u2191s (i \u2229 j)) } ** simp_rw [toMeasureOfZeroLE, Measure.ofMeasurable_apply _ hj\u2081, toMeasureOfZeroLE',\n  s.restrict_apply hi\u2081 hj\u2081, Set.inter_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.ofDual_max' ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1\u1d52\u1d48 hs : Finset.Nonempty s \u22a2 \u2191ofDual (max' s hs) = min' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s)) ** rw [\u2190 WithTop.coe_eq_coe] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1\u1d52\u1d48 hs : Finset.Nonempty s \u22a2 \u2191(\u2191ofDual (max' s hs)) = \u2191(min' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s))) ** simp only [max'_eq_sup', id_eq, ofDual_sup', Function.comp_apply, coe_inf', min'_eq_inf',\n  inf_image] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1\u1d52\u1d48 hs : Finset.Nonempty s \u22a2 inf s (WithTop.some \u2218 fun x => \u2191ofDual x) = inf s ((WithTop.some \u2218 fun x => x) \u2218 \u2191ofDual) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.setToFun_eq_setToL1 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C f : { x // x \u2208 Lp E 1 } \u22a2 setToFun \u03bc T hT \u2191\u2191f = \u2191(setToL1 hT) f ** rw [setToFun_eq hT (L1.integrable_coeFn f), Integrable.toL1_coeFn] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpL2_indicator_of_measurable ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E \u22a2 \u2191(\u2191(condexpL2 E \ud835\udd5c hm) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =     indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c ** rw [condexpL2] ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E \u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =     indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c ** haveI : Fact (m \u2264 m0) := \u27e8hm\u27e9 ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E this : Fact (m \u2264 m0) \u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =     indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c ** have h_mem : indicatorConstLp 2 (hm s hs) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc :=\n  mem_lpMeas_indicatorConstLp hm hs h\u03bcs ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E this : Fact (m \u2264 m0) h_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc \u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =     indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c ** let ind := (\u27e8indicatorConstLp 2 (hm s hs) h\u03bcs c, h_mem\u27e9 : lpMeas E \ud835\udd5c m 2 \u03bc) ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E this : Fact (m \u2264 m0) h_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc ind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem } \u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =     indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c ** have h_coe_ind : (ind : \u03b1 \u2192\u2082[\u03bc] E) = indicatorConstLp 2 (hm s hs) h\u03bcs c := rfl ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E this : Fact (m \u2264 m0) h_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc ind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem } h_coe_ind : \u2191ind = indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =     indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c ** have h_orth_mem := orthogonalProjection_mem_subspace_eq_self ind ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E this : Fact (m \u2264 m0) h_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc ind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem } h_coe_ind : \u2191ind = indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c h_orth_mem : \u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) \u2191ind = ind \u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =     indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c ** rw [\u2190 h_coe_ind, h_orth_mem] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.cast_le ** \u03b1 : Type u_1 inst\u271d : LinearOrderedRing \u03b1 m n : ZNum \u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n ** rw [\u2190 not_lt] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedRing \u03b1 m n : ZNum \u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n ** exact not_congr cast_lt ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.laverage_add_measure ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** by_cases h\u03bc : IsFiniteMeasure \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : IsFiniteMeasure \u03bc \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd  case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : \u00acIsFiniteMeasure \u03bc \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** swap ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : IsFiniteMeasure \u03bc \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** by_cases h\u03bd : IsFiniteMeasure \u03bd ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : IsFiniteMeasure \u03bc h\u03bd : IsFiniteMeasure \u03bd \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd  case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : IsFiniteMeasure \u03bc h\u03bd : \u00acIsFiniteMeasure \u03bd \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** swap ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : IsFiniteMeasure \u03bc h\u03bd : IsFiniteMeasure \u03bd \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** haveI := h\u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : IsFiniteMeasure \u03bc h\u03bd : IsFiniteMeasure \u03bd this : IsFiniteMeasure \u03bc \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** haveI := h\u03bd ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : IsFiniteMeasure \u03bc h\u03bd : IsFiniteMeasure \u03bd this\u271d : IsFiniteMeasure \u03bc this : IsFiniteMeasure \u03bd \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** simp only [\u2190ENNReal.mul_div_right_comm, measure_mul_laverage, \u2190ENNReal.add_div,\n  \u2190lintegral_add_measure, \u2190Measure.add_apply, \u2190laverage_eq] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : \u00acIsFiniteMeasure \u03bc \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** rw [not_isFiniteMeasure_iff] at h\u03bc ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : \u2191\u2191\u03bc univ = \u22a4 \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** simp [laverage_eq, h\u03bc] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : IsFiniteMeasure \u03bc h\u03bd : \u00acIsFiniteMeasure \u03bd \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** rw [not_isFiniteMeasure_iff] at h\u03bd ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : IsFiniteMeasure \u03bc h\u03bd : \u2191\u2191\u03bd univ = \u22a4 \u22a2 \u2a0d\u207b (x : \u03b1), f x \u2202(\u03bc + \u03bd) =     \u2191\u2191\u03bc univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc + \u2191\u2191\u03bd univ / (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ) * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bd ** simp [laverage_eq, h\u03bd] ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.SatisfiesM_mapM ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) \u22a2 SatisfiesM (fun arr => motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } arr[i])     (mapM f as) ** rw [mapM_eq_foldlM] ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) \u22a2 SatisfiesM (fun arr => motive (size as) \u2227 \u2203 eq, \u2200 (i : Nat) (h : i < size as), p { val := i, isLt := h } arr[i])     (foldlM (fun bs a => push bs <$> f a) #[] as 0 (size as)) ** refine SatisfiesM_foldlM (m := m) (\u03b2 := Array \u03b2)\n  (motive := fun i arr => motive i \u2227 arr.size = i \u2227 \u2200 i h2, p i (arr[i.1]'h2)) ?z ?s\n  |>.imp fun \u27e8h\u2081, eq, h\u2082\u27e9 => \u27e8h\u2081, eq, fun _ _ => h\u2082 ..\u27e9 ** case z m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) \u22a2 (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) 0 #[] ** case z => exact \u27e8h0, rfl, fun.\u27e9 ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) \u22a2 (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) 0 #[] ** exact \u27e8h0, rfl, fun.\u27e9 ** case s m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) \u22a2 \u2200 (i : Fin (size as)) (b : Array \u03b2),     (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) i.val b \u2192       SatisfiesM         ((fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val])           (i.val + 1))         (push b <$> f as[i]) ** case s =>\nintro \u27e8i, hi\u27e9 arr \u27e8ih\u2081, eq, ih\u2082\u27e9\nrefine (hs _ ih\u2081).map fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2082, by simp [eq], fun j hj => ?_\u27e9\nsimp [get_push] at hj \u22a2; split; {apply ih\u2082}\ncases j; cases (Nat.le_or_eq_of_le_succ hj).resolve_left \u2039_\u203a; cases eq; exact h\u2081 ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) \u22a2 \u2200 (i : Fin (size as)) (b : Array \u03b2),     (fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val]) i.val b \u2192       SatisfiesM         ((fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val])           (i.val + 1))         (push b <$> f as[i]) ** intro \u27e8i, hi\u27e9 arr \u27e8ih\u2081, eq, ih\u2082\u27e9 ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) i : Nat hi : i < size as arr : Array \u03b2 ih\u2081 : motive { val := i, isLt := hi }.val eq : size arr = { val := i, isLt := hi }.val ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] \u22a2 SatisfiesM     ((fun i arr => motive i \u2227 size arr = i \u2227 \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val])       ({ val := i, isLt := hi }.val + 1))     (push arr <$> f as[{ val := i, isLt := hi }]) ** refine (hs _ ih\u2081).map fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2082, by simp [eq], fun j hj => ?_\u27e9 ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) i : Nat hi : i < size as arr : Array \u03b2 ih\u2081 : motive { val := i, isLt := hi }.val eq : size arr = { val := i, isLt := hi }.val ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] a\u271d : \u03b2 x\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1) h\u2081 : p { val := i, isLt := hi } a\u271d h\u2082 : motive ({ val := i, isLt := hi }.val + 1) j : Fin (size as) hj : j.val < size (push arr a\u271d) \u22a2 p j (push arr a\u271d)[j.val] ** simp [get_push] at hj \u22a2 ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) i : Nat hi : i < size as arr : Array \u03b2 ih\u2081 : motive { val := i, isLt := hi }.val eq : size arr = { val := i, isLt := hi }.val ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] a\u271d : \u03b2 x\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1) h\u2081 : p { val := i, isLt := hi } a\u271d h\u2082 : motive ({ val := i, isLt := hi }.val + 1) j : Fin (size as) hj : j.val < size arr + 1 \u22a2 p j (if h : j.val < size arr then arr[j.val] else a\u271d) ** split ** case inl m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) i : Nat hi : i < size as arr : Array \u03b2 ih\u2081 : motive { val := i, isLt := hi }.val eq : size arr = { val := i, isLt := hi }.val ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] a\u271d : \u03b2 x\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1) h\u2081 : p { val := i, isLt := hi } a\u271d h\u2082 : motive ({ val := i, isLt := hi }.val + 1) j : Fin (size as) hj : j.val < size arr + 1 h\u271d : j.val < size arr \u22a2 p j arr[j.val]  case inr m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) i : Nat hi : i < size as arr : Array \u03b2 ih\u2081 : motive { val := i, isLt := hi }.val eq : size arr = { val := i, isLt := hi }.val ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] a\u271d : \u03b2 x\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1) h\u2081 : p { val := i, isLt := hi } a\u271d h\u2082 : motive ({ val := i, isLt := hi }.val + 1) j : Fin (size as) hj : j.val < size arr + 1 h\u271d : \u00acj.val < size arr \u22a2 p j a\u271d ** {apply ih\u2082} ** case inr m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) i : Nat hi : i < size as arr : Array \u03b2 ih\u2081 : motive { val := i, isLt := hi }.val eq : size arr = { val := i, isLt := hi }.val ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] a\u271d : \u03b2 x\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1) h\u2081 : p { val := i, isLt := hi } a\u271d h\u2082 : motive ({ val := i, isLt := hi }.val + 1) j : Fin (size as) hj : j.val < size arr + 1 h\u271d : \u00acj.val < size arr \u22a2 p j a\u271d ** cases j ** case inr.mk m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) i : Nat hi : i < size as arr : Array \u03b2 ih\u2081 : motive { val := i, isLt := hi }.val eq : size arr = { val := i, isLt := hi }.val ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] a\u271d : \u03b2 x\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1) h\u2081 : p { val := i, isLt := hi } a\u271d h\u2082 : motive ({ val := i, isLt := hi }.val + 1) val\u271d : Nat isLt\u271d : val\u271d < size as hj : { val := val\u271d, isLt := isLt\u271d }.val < size arr + 1 h\u271d : \u00ac{ val := val\u271d, isLt := isLt\u271d }.val < size arr \u22a2 p { val := val\u271d, isLt := isLt\u271d } a\u271d ** cases (Nat.le_or_eq_of_le_succ hj).resolve_left \u2039_\u203a ** case inr.mk.refl m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) i : Nat hi : i < size as arr : Array \u03b2 ih\u2081 : motive { val := i, isLt := hi }.val eq : size arr = { val := i, isLt := hi }.val ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] a\u271d : \u03b2 x\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1) h\u2081 : p { val := i, isLt := hi } a\u271d h\u2082 : motive ({ val := i, isLt := hi }.val + 1) isLt\u271d : List.length arr.data < size as hj : { val := List.length arr.data, isLt := isLt\u271d }.val < size arr + 1 h\u271d : \u00ac{ val := List.length arr.data, isLt := isLt\u271d }.val < size arr \u22a2 p { val := List.length arr.data, isLt := isLt\u271d } a\u271d ** cases eq ** case inr.mk.refl.refl m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) arr : Array \u03b2 ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] a\u271d : \u03b2 isLt\u271d : List.length arr.data < size as hj : { val := List.length arr.data, isLt := isLt\u271d }.val < size arr + 1 h\u271d : \u00ac{ val := List.length arr.data, isLt := isLt\u271d }.val < size arr hi : List.length arr.data < size as ih\u2081 : motive { val := List.length arr.data, isLt := hi }.val x\u271d : p { val := List.length arr.data, isLt := hi } a\u271d \u2227 motive ({ val := List.length arr.data, isLt := hi }.val + 1) h\u2081 : p { val := List.length arr.data, isLt := hi } a\u271d h\u2082 : motive ({ val := List.length arr.data, isLt := hi }.val + 1) \u22a2 p { val := List.length arr.data, isLt := isLt\u271d } a\u271d ** exact h\u2081 ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m as : Array \u03b1 f : \u03b1 \u2192 m \u03b2 motive : Nat \u2192 Prop h0 : motive 0 p : Fin (size as) \u2192 \u03b2 \u2192 Prop hs : \u2200 (i : Fin (size as)), motive i.val \u2192 SatisfiesM (fun x => p i x \u2227 motive (i.val + 1)) (f as[i]) i : Nat hi : i < size as arr : Array \u03b2 ih\u2081 : motive { val := i, isLt := hi }.val eq : size arr = { val := i, isLt := hi }.val ih\u2082 : \u2200 (i : Fin (size as)) (h2 : i.val < size arr), p i arr[i.val] a\u271d : \u03b2 x\u271d : p { val := i, isLt := hi } a\u271d \u2227 motive ({ val := i, isLt := hi }.val + 1) h\u2081 : p { val := i, isLt := hi } a\u271d h\u2082 : motive ({ val := i, isLt := hi }.val + 1) \u22a2 size (push arr a\u271d) = { val := i, isLt := hi }.val + 1 ** simp [eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "RegularExpression.matches'_map ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b : \u03b1 f : \u03b1 \u2192 \u03b2 a : \u03b1 \u22a2 matches' (map f (char a)) = \u2191(Language.map f) (matches' (char a)) ** rw [eq_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a\u271d b : \u03b1 f : \u03b1 \u2192 \u03b2 a : \u03b1 \u22a2 \u2191(Language.map f) (matches' (char a)) = matches' (map f (char a)) ** exact image_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 f : \u03b1 \u2192 \u03b2 R S : RegularExpression \u03b1 \u22a2 matches' (map f (R + S)) = \u2191(Language.map f) (matches' (R + S)) ** simp only [matches'_map, map, matches'_add] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 f : \u03b1 \u2192 \u03b2 R S : RegularExpression \u03b1 \u22a2 \u2191(Language.map f) (matches' R) + \u2191(Language.map f) (matches' S) = \u2191(Language.map f) (matches' R + matches' S) ** rw [map_add] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 f : \u03b1 \u2192 \u03b2 R S : RegularExpression \u03b1 \u22a2 matches' (map f (R * S)) = \u2191(Language.map f) (matches' (R * S)) ** simp only [matches'_map, map, matches'_mul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 f : \u03b1 \u2192 \u03b2 R S : RegularExpression \u03b1 \u22a2 \u2191(Language.map f) (matches' R) * \u2191(Language.map f) (matches' S) = \u2191(Language.map f) (matches' R * matches' S) ** rw [map_mul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 f : \u03b1 \u2192 \u03b2 R : RegularExpression \u03b1 \u22a2 matches' (map f (star R)) = \u2191(Language.map f) (matches' (star R)) ** simp_rw [map, matches', matches'_map] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 f : \u03b1 \u2192 \u03b2 R : RegularExpression \u03b1 \u22a2 (\u2191(Language.map f) (matches' R))\u2217 = \u2191(Language.map f) (matches' R)\u2217 ** rw [Language.kstar_eq_iSup_pow, Language.kstar_eq_iSup_pow] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 f : \u03b1 \u2192 \u03b2 R : RegularExpression \u03b1 \u22a2 \u2a06 i, \u2191(Language.map f) (matches' R) ^ i = \u2191(Language.map f) (\u2a06 i, matches' R ^ i) ** simp_rw [\u2190 map_pow] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 f : \u03b1 \u2192 \u03b2 R : RegularExpression \u03b1 \u22a2 \u2a06 i, \u2191(Language.map f) (matches' R ^ i) = \u2191(Language.map f) (\u2a06 i, matches' R ^ i) ** exact image_iUnion.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.pi_eq_generateFrom ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) i : \u03b9 \u22a2 MeasurableSpace (\u03b1 i) ** apply_assumption ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i h2C : \u2200 (i : \u03b9), IsPiSystem (C i) h3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i) \u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i) h\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) \u22a2 Measure.pi \u03bc = \u03bc\u03bd ** have h4C : \u2200 (i) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s := by\n  intro i s hs; rw [\u2190 hC]; exact measurableSet_generateFrom hs ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i h2C : \u2200 (i : \u03b9), IsPiSystem (C i) h3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i) \u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i) h\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) h4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s \u22a2 Measure.pi \u03bc = \u03bc\u03bd ** refine'\n  (FiniteSpanningSetsIn.pi h3C).ext\n    (generateFrom_eq_pi hC fun i => (h3C i).isCountablySpanning).symm (IsPiSystem.pi h2C) _ ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i h2C : \u2200 (i : \u03b9), IsPiSystem (C i) h3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i) \u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i) h\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) h4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s \u22a2 \u2200 (s : Set ((i : \u03b9) \u2192 \u03b1 i)), (s \u2208 Set.pi univ '' Set.pi univ fun i => C i) \u2192 \u2191\u2191(Measure.pi \u03bc) s = \u2191\u2191\u03bc\u03bd s ** rintro _ \u27e8s, hs, rfl\u27e9 ** case intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i h2C : \u2200 (i : \u03b9), IsPiSystem (C i) h3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i) \u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i) h\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) h4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s s : (i : \u03b9) \u2192 Set (\u03b1 i) hs : s \u2208 Set.pi univ fun i => C i \u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s) ** rw [mem_univ_pi] at hs ** case intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i h2C : \u2200 (i : \u03b9), IsPiSystem (C i) h3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i) \u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i) h\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) h4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s s : (i : \u03b9) \u2192 Set (\u03b1 i) hs : \u2200 (i : \u03b9), s i \u2208 C i \u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s) ** haveI := fun i => (h3C i).sigmaFinite ** case intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i h2C : \u2200 (i : \u03b9), IsPiSystem (C i) h3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i) \u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i) h\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) h4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s s : (i : \u03b9) \u2192 Set (\u03b1 i) hs : \u2200 (i : \u03b9), s i \u2208 C i this : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) \u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s) ** simp_rw [h\u2081 s hs, pi_pi_aux \u03bc s fun i => h4C i _ (hs i)] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i h2C : \u2200 (i : \u03b9), IsPiSystem (C i) h3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i) \u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i) h\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) \u22a2 \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s ** intro i s hs ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i h2C : \u2200 (i : \u03b9), IsPiSystem (C i) h3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i) \u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i) h\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) i : \u03b9 s : Set (\u03b1 i) hs : s \u2208 C i \u22a2 MeasurableSet s ** exact measurableSet_generateFrom hs ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.filter_apply_eq_zero_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s : Set \u03b1 h : \u2203 a, a \u2208 s \u2227 a \u2208 support p a : \u03b1 \u22a2 \u2191(filter p s h) a = 0 \u2194 \u00aca \u2208 s \u2228 \u00aca \u2208 support p ** erw [apply_eq_zero_iff, support_filter, Set.mem_inter_iff, not_and_or] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.withDensity_sum ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b9 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : \u03b9 \u2192 Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 withDensity (sum \u03bc) f = sum fun n => withDensity (\u03bc n) f ** ext1 s hs ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b9 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : \u03b9 \u2192 Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(withDensity (sum \u03bc) f) s = \u2191\u2191(sum fun n => withDensity (\u03bc n) f) s ** simp_rw [sum_apply _ hs, withDensity_apply f hs, restrict_sum \u03bc hs, lintegral_sum_measure] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.Heap.deleteMin_fst ** \u03b1 : Type u_1 s\u271d : Heap \u03b1 le : \u03b1 \u2192 \u03b1 \u2192 Bool r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 \u22a2 Option.map (fun x => x.fst) (deleteMin le (cons r a c s)) = head? le (cons r a c s) ** simp only [deleteMin, findMin_val, Option.map, head?] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 M : \u2115 hNM : N \u2264 M h : upperCrossingTime a b f N n \u03c9 < N \u22a2 upperCrossingTime a b f M n \u03c9 = upperCrossingTime a b f N n \u03c9 ** cases n ** case zero \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 M : \u2115 hNM : N \u2264 M h : upperCrossingTime a b f N Nat.zero \u03c9 < N \u22a2 upperCrossingTime a b f M Nat.zero \u03c9 = upperCrossingTime a b f N Nat.zero \u03c9 ** simp ** case succ \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 M : \u2115 hNM : N \u2264 M n\u271d : \u2115 h : upperCrossingTime a b f N (Nat.succ n\u271d) \u03c9 < N \u22a2 upperCrossingTime a b f M (Nat.succ n\u271d) \u03c9 = upperCrossingTime a b f N (Nat.succ n\u271d) \u03c9 ** exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1 ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.ToPartrec.Code.pred_eval ** v : List \u2115 \u22a2 eval pred v = pure [Nat.pred (List.headI v)] ** simp [pred] ** v : List \u2115 \u22a2 Nat.rec (Part.some [0]) (fun y x => Part.some [y]) (List.headI v) = Part.some [Nat.pred (List.headI v)] ** cases v.headI <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u22a2 \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** induction' f using MeasureTheory.SimpleFunc.induction with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing \u03b5 ** case h_ind \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** by_cases hc : c = 0 ** case neg \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** have \u03bcs_lt_top : \u03bc s < \u221e := by\n  classical\n  simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,\n    lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,\n    Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n    Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n    SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,\n    false_and_iff] using int_f ** case neg \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** have : (0 : \u211d\u22650\u221e) < \u03b5 / c := ENNReal.div_pos_iff.2 \u27e8\u03b50, ENNReal.coe_ne_top\u27e9 ** case neg \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this : 0 < \u03b5 / \u2191c \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** obtain \u27e8F, Fs, F_closed, \u03bcF\u27e9 : \u2203 (F : _), F \u2286 s \u2227 IsClosed F \u2227 \u03bc s < \u03bc F + \u03b5 / c :=\n  hs.exists_isClosed_lt_add \u03bcs_lt_top.ne this.ne' ** case neg.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** refine'\n  \u27e8Set.indicator F fun _ => c, fun x => _, F_closed.upperSemicontinuous_indicator (zero_le _),\n    _\u27e9 ** case pos \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : c = 0 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2227       UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** refine' \u27e8fun _ => 0, _, upperSemicontinuous_const, _\u27e9 ** case pos.refine'_1 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : c = 0 \u22a2 \u2200 (x : \u03b1), (fun x => 0) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x ** classical\nsimp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,\n  eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n  SimpleFunc.coe_piecewise, le_zero_iff] ** case pos.refine'_1 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : c = 0 \u22a2 \u2200 (x : \u03b1), (fun x => 0) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x ** simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,\n  eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n  SimpleFunc.coe_piecewise, le_zero_iff] ** case pos.refine'_2 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : c = 0 \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc + \u03b5 ** classical\nsimp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply,\n  SimpleFunc.const_zero, zero_add, zero_le', SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le] ** case pos.refine'_2 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : c = 0 \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc + \u03b5 ** simp only [hc, Set.indicator_zero', lintegral_const, zero_mul, Pi.zero_apply,\n  SimpleFunc.const_zero, zero_add, zero_le', SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le] ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u22a2 \u2191\u2191\u03bc s < \u22a4 ** classical\nsimpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,\n  lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,\n  Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n  Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n  SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,\n  false_and_iff] using int_f ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u22a2 \u2191\u2191\u03bc s < \u22a4 ** simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,\n  lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,\n  Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n  Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n  SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,\n  false_and_iff] using int_f ** case neg.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c x : \u03b1 \u22a2 Set.indicator F (fun x => c) x \u2264 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x ** simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise] ** case neg.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c x : \u03b1 \u22a2 Set.indicator F (fun x => c) x \u2264 Set.piecewise s (Function.const \u03b1 c) 0 x ** exact Set.indicator_le_indicator_of_subset Fs (fun x => zero_le _) _ ** case neg.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5 ** suffices (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * \u03bc F + \u03b5 by\n  classical\n  simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,\n    lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,\n    SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,\n    Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply] ** case neg.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c \u22a2 \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5 ** calc\n  (c : \u211d\u22650\u221e) * \u03bc s \u2264 c * (\u03bc F + \u03b5 / c) := mul_le_mul_left' \u03bcF.le _\n  _ = c * \u03bc F + \u03b5 := by\n    simp_rw [mul_add]\n    rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]\n    simpa using hc ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this\u271d : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c this : \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5 \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5 ** classical\nsimpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,\n  lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,\n  SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply] ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this\u271d : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c this : \u2191c * \u2191\u2191\u03bc s \u2264 \u2191c * \u2191\u2191\u03bc F + \u03b5 \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(Set.indicator F (fun x => c) x) \u2202\u03bc + \u03b5 ** simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,\n  lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,\n  SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply] ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c \u22a2 \u2191c * (\u2191\u2191\u03bc F + \u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc F + \u03b5 ** simp_rw [mul_add] ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c \u22a2 \u2191c * \u2191\u2191\u03bc F + \u2191c * (\u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc F + \u03b5 ** rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top] ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 c : \u211d\u22650 s : Set \u03b1 hs : MeasurableSet s int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 hc : \u00acc = 0 \u03bcs_lt_top : \u2191\u2191\u03bc s < \u22a4 this : 0 < \u03b5 / \u2191c F : Set \u03b1 Fs : F \u2286 s F_closed : IsClosed F \u03bcF : \u2191\u2191\u03bc s < \u2191\u2191\u03bc F + \u03b5 / \u2191c \u22a2 \u2191c \u2260 0 ** simpa using hc ** case h_add \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** have A : ((\u222b\u207b x : \u03b1, f\u2081 x \u2202\u03bc) + \u222b\u207b x : \u03b1, f\u2082 x \u2202\u03bc) \u2260 \u22a4 := by\n  rwa [\u2190 lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal] ** case h_add \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 A : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** rcases h\u2081 (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos \u03b50).ne' with\n  \u27e8g\u2081, f\u2081_le_g\u2081, g\u2081cont, g\u2081int\u27e9 ** case h_add.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 A : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 g\u2081 : \u03b1 \u2192 \u211d\u22650 f\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x g\u2081cont : UpperSemicontinuous g\u2081 g\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** rcases h\u2082 (ENNReal.add_ne_top.1 A).2 (ENNReal.half_pos \u03b50).ne' with\n  \u27e8g\u2082, f\u2082_le_g\u2082, g\u2082cont, g\u2082int\u27e9 ** case h_add.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 A : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 g\u2081 : \u03b1 \u2192 \u211d\u22650 f\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x g\u2081cont : UpperSemicontinuous g\u2081 g\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 g\u2082 : \u03b1 \u2192 \u211d\u22650 f\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x g\u2082cont : UpperSemicontinuous g\u2082 g\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x \u2264 \u2191(f\u2081 + f\u2082) x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 ** refine'\n  \u27e8fun x => g\u2081 x + g\u2082 x, fun x => add_le_add (f\u2081_le_g\u2081 x) (f\u2082_le_g\u2082 x), g\u2081cont.add g\u2082cont, _\u27e9 ** case h_add.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 A : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 g\u2081 : \u03b1 \u2192 \u211d\u22650 f\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x g\u2081cont : UpperSemicontinuous g\u2081 g\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 g\u2082 : \u03b1 \u2192 \u211d\u22650 f\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x g\u2082cont : UpperSemicontinuous g\u2082 g\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191((fun x => g\u2081 x + g\u2082 x) x) \u2202\u03bc + \u03b5 ** simp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply] ** case h_add.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 A : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 g\u2081 : \u03b1 \u2192 \u211d\u22650 f\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x g\u2081cont : UpperSemicontinuous g\u2081 g\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 g\u2082 : \u03b1 \u2192 \u211d\u22650 f\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x g\u2082cont : UpperSemicontinuous g\u2082 g\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) + \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) + \u2191(g\u2082 x) \u2202\u03bc + \u03b5 ** rw [lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal,\n  lintegral_add_left g\u2081cont.measurable.coe_nnreal_ennreal] ** case h_add.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 A : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 g\u2081 : \u03b1 \u2192 \u211d\u22650 f\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x g\u2081cont : UpperSemicontinuous g\u2081 g\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 g\u2082 : \u03b1 \u2192 \u211d\u22650 f\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x g\u2082cont : UpperSemicontinuous g\u2082 g\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5 ** convert add_le_add g\u2081int g\u2082int using 1 ** case h.e'_4 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 A : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 g\u2081 : \u03b1 \u2192 \u211d\u22650 f\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x g\u2081cont : UpperSemicontinuous g\u2081 g\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 g\u2082 : \u03b1 \u2192 \u211d\u22650 f\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x g\u2082cont : UpperSemicontinuous g\u2082 g\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + \u03b5 =     \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2) ** conv_lhs => rw [\u2190 ENNReal.add_halves \u03b5] ** case h.e'_4 \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 A : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 g\u2081 : \u03b1 \u2192 \u211d\u22650 f\u2081_le_g\u2081 : \u2200 (x : \u03b1), g\u2081 x \u2264 \u2191f\u2081 x g\u2081cont : UpperSemicontinuous g\u2081 g\u2081int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 g\u2082 : \u03b1 \u2192 \u211d\u22650 f\u2082_le_g\u2082 : \u2200 (x : \u03b1), g\u2082 x \u2264 \u2191f\u2082 x g\u2082cont : UpperSemicontinuous g\u2082 g\u2082int : \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2 \u22a2 \u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc + (\u03b5 / 2 + \u03b5 / 2) =     \u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc + \u03b5 / 2 + (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc + \u03b5 / 2) ** abel ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : WeaklyRegular \u03bc f : \u03b1 \u2192\u209b \u211d\u22650 int_f\u271d : \u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc \u2260 \u22a4 \u03b5\u271d : \u211d\u22650\u221e \u03b50\u271d : \u03b5\u271d \u2260 0 f\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650 a\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082) h\u2081 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2081 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 h\u2082 :   \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 \u2192     \u2200 {\u03b5 : \u211d\u22650\u221e},       \u03b5 \u2260 0 \u2192         \u2203 g, (\u2200 (x : \u03b1), g x \u2264 \u2191f\u2082 x) \u2227 UpperSemicontinuous g \u2227 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u03b5 int_f : \u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b50 : \u03b5 \u2260 0 \u22a2 \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc + \u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc \u2260 \u22a4 ** rwa [\u2190 lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal] ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.map_data ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 \u22a2 (map f arr).data = List.map f arr.data ** rw [map, mapM_eq_foldlM] ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 \u22a2 (Id.run (foldlM (fun bs a => push bs <$> f a) #[] arr 0 (size arr))).data = List.map f arr.data ** apply congrArg data (foldl_eq_foldl_data (fun bs a => push bs (f a)) #[] arr) |>.trans ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 \u22a2 (List.foldl (fun bs a => push bs (f a)) #[] arr.data).data = List.map f arr.data ** have H (l arr) : List.foldl (fun bs a => push bs (f a)) arr l = \u27e8arr.data ++ l.map f\u27e9 := by\n  induction l generalizing arr <;> simp [*] ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 H : \u2200 (l : List \u03b1) (arr : Array \u03b2), List.foldl (fun bs a => push bs (f a)) arr l = { data := arr.data ++ List.map f l } \u22a2 (List.foldl (fun bs a => push bs (f a)) #[] arr.data).data = List.map f arr.data ** simp [H] ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr\u271d : Array \u03b1 l : List \u03b1 arr : Array \u03b2 \u22a2 List.foldl (fun bs a => push bs (f a)) arr l = { data := arr.data ++ List.map f l } ** induction l generalizing arr <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.crossing_pos_eq ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b \u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = upperCrossingTime a b f N n \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = lowerCrossingTime a b f N n ** have hab' : 0 < b - a := sub_pos.2 hab ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 \u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = upperCrossingTime a b f N n \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = lowerCrossingTime a b f N n ** have hf' : \u2200 \u03c9 i, (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a := by\n  intro \u03c9 i\n  rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = upperCrossingTime a b f N n \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N n = lowerCrossingTime a b f N n ** induction' n with k ih ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a \u22a2 \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 ** intro i \u03c9 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a i : \u03a9 \u03c9 : \u2115 \u22a2 b - a \u2264 (f \u03c9 i - a)\u207a \u2194 b \u2264 f \u03c9 i ** refine' \u27e8fun h => _, fun h => _\u27e9 ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a i : \u03a9 \u03c9 : \u2115 h : b - a \u2264 (f \u03c9 i - a)\u207a \u22a2 b \u2264 f \u03c9 i ** rwa [\u2190 sub_le_sub_iff_right a, \u2190\n  LatticeOrderedGroup.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)] ** case refine'_2 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a i : \u03a9 \u03c9 : \u2115 h : b \u2264 f \u03c9 i \u22a2 b - a \u2264 (f \u03c9 i - a)\u207a ** rw [\u2190 sub_le_sub_iff_right a] at h ** case refine'_2 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a i : \u03a9 \u03c9 : \u2115 h : b - a \u2264 f \u03c9 i - a \u22a2 b - a \u2264 (f \u03c9 i - a)\u207a ** rwa [LatticeOrderedGroup.pos_of_nonneg _ (le_trans hab'.le h)] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 \u22a2 \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a ** intro \u03c9 i ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 \u03c9 : \u03a9 i : \u2115 \u22a2 (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a ** rw [LatticeOrderedGroup.pos_nonpos_iff, sub_nonpos] ** case zero \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N Nat.zero = upperCrossingTime a b f N Nat.zero \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N Nat.zero = lowerCrossingTime a b f N Nat.zero ** refine' \u27e8rfl, _\u27e9 ** case zero \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u22a2 lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N Nat.zero = lowerCrossingTime a b f N Nat.zero ** simp only [lowerCrossingTime_zero, hitting, Set.mem_Icc, Set.mem_Iic, Nat.zero_eq] ** case zero \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u22a2 (fun x =>       if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j x - a)\u207a \u2208 Set.Iic 0 then sInf (Set.Icc \u22a5 N \u2229 {i | (f i x - a)\u207a \u2264 0}) else N) =     fun x => if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j x \u2208 Set.Iic a then sInf (Set.Icc \u22a5 N \u2229 {i | f i x \u2264 a}) else N ** ext \u03c9 ** case zero.h \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u03c9 : \u03a9 \u22a2 (if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 then sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) else N) =     if \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a then sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a}) else N ** split_ifs with h\u2081 h\u2082 h\u2082 ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u03c9 : \u03a9 h\u2081 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a}) ** simp_rw [hf'] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u03c9 : \u03a9 h\u2081 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = N ** simp_rw [Set.mem_Iic, \u2190 hf' _ _] at h\u2082 ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u03c9 : \u03a9 h\u2081 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2264 0 \u22a2 sInf (Set.Icc \u22a5 N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = N ** exact False.elim (h\u2082 h\u2081) ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u03c9 : \u03a9 h\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 N = sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a}) ** simp_rw [Set.mem_Iic, hf' _ _] at h\u2081 ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u03c9 : \u03a9 h\u2082 : \u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a h\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2264 a \u22a2 N = sInf (Set.Icc \u22a5 N \u2229 {i | f i \u03c9 \u2264 a}) ** exact False.elim (h\u2081 h\u2082) ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a \u03c9 : \u03a9 h\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc \u22a5 N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 N = N ** rfl ** case succ \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) = upperCrossingTime a b f N (Nat.succ k) \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) = lowerCrossingTime a b f N (Nat.succ k) ** refine' \u27e8this, _\u27e9 ** case succ \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u22a2 lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) = lowerCrossingTime a b f N (Nat.succ k) ** ext \u03c9 ** case succ.h \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u03c9 : \u03a9 \u22a2 lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9 = lowerCrossingTime a b f N (Nat.succ k) \u03c9 ** simp only [lowerCrossingTime, this, hitting, Set.mem_Iic] ** case succ.h \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u03c9 : \u03a9 \u22a2 (if         \u2203 j,           j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227             (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 then       sInf (Set.Icc (upperCrossingTime a b f N (k + 1) \u03c9) N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0})     else N) =     if \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a then       sInf (Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2229 {i | f i \u03c9 \u2264 a})     else N ** split_ifs with h\u2081 h\u2082 h\u2082 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k \u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) ** ext \u03c9 ** case h \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k \u03c9 : \u03a9 \u22a2 upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) \u03c9 = upperCrossingTime a b f N (k + 1) \u03c9 ** simp only [upperCrossingTime_succ_eq, \u2190 ih.2, hitting, Set.mem_Ici, tsub_le_iff_right] ** case h \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k \u03c9 : \u03a9 \u22a2 (if         \u2203 j,           j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227             (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) then       sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 (f i \u03c9 - a)\u207a + a})     else N) =     if \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b then       sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 f i \u03c9})     else N ** split_ifs with h\u2081 h\u2082 h\u2082 ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k \u03c9 : \u03a9 h\u2081 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) h\u2082 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b \u22a2 sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 (f i \u03c9 - a)\u207a + a}) =     sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 f i \u03c9}) ** simp_rw [\u2190 sub_le_iff_le_add, hf \u03c9] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k \u03c9 : \u03a9 h\u2081 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) h\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b \u22a2 sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 (f i \u03c9 - a)\u207a + a}) = N ** refine' False.elim (h\u2082 _) ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k \u03c9 : \u03a9 h\u2081 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) h\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b \u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b ** simp_all only [Set.mem_Ici] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k \u03c9 : \u03a9 h\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) h\u2082 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b \u22a2 N = sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2229 {i | b \u2264 f i \u03c9}) ** refine' False.elim (h\u2081 _) ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k \u03c9 : \u03a9 h\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) h\u2082 : \u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b \u22a2 \u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) ** simp_all only [Set.mem_Ici] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k \u03c9 : \u03a9 h\u2081 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Ici (b - a) h\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (lowerCrossingTime a b f N k \u03c9) N \u2227 f j \u03c9 \u2208 Set.Ici b \u22a2 N = N ** rfl ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u03c9 : \u03a9 h\u2081 :   \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 sInf (Set.Icc (upperCrossingTime a b f N (k + 1) \u03c9) N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) =     sInf (Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2229 {i | f i \u03c9 \u2264 a}) ** simp_rw [hf' \u03c9] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u03c9 : \u03a9 h\u2081 :   \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 sInf (Set.Icc (upperCrossingTime a b f N (k + 1) \u03c9) N \u2229 {i | (f i \u03c9 - a)\u207a \u2264 0}) = N ** refine' False.elim (h\u2082 _) ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u03c9 : \u03a9 h\u2081 :   \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a ** simp_all only [Set.mem_Iic] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u03c9 : \u03a9 h\u2081 :   \u00ac\u2203 j,       j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 N = sInf (Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2229 {i | f i \u03c9 \u2264 a}) ** refine' False.elim (h\u2081 _) ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u03c9 : \u03a9 h\u2081 :   \u00ac\u2203 j,       j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 \u2203 j, j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 ** simp_all only [Set.mem_Iic] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hab' : 0 < b - a hf : \u2200 (\u03c9 : \u03a9) (i : \u2115), b - a \u2264 (f i \u03c9 - a)\u207a \u2194 b \u2264 f i \u03c9 hf' : \u2200 (\u03c9 : \u03a9) (i : \u2115), (f i \u03c9 - a)\u207a \u2264 0 \u2194 f i \u03c9 \u2264 a k : \u2115 ih :   upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = upperCrossingTime a b f N k \u2227     lowerCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N k = lowerCrossingTime a b f N k this : upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (k + 1) = upperCrossingTime a b f N (k + 1) \u03c9 : \u03a9 h\u2081 :   \u00ac\u2203 j,       j \u2208 Set.Icc (upperCrossingTime 0 (b - a) (fun n \u03c9 => (f n \u03c9 - a)\u207a) N (Nat.succ k) \u03c9) N \u2227 (f j \u03c9 - a)\u207a \u2208 Set.Iic 0 h\u2082 : \u00ac\u2203 j, j \u2208 Set.Icc (upperCrossingTime a b f N (Nat.succ k) \u03c9) N \u2227 f j \u03c9 \u2208 Set.Iic a \u22a2 N = N ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.Cofix.dest_corec ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: \u03b2) x : \u03b2 \u22a2 dest (corec g x) = (TypeVec.id ::: corec g) <$$> g x ** conv =>\n  lhs\n  rw [Cofix.dest, Cofix.corec]; ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: \u03b2) x : \u03b2 \u22a2 Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))       (_ :         \u2200 (x y : M (P F) \u03b1),           Mcongr x y \u2192             (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =               (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)       (Quot.mk Mcongr (corecF g x)) =     (TypeVec.id ::: corec g) <$$> g x ** dsimp ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: \u03b2) x : \u03b2 \u22a2 (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (corecF g x)) = (TypeVec.id ::: corec g) <$$> g x ** rw [corecF_eq, abs_map, abs_repr, \u2190 comp_map, \u2190 appendFun_comp] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u g : \u03b2 \u2192 F (\u03b1 ::: \u03b2) x : \u03b2 \u22a2 (TypeVec.id \u229a TypeVec.id ::: Quot.mk Mcongr \u2218 corecF g) <$$> g x = (TypeVec.id ::: corec g) <$$> g x ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.zero_prod ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2' inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd\u271d \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : SigmaFinite \u03bd\u271d inst\u271d : SigmaFinite \u03bc \u03bd : Measure \u03b2 \u22a2 Measure.prod 0 \u03bd = 0 ** rw [Measure.prod] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2' inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd\u271d \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : SigmaFinite \u03bd\u271d inst\u271d : SigmaFinite \u03bc \u03bd : Measure \u03b2 \u22a2 (bind 0 fun x => map (Prod.mk x) \u03bd) = 0 ** exact bind_zero_left _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_upperSemicontinuous_lt_integral_gt ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x < \u2191(f x)) \u2227       UpperSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < g x) \u2227 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u03b5 ** rcases exists_lt_lowerSemicontinuous_integral_lt (fun x => -f x) hf.neg \u03b5pos with\n  \u27e8g, g_lt_f, gcont, g_integrable, g_lt_top, gint\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), g x < \u2191(f x)) \u2227       UpperSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < g x) \u2227 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u03b5 ** refine' \u27e8fun x => -g x, _, _, _, _, _\u27e9 ** case intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5 \u22a2 \u2200 (x : \u03b1), (fun x => -g x) x < \u2191(f x) ** exact fun x => EReal.neg_lt_iff_neg_lt.1 (by simpa only [EReal.coe_neg] using g_lt_f x) ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5 x : \u03b1 \u22a2 -\u2191(f x) < g x ** simpa only [EReal.coe_neg] using g_lt_f x ** case intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5 \u22a2 UpperSemicontinuous fun x => -g x ** exact\n  continuous_neg.comp_lowerSemicontinuous_antitone gcont fun x y hxy =>\n    EReal.neg_le_neg_iff.2 hxy ** case intro.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5 \u22a2 Integrable fun x => EReal.toReal ((fun x => -g x) x) ** convert g_integrable.neg ** case h.e'_5.h \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5 x\u271d : \u03b1 \u22a2 EReal.toReal ((fun x => -g x) x\u271d) = (-fun x => EReal.toReal (g x)) x\u271d ** simp ** case intro.intro.intro.intro.intro.refine'_4 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u22a5 < (fun x => -g x) x ** simpa [bot_lt_iff_ne_bot, lt_top_iff_ne_top] using g_lt_top ** case intro.intro.intro.intro.intro.refine'_5 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), -f x \u2202\u03bc + \u03b5 \u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5 ** simp_rw [integral_neg, lt_neg_add_iff_add_lt] at gint ** case intro.intro.intro.intro.intro.refine'_5 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (a : \u03b1), f a \u2202\u03bc + \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u03b5 \u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5 ** rw [add_comm] at gint ** case intro.intro.intro.intro.intro.refine'_5 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 g : \u03b1 \u2192 EReal g_lt_f : \u2200 (x : \u03b1), \u2191(-f x) < g x gcont : LowerSemicontinuous g g_integrable : Integrable fun x => EReal.toReal (g x) g_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 gint : \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc + \u222b (a : \u03b1), f a \u2202\u03bc < \u03b5 \u22a2 \u222b (x : \u03b1), f x \u2202\u03bc < \u222b (x : \u03b1), EReal.toReal ((fun x => -g x) x) \u2202\u03bc + \u03b5 ** simpa [integral_neg] using gint ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.integral_eq_lintegral ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup F' inst\u271d : NormedSpace \u211d F' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } h_pos : 0 \u2264\u1d50[\u03bc] \u2191(toSimpleFunc f) \u22a2 integral f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191(toSimpleFunc f) a) \u2202\u03bc) ** rw [integral, SimpleFunc.integral_eq_lintegral (SimpleFunc.integrable f) h_pos] ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.essSup_liminf_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f\u271d : \u03b1 \u2192 \u211d\u22650\u221e \u03b9 : Type u_3 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u22a2 essSup (fun x => liminf (fun n => f n x) atTop) \u03bc \u2264 liminf (fun n => essSup (fun x => f n x) \u03bc) atTop ** simp_rw [essSup] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f\u271d : \u03b1 \u2192 \u211d\u22650\u221e \u03b9 : Type u_3 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : LinearOrder \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u22a2 limsup (fun x => liminf (fun n => f n x) atTop) (Measure.ae \u03bc) \u2264     liminf (fun n => limsup (fun x => f n x) (Measure.ae \u03bc)) atTop ** exact ENNReal.limsup_liminf_le_liminf_limsup fun a b => f b a ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.smul_neg ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : Monoid \u03b1 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : DistribMulAction \u03b1 \u03b2 a : \u03b1 s : Set \u03b1 t : Set \u03b2 \u22a2 s \u2022 -t = -(s \u2022 t) ** simp_rw [\u2190 image_neg] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : Monoid \u03b1 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : DistribMulAction \u03b1 \u03b2 a : \u03b1 s : Set \u03b1 t : Set \u03b2 \u22a2 s \u2022 Neg.neg '' t = Neg.neg '' (s \u2022 t) ** exact image_image2_right_comm smul_neg ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.all ** l m r : List Char f : Char \u2192 Bool x\u271d : Substring h : ValidFor l m r x\u271d \u22a2 Substring.all x\u271d f = List.all m f ** simp [Substring.all, h.any, List.all_eq_not_any_not] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IntegrableOn.restrict_toMeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 \u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s ** rcases exists_seq_strictAnti_tendsto (0 : \u211d) with \u27e8u, _, u_pos, u_lim\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) \u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s ** let v n := toMeasurable (\u03bc.restrict s) { x | u n \u2264 \u2016f x\u2016 } ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} \u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s ** have A : \u2200 n, \u03bc (s \u2229 v n) \u2260 \u221e := by\n  intro n\n  rw [inter_comm, \u2190 Measure.restrict_apply (measurableSet_toMeasurable _ _),\n    measure_toMeasurable]\n  exact (hf.measure_norm_ge_lt_top (u_pos n)).ne ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 \u22a2 Measure.restrict \u03bc (toMeasurable \u03bc s) = Measure.restrict \u03bc s ** apply Measure.restrict_toMeasurable_of_cover _ A ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 \u22a2 s \u2286 \u22c3 n, v n ** intro x hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 x : \u03b1 hx : x \u2208 s \u22a2 x \u2208 \u22c3 n, v n ** have : 0 < \u2016f x\u2016 := by simp only [h's x hx, norm_pos_iff, Ne.def, not_false_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 x : \u03b1 hx : x \u2208 s this : 0 < \u2016f x\u2016 \u22a2 x \u2208 \u22c3 n, v n ** obtain \u27e8n, hn\u27e9 : \u2203 n, u n < \u2016f x\u2016 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 x : \u03b1 hx : x \u2208 s this : 0 < \u2016f x\u2016 \u22a2 \u2203 n, u n < \u2016f x\u2016  case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 x : \u03b1 hx : x \u2208 s this : 0 < \u2016f x\u2016 n : \u2115 hn : u n < \u2016f x\u2016 \u22a2 x \u2208 \u22c3 n, v n ** exact ((tendsto_order.1 u_lim).2 _ this).exists ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 x : \u03b1 hx : x \u2208 s this : 0 < \u2016f x\u2016 n : \u2115 hn : u n < \u2016f x\u2016 \u22a2 x \u2208 \u22c3 n, v n ** refine' mem_iUnion.2 \u27e8n, _\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 x : \u03b1 hx : x \u2208 s this : 0 < \u2016f x\u2016 n : \u2115 hn : u n < \u2016f x\u2016 \u22a2 x \u2208 v n ** exact subset_toMeasurable _ _ hn.le ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} \u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} n : \u2115 \u22a2 \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 ** rw [inter_comm, \u2190 Measure.restrict_apply (measurableSet_toMeasurable _ _),\n  measure_toMeasurable] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} n : \u2115 \u22a2 \u2191\u2191(Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} \u2260 \u22a4 ** exact (hf.measure_norm_ge_lt_top (u_pos n)).ne ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h's : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0 u : \u2115 \u2192 \u211d left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) v : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable (Measure.restrict \u03bc s) {x | u n \u2264 \u2016f x\u2016} A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \u2229 v n) \u2260 \u22a4 x : \u03b1 hx : x \u2208 s \u22a2 0 < \u2016f x\u2016 ** simp only [h's x hx, norm_pos_iff, Ne.def, not_false_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } \u22a2 g \u2208 closure (Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) ** borelize G ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G \u22a2 g \u2208 closure (Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) ** rw [mem_closure_iff_seq_limit] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** have hg_mem\u2112p : Mem\u2112p (g : \u03b1 \u2192 G) p \u03bc := Lp.mem\u2112p (g : Lp G p \u03bc) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** have zero_mem : (0 : G) \u2208 (range (g : \u03b1 \u2192 G) \u222a {0} : Set G) \u2229 { y | 0 \u2264 y } := by\n  simp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or_iff,\n    mem_setOf_eq, le_refl, and_self_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** have : SeparableSpace ((range (g : \u03b1 \u2192 G) \u222a {0}) \u2229 { y | 0 \u2264 y } : Set G) := by\n  apply IsSeparable.separableSpace\n  apply IsSeparable.mono _ (Set.inter_subset_left _ _)\n  exact\n    (Lp.stronglyMeasurable (g : Lp G p \u03bc)).isSeparable_range.union\n      (finite_singleton _).isSeparable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** have g_meas : Measurable (g : \u03b1 \u2192 G) := (Lp.stronglyMeasurable (g : Lp G p \u03bc)).measurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** let x n := SimpleFunc.approxOn g g_meas ((range (g : \u03b1 \u2192 G) \u222a {0}) \u2229 { y | 0 \u2264 y }) 0 zero_mem n ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** have hx_nonneg : \u2200 n, 0 \u2264 x n := by\n  intro n a\n  change x n a \u2208 { y : G | 0 \u2264 y }\n  have A : (range (g : \u03b1 \u2192 G) \u222a {0} : Set G) \u2229 { y | 0 \u2264 y } \u2286 { y | 0 \u2264 y } :=\n    inter_subset_right _ _\n  apply A\n  exact SimpleFunc.approxOn_mem g_meas _ n a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** have hx_mem\u2112p : \u2200 n, Mem\u2112p (x n) p \u03bc :=\n  SimpleFunc.mem\u2112p_approxOn _ hg_mem\u2112p _ \u27e8aestronglyMeasurable_const, by simp\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** have h_toLp := fun n => Mem\u2112p.coeFn_toLp (hx_mem\u2112p n) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** have hx_nonneg_Lp : \u2200 n, 0 \u2264 toLp (x n) (hx_mem\u2112p n) := by\n  intro n\n  rw [\u2190 Lp.simpleFunc.coeFn_le, Lp.simpleFunc.toLp_eq_toLp]\n  filter_upwards [Lp.simpleFunc.coeFn_zero p \u03bc G, h_toLp n] with a ha0 ha_toLp\n  rw [ha0, ha_toLp]\n  exact hx_nonneg n a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) \u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g) ** refine'\n  \u27e8fun n =>\n    (coeSimpleFuncNonnegToLpNonneg p \u03bc G) \u27e8toLp (x n) (hx_mem\u2112p n), hx_nonneg_Lp n\u27e9,\n    fun n => mem_range_self _, _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) \u22a2 Tendsto     (fun n =>       coeSimpleFuncNonnegToLpNonneg p \u03bc G         { val := toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p), property := (_ : 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) })     atTop (\ud835\udcdd g) ** suffices Tendsto (fun n : \u2115 => (toLp (x n) (hx_mem\u2112p n) : Lp G p \u03bc)) atTop (\ud835\udcdd (g : Lp G p \u03bc)) by\n  rw [tendsto_iff_dist_tendsto_zero] at this \u22a2\n  simp_rw [Subtype.dist_eq]\n  exact this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => \u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p))) atTop (\ud835\udcdd \u2191g) ** rw [Lp.tendsto_Lp_iff_tendsto_\u2112p'] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => snorm (\u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) ** refine Filter.Tendsto.congr (fun n => snorm_congr_ae (EventuallyEq.sub ?_ ?_)) hx_tendsto ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p \u22a2 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} ** simp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or_iff,\n  mem_setOf_eq, le_refl, and_self_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u22a2 SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) ** apply IsSeparable.separableSpace ** case hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u22a2 IsSeparable ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) ** apply IsSeparable.mono _ (Set.inter_subset_left _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u22a2 IsSeparable (Set.range \u2191\u2191\u2191g \u222a {0}) ** exact\n  (Lp.stronglyMeasurable (g : Lp G p \u03bc)).isSeparable_range.union\n    (finite_singleton _).isSeparable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n \u22a2 \u2200 (n : \u2115), 0 \u2264 x n ** intro n a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n n : \u2115 a : \u03b1 \u22a2 \u21910 a \u2264 \u2191(x n) a ** change x n a \u2208 { y : G | 0 \u2264 y } ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n n : \u2115 a : \u03b1 \u22a2 \u2191(x n) a \u2208 {y | 0 \u2264 y} ** have A : (range (g : \u03b1 \u2192 G) \u222a {0} : Set G) \u2229 { y | 0 \u2264 y } \u2286 { y | 0 \u2264 y } :=\n  inter_subset_right _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n n : \u2115 a : \u03b1 A : (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y} \u22a2 \u2191(x n) a \u2208 {y | 0 \u2264 y} ** apply A ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n n : \u2115 a : \u03b1 A : (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y} \u22a2 \u2191(x n) a \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} ** exact SimpleFunc.approxOn_mem g_meas _ n a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n \u22a2 snorm (fun x => 0) p \u03bc < \u22a4 ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) \u22a2 \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) n : \u2115 \u22a2 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) ** rw [\u2190 Lp.simpleFunc.coeFn_le, Lp.simpleFunc.toLp_eq_toLp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) n : \u2115 \u22a2 \u2191\u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) ** filter_upwards [Lp.simpleFunc.coeFn_zero p \u03bc G, h_toLp n] with a ha0 ha_toLp ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) n : \u2115 a : \u03b1 ha0 : \u2191\u2191\u21910 a = OfNat.ofNat 0 a ha_toLp : \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) a = \u2191(x n) a \u22a2 \u2191\u2191\u21910 a \u2264 \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) a ** rw [ha0, ha_toLp] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) n : \u2115 a : \u03b1 ha0 : \u2191\u2191\u21910 a = OfNat.ofNat 0 a ha_toLp : \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) a = \u2191(x n) a \u22a2 OfNat.ofNat 0 a \u2264 \u2191(x n) a ** exact hx_nonneg n a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) \u22a2 Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) ** apply SimpleFunc.tendsto_approxOn_Lp_snorm g_meas zero_mem hp_ne_top ** case h\u03bc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191g x \u2208 closure ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) ** have hg_nonneg : (0 : \u03b1 \u2192 G) \u2264\u1d50[\u03bc] g := (Lp.coeFn_nonneg _).mpr g.2 ** case h\u03bc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hg_nonneg : 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191g x \u2208 closure ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) ** refine' hg_nonneg.mono fun a ha => subset_closure _ ** case h\u03bc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hg_nonneg : 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g a : \u03b1 ha : OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191g a \u22a2 \u2191\u2191\u2191g a \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} ** simpa using ha ** case hi \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) \u22a2 snorm (fun x => \u2191\u2191\u2191g x - 0) p \u03bc < \u22a4 ** simp_rw [sub_zero] ** case hi \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) \u22a2 snorm (fun x => \u2191\u2191\u2191g x) p \u03bc < \u22a4 ** exact hg_mem\u2112p.snorm_lt_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b2 : MeasurableSpace G := borel G this\u271d\u00b9 : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) this : Tendsto (fun n => \u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p))) atTop (\ud835\udcdd \u2191g) \u22a2 Tendsto     (fun n =>       coeSimpleFuncNonnegToLpNonneg p \u03bc G         { val := toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p), property := (_ : 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) })     atTop (\ud835\udcdd g) ** rw [tendsto_iff_dist_tendsto_zero] at this \u22a2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b2 : MeasurableSpace G := borel G this\u271d\u00b9 : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) this : Tendsto (fun b => dist \u2191(toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) \u2191g) atTop (\ud835\udcdd 0) \u22a2 Tendsto     (fun b =>       dist         (coeSimpleFuncNonnegToLpNonneg p \u03bc G           { val := toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p), property := (_ : 0 \u2264 toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) })         g)     atTop (\ud835\udcdd 0) ** simp_rw [Subtype.dist_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b2 : MeasurableSpace G := borel G this\u271d\u00b9 : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) this : Tendsto (fun b => dist \u2191(toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) \u2191g) atTop (\ud835\udcdd 0) \u22a2 Tendsto     (fun b =>       dist         \u2191(coeSimpleFuncNonnegToLpNonneg p \u03bc G             {               val :=                 toLp (SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem b)                   (_ : Mem\u2112p (\u2191(x b)) p),               property := (_ : 0 \u2264 toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) })         \u2191g)     atTop (\ud835\udcdd 0) ** exact this ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 (fun x_1 => \u2191(x n) x_1) =\u1d50[\u03bc] fun x_1 => \u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1 ** symm ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 (fun x_1 => \u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1) =\u1d50[\u03bc] fun x_1 => \u2191(x n) x_1 ** rw [Lp.simpleFunc.toLp_eq_toLp] ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 (fun x_1 => \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1) =\u1d50[\u03bc] fun x_1 => \u2191(x n) x_1 ** exact h_toLp n ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 g : { g // 0 \u2264 g } this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G hg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p zero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} this : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) g_meas : Measurable \u2191\u2191\u2191g x : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n hx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n hx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p h_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n) hx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p) hx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 (fun x => \u2191\u2191\u2191g x) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191g x ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.norm_stoppedValue_leastGE_le ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, stoppedValue f (leastGE f r i) \u03c9 \u2264 r + \u2191R ** filter_upwards [hbdd] with \u03c9 hbdd\u03c9 ** case h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u03c9 : \u03a9 hbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u22a2 stoppedValue f (leastGE f r i) \u03c9 \u2264 r + \u2191R ** change f (leastGE f r i \u03c9) \u03c9 \u2264 r + R ** case h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u03c9 : \u03a9 hbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R ** by_cases heq : leastGE f r i \u03c9 = 0 ** case pos \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u03c9 : \u03a9 hbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R heq : leastGE f r i \u03c9 = 0 \u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R ** rw [heq, hf0, Pi.zero_apply] ** case pos \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u03c9 : \u03a9 hbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R heq : leastGE f r i \u03c9 = 0 \u22a2 0 \u2264 r + \u2191R ** exact add_nonneg hr R.coe_nonneg ** case neg \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u03c9 : \u03a9 hbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R heq : \u00acleastGE f r i \u03c9 = 0 \u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R ** obtain \u27e8k, hk\u27e9 := Nat.exists_eq_succ_of_ne_zero heq ** case neg.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u03c9 : \u03a9 hbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R heq : \u00acleastGE f r i \u03c9 = 0 k : \u2115 hk : leastGE f r i \u03c9 = Nat.succ k \u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R ** rw [hk, add_comm, \u2190 sub_le_iff_le_add] ** case neg.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u03c9 : \u03a9 hbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R heq : \u00acleastGE f r i \u03c9 = 0 k : \u2115 hk : leastGE f r i \u03c9 = Nat.succ k \u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R ** have := not_mem_of_lt_hitting (hk.symm \u25b8 k.lt_succ_self : k < leastGE f r i \u03c9) (zero_le _) ** case neg.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u03c9 : \u03a9 hbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R heq : \u00acleastGE f r i \u03c9 = 0 k : \u2115 hk : leastGE f r i \u03c9 = Nat.succ k this : \u00acf k \u03c9 \u2208 Set.Ici r \u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R ** simp only [Set.mem_union, Set.mem_Iic, Set.mem_Ici, not_or, not_le] at this ** case neg.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 hr : 0 \u2264 r hf0 : f 0 = 0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R i : \u2115 \u03c9 : \u03a9 hbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R heq : \u00acleastGE f r i \u03c9 = 0 k : \u2115 hk : leastGE f r i \u03c9 = Nat.succ k this : f k \u03c9 < r \u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R ** exact (sub_lt_sub_left this _).le.trans ((le_abs_self _).trans (hbdd\u03c9 _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.univ_pi_subset_univ_pi_iff ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 \u22a2 pi univ t\u2081 \u2286 pi univ t\u2082 \u2194 (\u2200 (i : \u03b9), t\u2081 i \u2286 t\u2082 i) \u2228 \u2203 i, t\u2081 i = \u2205 ** simp [pi_subset_pi_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.noncommProd_mul_distrib ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y) \u22a2 noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd s f comm_ff * noncommProd s g comm_gg ** induction' s using Finset.induction_on with x s hnmem ih ** case insert F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s\u271d : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y) x : \u03b1 s : Finset \u03b1 hnmem : \u00acx \u2208 s ih :   \u2200 (comm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y))     (comm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))     (comm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)),     noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =       noncommProd s f comm_ff * noncommProd s g comm_gg comm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y) \u22a2 noncommProd (insert x s) (f * g) (_ : Set.Pairwise \u2191(insert x s) fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd (insert x s) f comm_ff * noncommProd (insert x s) g comm_gg ** simp only [Finset.noncommProd_insert_of_not_mem _ _ _ _ hnmem] ** case insert F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s\u271d : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y) x : \u03b1 s : Finset \u03b1 hnmem : \u00acx \u2208 s ih :   \u2200 (comm_ff : Set.Pairwise \u2191s fun x y => Commute (f x) (f y))     (comm_gg : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))     (comm_gf : Set.Pairwise \u2191s fun x y => Commute (g x) (f y)),     noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =       noncommProd s f comm_ff * noncommProd s g comm_gg comm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y) \u22a2 (f * g) x * noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun a b => Commute ((f * g) a) ((f * g) b)) =     f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *       (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b))) ** specialize\n  ih (comm_ff.mono fun _ => mem_insert_of_mem) (comm_gg.mono fun _ => mem_insert_of_mem)\n    (comm_gf.mono fun _ => mem_insert_of_mem) ** case insert F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s\u271d : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y) x : \u03b1 s : Finset \u03b1 hnmem : \u00acx \u2208 s comm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y) ih :   noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *       noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) \u22a2 (f * g) x * noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun a b => Commute ((f * g) a) ((f * g) b)) =     f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *       (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b))) ** rw [ih, Pi.mul_apply] ** case insert F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s\u271d : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y) x : \u03b1 s : Finset \u03b1 hnmem : \u00acx \u2208 s comm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y) ih :   noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *       noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) \u22a2 f x * g x *       (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *         noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))) =     f x * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *       (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun a b => Commute (g a) (g b))) ** simp only [mul_assoc] ** case insert F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s\u271d : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y) x : \u03b1 s : Finset \u03b1 hnmem : \u00acx \u2208 s comm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y) ih :   noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *       noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) \u22a2 f x *       (g x *         (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *           noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)))) =     f x *       (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *         (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)))) ** congr 1 ** case insert.e_a F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s\u271d : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y) x : \u03b1 s : Finset \u03b1 hnmem : \u00acx \u2208 s comm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y) ih :   noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *       noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) \u22a2 g x *       (noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *         noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *       (g x * noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y))) ** simp only [\u2190 mul_assoc] ** case insert.e_a F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s\u271d : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y) x : \u03b1 s : Finset \u03b1 hnmem : \u00acx \u2208 s comm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y) ih :   noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *       noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) \u22a2 g x * noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *       noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) * g x *       noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) ** congr 1 ** case insert.e_a.e_a F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s\u271d : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y) x : \u03b1 s : Finset \u03b1 hnmem : \u00acx \u2208 s comm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y) ih :   noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *       noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) \u22a2 g x * noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) * g x ** refine' noncommProd_commute _ _ _ _ fun y hy => _ ** case insert.e_a.e_a F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s\u271d : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s\u271d fun x y => Commute (g x) (f y) x : \u03b1 s : Finset \u03b1 hnmem : \u00acx \u2208 s comm_ff : Set.Pairwise \u2191(insert x s) fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191(insert x s) fun x y => Commute (g x) (f y) ih :   noncommProd s (f * g) (_ : Set.Pairwise \u2191s fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd s f (_ : Set.Pairwise \u2191s fun x y => Commute (f x) (f y)) *       noncommProd s g (_ : Set.Pairwise \u2191s fun x y => Commute (g x) (g y)) y : \u03b1 hy : y \u2208 s \u22a2 Commute (g x) (f y) ** exact comm_gf (mem_insert_self x s) (mem_insert_of_mem hy) (ne_of_mem_of_not_mem hy hnmem).symm ** case empty F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s : Finset \u03b1 f g : \u03b1 \u2192 \u03b2 comm_ff\u271d : Set.Pairwise \u2191s fun x y => Commute (f x) (f y) comm_gg\u271d : Set.Pairwise \u2191s fun x y => Commute (g x) (g y) comm_gf\u271d : Set.Pairwise \u2191s fun x y => Commute (g x) (f y) comm_ff : Set.Pairwise \u2191\u2205 fun x y => Commute (f x) (f y) comm_gg : Set.Pairwise \u2191\u2205 fun x y => Commute (g x) (g y) comm_gf : Set.Pairwise \u2191\u2205 fun x y => Commute (g x) (f y) \u22a2 noncommProd \u2205 (f * g) (_ : Set.Pairwise \u2191\u2205 fun x y => Commute ((f * g) x) ((f * g) y)) =     noncommProd \u2205 f comm_ff * noncommProd \u2205 g comm_gg ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.set_integral_le ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f : \u03b9 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j s : Set \u03a9 hs : MeasurableSet s \u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc ** rw [\u2190 neg_le_neg_iff, \u2190 integral_neg, \u2190 integral_neg] ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f : \u03b9 \u2192 \u03a9 \u2192 \u211d hf : Submartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j s : Set \u03a9 hs : MeasurableSet s \u22a2 \u222b (a : \u03a9) in s, -f j a \u2202\u03bc \u2264 \u222b (a : \u03a9) in s, -f i a \u2202\u03bc ** exact Supermartingale.set_integral_le hf.neg hij hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_normalize_testAgainstNN_of_tendsto ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F     (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) ** have lim_mass := \u03bcs_lim.mass ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) \u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F     (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) ** have aux : {(0 : \u211d\u22650)}\u1d9c \u2208 \ud835\udcdd \u03bc.mass :=\n  isOpen_compl_singleton.mem_nhds (\u03bc.mass_nonzero_iff.mpr nonzero) ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) \u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F     (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) ** have eventually_nonzero : \u2200\u1da0 i in F, \u03bcs i \u2260 0 := by\n  simp_rw [\u2190 mass_nonzero_iff]\n  exact lim_mass aux ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) eventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 \u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F     (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) ** have eve : \u2200\u1da0 i in F,\n    (\u03bcs i).normalize.toFiniteMeasure.testAgainstNN f =\n      (\u03bcs i).mass\u207b\u00b9 * (\u03bcs i).testAgainstNN f := by\n  filter_upwards [eventually_iff.mp eventually_nonzero]\n  intro i hi\n  apply normalize_testAgainstNN _ hi ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) eventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 eve :   \u2200\u1da0 (i : \u03b3) in F,     testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f \u22a2 Tendsto (fun i => testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F     (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) ** simp_rw [tendsto_congr' eve, \u03bc.normalize_testAgainstNN nonzero] ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) eventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 eve :   \u2200\u1da0 (i : \u03b3) in F,     testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f lim_pair : Tendsto (fun i => ((mass (\u03bcs i))\u207b\u00b9, testAgainstNN (\u03bcs i) f)) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f)) \u22a2 Tendsto (fun x => (mass (\u03bcs x))\u207b\u00b9 * testAgainstNN (\u03bcs x) f) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9 * testAgainstNN \u03bc f)) ** exact tendsto_mul.comp lim_pair ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) \u22a2 \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 ** simp_rw [\u2190 mass_nonzero_iff] ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) \u22a2 \u2200\u1da0 (i : \u03b3) in F, mass (\u03bcs i) \u2260 0 ** exact lim_mass aux ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) eventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 \u22a2 \u2200\u1da0 (i : \u03b3) in F,     testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f ** filter_upwards [eventually_iff.mp eventually_nonzero] ** case h \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) eventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 \u22a2 \u2200 (a : \u03b3),     \u03bcs a \u2260 0 \u2192       testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs a))) f = (mass (\u03bcs a))\u207b\u00b9 * testAgainstNN (\u03bcs a) f ** intro i hi ** case h \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) eventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 i : \u03b3 hi : \u03bcs i \u2260 0 \u22a2 testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f ** apply normalize_testAgainstNN _ hi ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) eventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 eve :   \u2200\u1da0 (i : \u03b3) in F,     testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f \u22a2 Tendsto (fun i => ((mass (\u03bcs i))\u207b\u00b9, testAgainstNN (\u03bcs i) f)) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f)) ** refine' (Prod.tendsto_iff _ _).mpr \u27e8_, _\u27e9 ** case refine'_1 \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) eventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 eve :   \u2200\u1da0 (i : \u03b3) in F,     testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f \u22a2 Tendsto (fun n => ((mass (\u03bcs n))\u207b\u00b9, testAgainstNN (\u03bcs n) f).1) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f).1) ** exact (continuousOn_inv\u2080.continuousAt aux).tendsto.comp lim_mass ** case refine'_2 \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs F (\ud835\udcdd \u03bc) nonzero : \u03bc \u2260 0 f : \u03a9 \u2192\u1d47 \u211d\u22650 lim_mass : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) aux : {0}\u1d9c \u2208 \ud835\udcdd (mass \u03bc) eventually_nonzero : \u2200\u1da0 (i : \u03b3) in F, \u03bcs i \u2260 0 eve :   \u2200\u1da0 (i : \u03b3) in F,     testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f = (mass (\u03bcs i))\u207b\u00b9 * testAgainstNN (\u03bcs i) f \u22a2 Tendsto (fun n => ((mass (\u03bcs n))\u207b\u00b9, testAgainstNN (\u03bcs n) f).2) F (\ud835\udcdd ((mass \u03bc)\u207b\u00b9, testAgainstNN \u03bc f).2) ** exact tendsto_iff_forall_testAgainstNN_tendsto.mp \u03bcs_lim f ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.finSuccEquiv_apply ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 p : MvPolynomial (Fin (n + 1)) R \u22a2 \u2191(finSuccEquiv R n) p =     \u2191(eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i) p ** rw [\u2190 finSuccEquiv_eq, RingHom.coe_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.iIndepFun.integrable_exp_mul_sum ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s : Finset \u03b9 h_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) \u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9) ** induction' s using Finset.induction_on with i s hi_notin_s h_rec h_int ** case empty \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) h_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) \u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum \u2205 (fun i => X i) \u03c9) ** simp only [Pi.zero_apply, sum_apply, sum_empty, mul_zero, exp_zero] ** case empty \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) h_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) \u22a2 Integrable fun \u03c9 => 1 ** exact integrable_const _ ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s\u271d : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_rec :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)) \u2192     Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9) h_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9) \u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9) ** have : \u2200 i : \u03b9, i \u2208 s \u2192 Integrable (fun \u03c9 : \u03a9 => exp (t * X i \u03c9)) \u03bc := fun i hi =>\n  h_int i (mem_insert_of_mem hi) ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s\u271d : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_rec :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9)) \u2192     Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9) h_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9) this : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) \u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9) ** specialize h_rec this ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s\u271d : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9) this : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) h_rec : Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9) \u22a2 Integrable fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9) ** rw [sum_insert hi_notin_s] ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s\u271d : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9) this : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) h_rec : Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9) \u22a2 Integrable fun \u03c9 => rexp (t * (X i + \u2211 x in s, X x) \u03c9) ** refine' IndepFun.integrable_exp_mul_add _ (h_int i (mem_insert_self _ _)) h_rec ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s\u271d : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 Integrable fun \u03c9 => rexp (t * X i_1 \u03c9) this : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) h_rec : Integrable fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9) \u22a2 IndepFun (X i) (\u2211 x in s, X x) ** exact (h_indep.indepFun_finset_sum_of_not_mem h_meas hi_notin_s).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.hasFiniteIntegral_compProd_iff' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 f : \u03b2 \u00d7 \u03b3 \u2192 E h1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a) \u22a2 HasFiniteIntegral f \u2194     (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => f (x, y)) \u2227       HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x) ** rw [hasFiniteIntegral_congr h1f.ae_eq_mk,\n  hasFiniteIntegral_compProd_iff h1f.stronglyMeasurable_mk] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 f : \u03b2 \u00d7 \u03b3 \u2192 E h1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a) \u22a2 ((\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2227       HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)) \u2194     (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => f (x, y)) \u2227       HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x) ** apply and_congr ** case h\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 f : \u03b2 \u00d7 \u03b3 \u2192 E h1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a) \u22a2 (\u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194     \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a, HasFiniteIntegral fun y => f (x, y) ** apply eventually_congr ** case h\u2081.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 f : \u03b2 \u00d7 \u03b3 \u2192 E h1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a) \u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u2191\u03ba a,     (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y) ** filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 f : \u03b2 \u00d7 \u03b3 \u2192 E h1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a) \u22a2 \u2200 (a_1 : \u03b2),     (\u2200\u1d50 (c : \u03b3) \u2202\u2191\u03b7 (a, a_1), AEStronglyMeasurable.mk f h1f (a_1, c) = f (a_1, c)) \u2192       ((HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (a_1, y)) \u2194 HasFiniteIntegral fun y => f (a_1, y)) ** intro x hx ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 f : \u03b2 \u00d7 \u03b3 \u2192 E h1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a) x : \u03b2 hx : \u2200\u1d50 (c : \u03b3) \u2202\u2191\u03b7 (a, x), AEStronglyMeasurable.mk f h1f (x, c) = f (x, c) \u22a2 (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y) ** exact hasFiniteIntegral_congr hx ** case h\u2082 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 f : \u03b2 \u00d7 \u03b3 \u2192 E h1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a) \u22a2 (HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)) \u2194     HasFiniteIntegral fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x) ** apply hasFiniteIntegral_congr ** case h\u2082.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 f : \u03b2 \u00d7 \u03b3 \u2192 E h1f : AEStronglyMeasurable f (\u2191(\u03ba \u2297\u2096 \u03b7) a) \u22a2 (fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u2191\u03b7 (a, x)) =\u1d50[\u2191\u03ba a] fun x =>     \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u2191\u03b7 (a, x) ** filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with _ hx using\n  integral_congr_ae (EventuallyEq.fun_comp hx _) ** Qed",
        "informal": ""
    },
    {
        "formal": "PEquiv.trans_symm_eq_iff_forall_isSome ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 \u22a2 PEquiv.trans f (PEquiv.symm f) = PEquiv.refl \u03b1 \u2194 \u2200 (a : \u03b1), isSome (\u2191f a) = true ** rw [self_trans_symm, ofSet_eq_refl, Set.eq_univ_iff_forall] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 \u22a2 (\u2200 (x : \u03b1), x \u2208 {a | isSome (\u2191f a) = true}) \u2194 \u2200 (a : \u03b1), isSome (\u2191f a) = true ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.pdf.IsUniform.hasPDF ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hu : IsUniform X s \u2119 \u22a2 pdf X \u2119 \u2260 0 ** intro hpdf ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hu : IsUniform X s \u2119 hpdf : pdf X \u2119 = 0 \u22a2 False ** simp only [IsUniform, hpdf] at hu ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hpdf : pdf X \u2119 = 0 hu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) \u22a2 False ** suffices \u03bc (s \u2229 Function.support ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e))) = 0 by\n  have heq : Function.support ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e)) = Set.univ := by\n    ext x\n    rw [Function.mem_support]\n    simp [hnt]\n  rw [heq, Set.inter_univ] at this\n  exact hns this ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hpdf : pdf X \u2119 = 0 hu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) \u22a2 \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0 ** exact Set.indicator_ae_eq_zero.1 hu.symm ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hpdf : pdf X \u2119 = 0 hu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) this : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0 \u22a2 False ** have heq : Function.support ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e)) = Set.univ := by\n  ext x\n  rw [Function.mem_support]\n  simp [hnt] ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hpdf : pdf X \u2119 = 0 hu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) this : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0 heq : Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) = Set.univ \u22a2 False ** rw [heq, Set.inter_univ] at this ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hpdf : pdf X \u2119 = 0 hu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) this : \u2191\u2191\u03bc s = 0 heq : Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) = Set.univ \u22a2 False ** exact hns this ** \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hpdf : pdf X \u2119 = 0 hu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) this : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0 \u22a2 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) = Set.univ ** ext x ** case h \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hpdf : pdf X \u2119 = 0 hu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) this : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0 x : E \u22a2 x \u2208 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) \u2194 x \u2208 Set.univ ** rw [Function.mem_support] ** case h \u03a9 : Type u_1 E : Type u_2 inst\u271d : MeasurableSpace E m\u271d : MeasurableSpace \u03a9 \u2119\u271d : Measure \u03a9 \u03bc\u271d : Measure E m : MeasurableSpace \u03a9 X : \u03a9 \u2192 E \u2119 : Measure \u03a9 \u03bc : Measure E s : Set E hns : \u2191\u2191\u03bc s \u2260 0 hnt : \u2191\u2191\u03bc s \u2260 \u22a4 hpdf : pdf X \u2119 = 0 hu : 0 =\u1da0[ae \u03bc] Set.indicator s ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) this : \u2191\u2191\u03bc (s \u2229 Function.support ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1)) = 0 x : E \u22a2 ((\u2191\u2191\u03bc s)\u207b\u00b9 \u2022 1) x \u2260 0 \u2194 x \u2208 Set.univ ** simp [hnt] ** Qed",
        "informal": ""
    },
    {
        "formal": "iUnion_Ici_eq_Ioi_iInf ** \u03b9 : Sort u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b9 : LinearOrder \u03b1 s : Set \u03b1 a : \u03b1 f\u271d : \u03b9 \u2192 \u03b1 R : Type u_1 inst\u271d : CompleteLinearOrder R f : \u03b9 \u2192 R no_least_elem : \u00ac\u2a05 i, f i \u2208 range f \u22a2 \u22c3 i, Ici (f i) = Ioi (\u2a05 i, f i) ** simp only [\u2190 IsGLB.biUnion_Ici_eq_Ioi (@isGLB_iInf _ _ _ f) no_least_elem, mem_range,\n  iUnion_exists, iUnion_iUnion_eq'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.mem\u2112p_of_exponent_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hfq : Mem\u2112p f q hpq : p \u2264 q \u22a2 Mem\u2112p f p ** cases' hfq with hfq_m hfq_lt_top ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 \u22a2 Mem\u2112p f p ** by_cases hp0 : p = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : \u00acp = 0 \u22a2 Mem\u2112p f p ** rw [\u2190 Ne.def] at hp0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 \u22a2 Mem\u2112p f p ** refine' \u27e8hfq_m, _\u27e9 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 \u22a2 snorm f p \u03bc < \u22a4 ** by_cases hp_top : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 \u22a2 snorm f p \u03bc < \u22a4 ** have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_top ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p \u22a2 snorm f p \u03bc < \u22a4 ** by_cases hq_top : q = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 \u22a2 snorm f p \u03bc < \u22a4 ** have hq0 : q \u2260 0 := by\n  by_contra hq_eq_zero\n  have hp_eq_zero : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq) (zero_le _)\n  rw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos\n  exact (lt_irrefl _) hp_pos ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 hq0 : q \u2260 0 \u22a2 snorm f p \u03bc < \u22a4 ** have hpq_real : p.toReal \u2264 q.toReal := by rwa [ENNReal.toReal_le_toReal hp_top hq_top] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 hq0 : q \u2260 0 hpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q \u22a2 snorm f p \u03bc < \u22a4 ** rw [snorm_eq_snorm' hp0 hp_top] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 hq0 : q \u2260 0 hpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q \u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4 ** rw [snorm_eq_snorm' hq0 hq_top] at hfq_lt_top ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm' f (ENNReal.toReal q) \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 hq0 : q \u2260 0 hpq_real : ENNReal.toReal p \u2264 ENNReal.toReal q \u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4 ** exact snorm'_lt_top_of_snorm'_lt_top_of_exponent_le hfq_m hfq_lt_top (le_of_lt hp_pos) hpq_real ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p = 0 \u22a2 Mem\u2112p f p ** rwa [hp0, mem\u2112p_zero_iff_aestronglyMeasurable] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : p = \u22a4 \u22a2 snorm f p \u03bc < \u22a4 ** have hq_top : q = \u221e := by rwa [hp_top, top_le_iff] at hpq ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : p = \u22a4 hq_top : q = \u22a4 \u22a2 snorm f p \u03bc < \u22a4 ** rw [hp_top] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : p = \u22a4 hq_top : q = \u22a4 \u22a2 snorm f \u22a4 \u03bc < \u22a4 ** rwa [hq_top] at hfq_lt_top ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : p = \u22a4 \u22a2 q = \u22a4 ** rwa [hp_top, top_le_iff] at hpq ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : q = \u22a4 \u22a2 snorm f p \u03bc < \u22a4 ** rw [snorm_eq_snorm' hp0 hp_top] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : q = \u22a4 \u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4 ** rw [hq_top, snorm_exponent_top] at hfq_lt_top ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snormEssSup f \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : q = \u22a4 \u22a2 snorm' f (ENNReal.toReal p) \u03bc < \u22a4 ** refine' lt_of_le_of_lt (snorm'_le_snormEssSup_mul_rpow_measure_univ hp_pos) _ ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snormEssSup f \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : q = \u22a4 \u22a2 snormEssSup f \u03bc * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) < \u22a4 ** refine' ENNReal.mul_lt_top hfq_lt_top.ne _ ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snormEssSup f \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : q = \u22a4 \u22a2 \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p) \u2260 \u22a4 ** exact (ENNReal.rpow_lt_top_of_nonneg (by simp [hp_pos.le]) (measure_ne_top \u03bc Set.univ)).ne ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snormEssSup f \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : q = \u22a4 \u22a2 0 \u2264 1 / ENNReal.toReal p ** simp [hp_pos.le] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 \u22a2 q \u2260 0 ** by_contra hq_eq_zero ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 hq_eq_zero : q = 0 \u22a2 False ** have hp_eq_zero : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq) (zero_le _) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 hq_eq_zero : q = 0 hp_eq_zero : p = 0 \u22a2 False ** rw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < 0 hq_top : \u00acq = \u22a4 hq_eq_zero : q = 0 hp_eq_zero : p = 0 \u22a2 False ** exact (lt_irrefl _) hp_pos ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 hq_eq_zero : q = 0 \u22a2 p \u2264 0 ** rwa [hq_eq_zero] at hpq ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G p q : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E hpq : p \u2264 q hfq_m : AEStronglyMeasurable f \u03bc hfq_lt_top : snorm f q \u03bc < \u22a4 hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p hq_top : \u00acq = \u22a4 hq0 : q \u2260 0 \u22a2 ENNReal.toReal p \u2264 ENNReal.toReal q ** rwa [ENNReal.toReal_le_toReal hp_top hq_top] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.average_union ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s\u271d t\u271d : Set \u03b1 f\u271d g f : \u03b1 \u2192 E s t : Set \u03b1 hd : AEDisjoint \u03bc s t ht : NullMeasurableSet t hs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4 ht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4 hfs : IntegrableOn f s hft : IntegrableOn f t \u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =     (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +       (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc ** haveI := Fact.mk hs\u03bc.lt_top ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s\u271d t\u271d : Set \u03b1 f\u271d g f : \u03b1 \u2192 E s t : Set \u03b1 hd : AEDisjoint \u03bc s t ht : NullMeasurableSet t hs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4 ht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4 hfs : IntegrableOn f s hft : IntegrableOn f t this : Fact (\u2191\u2191\u03bc s < \u22a4) \u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =     (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +       (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc ** haveI := Fact.mk ht\u03bc.lt_top ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s\u271d t\u271d : Set \u03b1 f\u271d g f : \u03b1 \u2192 E s t : Set \u03b1 hd : AEDisjoint \u03bc s t ht : NullMeasurableSet t hs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4 ht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4 hfs : IntegrableOn f s hft : IntegrableOn f t this\u271d : Fact (\u2191\u2191\u03bc s < \u22a4) this : Fact (\u2191\u2191\u03bc t < \u22a4) \u22a2 \u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc =     (ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc +       (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc ** rw [restrict_union\u2080 hd ht, average_add_measure hfs hft, restrict_apply_univ, restrict_apply_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_of_cont_bdd_of_tendsto_indicator ** \u03a9 : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9 inst\u271d\u00b2 : TopologicalSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc c : \u211d\u22650 E : Set \u03a9 E_mble : MeasurableSet E fs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650 fs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c fs_lim : Tendsto (fun n => \u2191(fs n)) atTop (\ud835\udcdd (indicator E fun x => 1)) \u22a2 Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(fs n) \u03c9) \u2202\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u03bc E)) ** have fs_lim' :\n  \u2200 \u03c9, Tendsto (fun n : \u2115 => (fs n \u03c9 : \u211d\u22650)) atTop (\ud835\udcdd (indicator E (fun _ => (1 : \u211d\u22650)) \u03c9)) := by\n  rw [tendsto_pi_nhds] at fs_lim\n  exact fun \u03c9 => fs_lim \u03c9 ** \u03a9 : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9 inst\u271d\u00b2 : TopologicalSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc c : \u211d\u22650 E : Set \u03a9 E_mble : MeasurableSet E fs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650 fs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c fs_lim : Tendsto (fun n => \u2191(fs n)) atTop (\ud835\udcdd (indicator E fun x => 1)) fs_lim' : \u2200 (\u03c9 : \u03a9), Tendsto (fun n => \u2191(fs n) \u03c9) atTop (\ud835\udcdd (indicator E (fun x => 1) \u03c9)) \u22a2 Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(fs n) \u03c9) \u2202\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u03bc E)) ** apply measure_of_cont_bdd_of_tendsto_filter_indicator \u03bc E_mble fs\n  (eventually_of_forall fun n => eventually_of_forall (fs_bdd n)) (eventually_of_forall fs_lim') ** \u03a9 : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9 inst\u271d\u00b2 : TopologicalSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc c : \u211d\u22650 E : Set \u03a9 E_mble : MeasurableSet E fs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650 fs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c fs_lim : Tendsto (fun n => \u2191(fs n)) atTop (\ud835\udcdd (indicator E fun x => 1)) \u22a2 \u2200 (\u03c9 : \u03a9), Tendsto (fun n => \u2191(fs n) \u03c9) atTop (\ud835\udcdd (indicator E (fun x => 1) \u03c9)) ** rw [tendsto_pi_nhds] at fs_lim ** \u03a9 : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9 inst\u271d\u00b2 : TopologicalSpace \u03a9 inst\u271d\u00b9 : OpensMeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc c : \u211d\u22650 E : Set \u03a9 E_mble : MeasurableSet E fs : \u2115 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650 fs_bdd : \u2200 (n : \u2115) (\u03c9 : \u03a9), \u2191(fs n) \u03c9 \u2264 c fs_lim : \u2200 (x : \u03a9), Tendsto (fun i => \u2191(fs i) x) atTop (\ud835\udcdd (indicator E (fun x => 1) x)) \u22a2 \u2200 (\u03c9 : \u03a9), Tendsto (fun n => \u2191(fs n) \u03c9) atTop (\ud835\udcdd (indicator E (fun x => 1) \u03c9)) ** exact fun \u03c9 => fs_lim \u03c9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.singularPart_totalVariation ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 totalVariation (singularPart s \u03bc) =     Measure.singularPart (toJordanDecomposition s).posPart \u03bc + Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** have :\n  (s.singularPart \u03bc).toJordanDecomposition =\n    \u27e8s.toJordanDecomposition.posPart.singularPart \u03bc,\n      s.toJordanDecomposition.negPart.singularPart \u03bc, singularPart_mutuallySingular s \u03bc\u27e9 := by\n  refine' JordanDecomposition.toSignedMeasure_injective _\n  rw [toSignedMeasure_toJordanDecomposition]\n  rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 toJordanDecomposition (singularPart s \u03bc) =     JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)       (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)       (_ :         Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098           Measure.singularPart (toJordanDecomposition s).negPart \u03bc) ** refine' JordanDecomposition.toSignedMeasure_injective _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 JordanDecomposition.toSignedMeasure (toJordanDecomposition (singularPart s \u03bc)) =     JordanDecomposition.toSignedMeasure       (JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)         (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)         (_ :           Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098             Measure.singularPart (toJordanDecomposition s).negPart \u03bc)) ** rw [toSignedMeasure_toJordanDecomposition] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 singularPart s \u03bc =     JordanDecomposition.toSignedMeasure       (JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)         (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)         (_ :           Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098             Measure.singularPart (toJordanDecomposition s).negPart \u03bc)) ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 this :   toJordanDecomposition (singularPart s \u03bc) =     JordanDecomposition.mk (Measure.singularPart (toJordanDecomposition s).posPart \u03bc)       (Measure.singularPart (toJordanDecomposition s).negPart \u03bc)       (_ :         Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098           Measure.singularPart (toJordanDecomposition s).negPart \u03bc) \u22a2 totalVariation (singularPart s \u03bc) =     Measure.singularPart (toJordanDecomposition s).posPart \u03bc + Measure.singularPart (toJordanDecomposition s).negPart \u03bc ** rw [totalVariation, this] ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.lintegral_Lp_add_le_of_le_one ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hp0 : 0 \u2264 p hp1 : p \u2264 1 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264     2 ^ (1 / p - 1) * ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) ** rcases eq_or_lt_of_le hp0 with (rfl | hp) ** case inr \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hp0 : 0 \u2264 p hp1 : p \u2264 1 hp : 0 < p \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264     2 ^ (1 / p - 1) * ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) ** calc\n  (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 ((\u222b\u207b a, f a ^ p \u2202\u03bc) + \u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p) := by\n    apply rpow_le_rpow _ (div_nonneg zero_le_one hp0)\n    rw [\u2190 lintegral_add_left' (hf.pow_const p)]\n    exact lintegral_mono fun a => rpow_add_le_add_rpow _ _ hp0 hp1\n  _ \u2264 (2 : \u211d\u22650\u221e) ^ (1 / p - 1) * ((\u222b\u207b a, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p)) :=\n    rpow_add_le_mul_rpow_add_rpow _ _ ((one_le_div hp).2 hp1) ** case inl \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hp0 : 0 \u2264 0 hp1 : 0 \u2264 1 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ 0 \u2202\u03bc) ^ (1 / 0) \u2264     2 ^ (1 / 0 - 1) * ((\u222b\u207b (a : \u03b1), f a ^ 0 \u2202\u03bc) ^ (1 / 0) + (\u222b\u207b (a : \u03b1), g a ^ 0 \u2202\u03bc) ^ (1 / 0)) ** simp only [Pi.add_apply, rpow_zero, lintegral_one, _root_.div_zero, zero_sub] ** case inl \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hp0 : 0 \u2264 0 hp1 : 0 \u2264 1 \u22a2 1 \u2264 2 ^ (-1) * (1 + 1) ** norm_num ** case inl \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hp0 : 0 \u2264 0 hp1 : 0 \u2264 1 \u22a2 1 \u2264 2 ^ (-1) * 2 ** rw [rpow_neg, rpow_one, ENNReal.inv_mul_cancel two_ne_zero two_ne_top] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hp0 : 0 \u2264 p hp1 : p \u2264 1 hp : 0 < p \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** apply rpow_le_rpow _ (div_nonneg zero_le_one hp0) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hp0 : 0 \u2264 p hp1 : p \u2264 1 hp : 0 < p \u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc ** rw [\u2190 lintegral_add_left' (hf.pow_const p)] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hp0 : 0 \u2264 p hp1 : p \u2264 1 hp : 0 < p \u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a ^ p + g a ^ p \u2202\u03bc ** exact lintegral_mono fun a => rpow_add_le_add_rpow _ _ hp0 hp1 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.coeFn_smul_apply ** \u03a9 : Type u_1 inst\u271d\u2075 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2074 : SMul R \u211d\u22650 inst\u271d\u00b3 : SMul R \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650 c : R \u03bc : FiniteMeasure \u03a9 s : Set \u03a9 \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) s)) s = c \u2022 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s ** rw [coeFn_smul, Pi.smul_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_preserving_transvectionStruct ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d \u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) ** let p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) ** let \u03b1 : Type _ := { x // p x } ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) ** let \u03b2 : Type _ := { x // \u00acp x } ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } \u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) ** let g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun _ => t.c * a \u27e8t.j, t.hij.symm\u27e9) + b ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b \u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) ** let F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) \u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) ** let e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun _ : \u03b9 => \u211d) p ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p this : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e \u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) ** rw [this] ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p this : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e \u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e) ** have A : MeasurePreserving e := by\n  convert volume_preserving_piEquivPiSubtypeProd (fun _ : \u03b9 => \u211d) p ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p this : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e A : MeasurePreserving \u2191e \u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e) ** have B : MeasurePreserving F :=\n  haveI g_meas : Measurable (Function.uncurry g) := by\n    have : Measurable fun c : \u03b1 \u2192 \u211d => c \u27e8t.j, t.hij.symm\u27e9 :=\n      measurable_pi_apply \u27e8t.j, t.hij.symm\u27e9\n    refine Measurable.add ?_ measurable_snd\n    refine measurable_pi_lambda _ fun _ => Measurable.const_mul ?_ _\n    exact this.comp measurable_fst\n  (MeasurePreserving.id _).skew_product g_meas\n    (eventually_of_forall fun a => map_add_left_eq_self\n      (Measure.pi fun _ => (stdOrthonormalBasis \u211d \u211d).toBasis.addHaar) _) ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p this : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e A : MeasurePreserving \u2191e B : MeasurePreserving F \u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e) ** exact ((A.symm e).comp B).comp A ** case mk \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t_i t_j : \u03b9 t_hij : t_i \u2260 t_j t_c : \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=   fun a b =>     (fun x =>         { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *           a             { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,               property :=                 (_ :                   { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260                     { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +       b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p \u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix { i := t_i, j := t_j, hij := t_hij, c := t_c })) =     \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e ** ext f k ** case mk.h.h \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t_i t_j : \u03b9 t_hij : t_i \u2260 t_j t_c : \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=   fun a b =>     (fun x =>         { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *           a             { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,               property :=                 (_ :                   { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260                     { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +       b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p f : \u03b9 \u2192 \u211d k : \u03b9 \u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix { i := t_i, j := t_j, hij := t_hij, c := t_c })) f k =     (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e) f k ** simp only [LinearEquiv.map_smul, dite_eq_ite, LinearMap.id_coe, ite_not,\n  Algebra.id.smul_eq_mul, one_mul, dotProduct, stdBasisMatrix,\n  MeasurableEquiv.piEquivPiSubtypeProd_symm_apply, id.def, transvection, Pi.add_apply,\n  zero_mul, LinearMap.smul_apply, Function.comp_apply,\n  MeasurableEquiv.piEquivPiSubtypeProd_apply, Matrix.TransvectionStruct.toMatrix_mk,\n  Matrix.mulVec, LinearEquiv.map_add, ite_mul, Matrix.toLin'_apply, Pi.smul_apply,\n  Subtype.coe_mk, LinearMap.add_apply, Finset.sum_congr, Matrix.toLin'_one] ** case mk.h.h \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t_i t_j : \u03b9 t_hij : t_i \u2260 t_j t_c : \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=   fun a b =>     (fun x =>         { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *           a             { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,               property :=                 (_ :                   { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260                     { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +       b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p f : \u03b9 \u2192 \u211d k : \u03b9 \u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k ** by_cases h : t_i = k ** case pos \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t_i t_j : \u03b9 t_hij : t_i \u2260 t_j t_c : \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=   fun a b =>     (fun x =>         { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *           a             { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,               property :=                 (_ :                   { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260                     { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +       b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p f : \u03b9 \u2192 \u211d k : \u03b9 h : t_i = k \u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k ** simp only [h, true_and_iff, Finset.mem_univ, if_true, eq_self_iff_true, Finset.sum_ite_eq,\n  one_apply, boole_mul, add_comm] ** case neg \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t_i t_j : \u03b9 t_hij : t_i \u2260 t_j t_c : \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=   fun a b =>     (fun x =>         { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *           a             { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,               property :=                 (_ :                   { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260                     { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +       b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p f : \u03b9 \u2192 \u211d k : \u03b9 h : \u00act_i = k \u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k ** simp only [h, Ne.symm h, add_zero, if_false, Finset.sum_const_zero, false_and_iff,\n  mul_zero] ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p this : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e \u22a2 MeasurePreserving \u2191e ** convert volume_preserving_piEquivPiSubtypeProd (fun _ : \u03b9 => \u211d) p ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p this : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e A : MeasurePreserving \u2191e \u22a2 Measurable (Function.uncurry g) ** have : Measurable fun c : \u03b1 \u2192 \u211d => c \u27e8t.j, t.hij.symm\u27e9 :=\n  measurable_pi_apply \u27e8t.j, t.hij.symm\u27e9 ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p this\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e A : MeasurePreserving \u2191e this : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) } \u22a2 Measurable (Function.uncurry g) ** refine Measurable.add ?_ measurable_snd ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p this\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e A : MeasurePreserving \u2191e this : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) } \u22a2 Measurable fun a x => t.c * a.1 { val := t.j, property := (_ : t.j \u2260 t.i) } ** refine measurable_pi_lambda _ fun _ => Measurable.const_mul ?_ _ ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 t : TransvectionStruct \u03b9 \u211d p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i \u03b1 : Type u_1 := { x // p x } \u03b2 : Type u_1 := { x // \u00acp x } g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2) e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p this\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e A : MeasurePreserving \u2191e this : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) } x\u271d : \u03b2 \u22a2 Measurable fun c => c.1 { val := t.j, property := (_ : t.j \u2260 t.i) } ** exact this.comp measurable_fst ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.exists_subset_or_subset_of_two_mul_lt_ncard ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hst : 2 * n < ncard (s \u222a t) \u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t) ** classical\nhave hu := finite_of_ncard_ne_zero ((Nat.zero_le _).trans_lt hst).ne.symm\nrw [ncard_eq_toFinset_card _ hu,\n  Finite.toFinset_union (hu.subset (subset_union_left _ _))\n    (hu.subset (subset_union_right _ _))] at hst\nobtain \u27e8r', hnr', hr'\u27e9 := Finset.exists_subset_or_subset_of_two_mul_lt_card hst\nexact \u27e8r', by simpa, by simpa using hr'\u27e9 ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hst : 2 * n < ncard (s \u222a t) \u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t) ** have hu := finite_of_ncard_ne_zero ((Nat.zero_le _).trans_lt hst).ne.symm ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hst : 2 * n < ncard (s \u222a t) hu : Set.Finite (s \u222a t) \u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t) ** rw [ncard_eq_toFinset_card _ hu,\n  Finite.toFinset_union (hu.subset (subset_union_left _ _))\n    (hu.subset (subset_union_right _ _))] at hst ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hu : Set.Finite (s \u222a t) hst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t)) \u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t) ** obtain \u27e8r', hnr', hr'\u27e9 := Finset.exists_subset_or_subset_of_two_mul_lt_card hst ** case intro.intro \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hu : Set.Finite (s \u222a t) hst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t)) r' : Finset \u03b1 hnr' : n < Finset.card r' hr' : r' \u2286 Finite.toFinset (_ : Set.Finite s) \u2228 r' \u2286 Finite.toFinset (_ : Set.Finite t) \u22a2 \u2203 r, n < ncard r \u2227 (r \u2286 s \u2228 r \u2286 t) ** exact \u27e8r', by simpa, by simpa using hr'\u27e9 ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hu : Set.Finite (s \u222a t) hst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t)) r' : Finset \u03b1 hnr' : n < Finset.card r' hr' : r' \u2286 Finite.toFinset (_ : Set.Finite s) \u2228 r' \u2286 Finite.toFinset (_ : Set.Finite t) \u22a2 n < ncard \u2191r' ** simpa ** \u03b1 : Type u_1 s t : Set \u03b1 n : \u2115 hu : Set.Finite (s \u222a t) hst : 2 * n < Finset.card (Finite.toFinset (_ : Set.Finite s) \u222a Finite.toFinset (_ : Set.Finite t)) r' : Finset \u03b1 hnr' : n < Finset.card r' hr' : r' \u2286 Finite.toFinset (_ : Set.Finite s) \u2228 r' \u2286 Finite.toFinset (_ : Set.Finite t) \u22a2 \u2191r' \u2286 s \u2228 \u2191r' \u2286 t ** simpa using hr' ** Qed",
        "informal": ""
    },
    {
        "formal": "DFA.evalFrom_split ** \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t \u22a2 \u2203 q a b c,     x = a ++ b ++ c \u2227       List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227         b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t ** obtain \u27e8n, m, hneq, heq\u27e9 :=\n  Fintype.exists_ne_map_eq_of_card_lt\n    (fun n : Fin (Fintype.card \u03c3 + 1) => M.evalFrom s (x.take n)) (by norm_num) ** case intro.intro.intro \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) \u22a2 \u2203 q a b c,     x = a ++ b ++ c \u2227       List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227         b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t ** wlog hle : (n : \u2115) \u2264 m ** \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m \u22a2 \u2203 q a b c,     x = a ++ b ++ c \u2227       List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227         b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t ** have hm : (m : \u2115) \u2264 Fintype.card \u03c3 := Fin.is_le m ** \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 \u22a2 \u2203 q a b c,     x = a ++ b ++ c \u2227       List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227         b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t ** refine'\n  \u27e8M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by\n    rfl, _\u27e9 ** case refine'_4 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 \u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =       evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) \u2227     evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191m) x) = t ** have hq : M.evalFrom (M.evalFrom s ((x.take m).take n)) ((x.take m).drop n) =\n    M.evalFrom s ((x.take m).take n) := by\n  rw [List.take_take, min_eq_left hle, \u2190 evalFrom_of_append, heq, \u2190 min_eq_left hle, \u2190\n    List.take_take, min_eq_left hle, List.take_append_drop] ** case refine'_4 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 hq :   evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =     evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) \u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =       evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) \u2227     evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191m) x) = t ** use hq ** case right \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 hq :   evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =     evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) \u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191m) x) = t ** rwa [\u2190 hq, \u2190 evalFrom_of_append, \u2190 evalFrom_of_append, \u2190 List.append_assoc,\n  List.take_append_drop, List.take_append_drop] ** \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t \u22a2 Fintype.card \u03c3 < Fintype.card (Fin (Fintype.card \u03c3 + 1)) ** norm_num ** case intro.intro.intro.inr \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) this :   \u2200 {\u03b1 : Type u} {\u03c3 : Type v} (M : DFA \u03b1 \u03c3) [inst : Fintype \u03c3] {x : List \u03b1} {s t : \u03c3},     Fintype.card \u03c3 \u2264 List.length x \u2192       evalFrom M s x = t \u2192         \u2200 (n m : Fin (Fintype.card \u03c3 + 1)),           n \u2260 m \u2192             evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) \u2192               \u2191n \u2264 \u2191m \u2192                 \u2203 q a b c,                   x = a ++ b ++ c \u2227                     List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227                       b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t hle : \u00ac\u2191n \u2264 \u2191m \u22a2 \u2203 q a b c,     x = a ++ b ++ c \u2227       List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227         b \u2260 [] \u2227 evalFrom M s a = q \u2227 evalFrom M q b = q \u2227 evalFrom M q c = t ** exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle) ** \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 \u22a2 evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) = evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) ** rfl ** case refine'_1 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 \u22a2 x = List.take (\u2191n) (List.take (\u2191m) x) ++ List.drop (\u2191n) (List.take (\u2191m) x) ++ List.drop (\u2191m) x ** rw [List.take_append_drop, List.take_append_drop] ** case refine'_2 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 \u22a2 List.length (List.take (\u2191n) (List.take (\u2191m) x)) + List.length (List.drop (\u2191n) (List.take (\u2191m) x)) \u2264 Fintype.card \u03c3 ** simp only [List.length_drop, List.length_take] ** case refine'_2 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 \u22a2 min (\u2191n) (min (\u2191m) (List.length x)) + (min (\u2191m) (List.length x) - \u2191n) \u2264 Fintype.card \u03c3 ** rw [min_eq_left (hm.trans hlen), min_eq_left hle, add_tsub_cancel_of_le hle] ** case refine'_2 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 \u22a2 \u2191m \u2264 Fintype.card \u03c3 ** exact hm ** case refine'_3 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 \u22a2 List.drop (\u2191n) (List.take (\u2191m) x) \u2260 [] ** intro h ** case refine'_3 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 h : List.drop (\u2191n) (List.take (\u2191m) x) = [] \u22a2 False ** have hlen' := congr_arg List.length h ** case refine'_3 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 h : List.drop (\u2191n) (List.take (\u2191m) x) = [] hlen' : List.length (List.drop (\u2191n) (List.take (\u2191m) x)) = List.length [] \u22a2 False ** simp only [List.length_drop, List.length, List.length_take] at hlen' ** case refine'_3 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 h : List.drop (\u2191n) (List.take (\u2191m) x) = [] hlen' : min (\u2191m) (List.length x) - \u2191n = 0 \u22a2 False ** rw [min_eq_left, tsub_eq_zero_iff_le] at hlen' ** case refine'_3 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 h : List.drop (\u2191n) (List.take (\u2191m) x) = [] hlen' : min (\u2191m) (List.length x) - \u2191n = 0 \u22a2 \u2191m \u2264 List.length x ** exact hm.trans hlen ** case refine'_3 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 h : List.drop (\u2191n) (List.take (\u2191m) x) = [] hlen' : \u2191m \u2264 \u2191n \u22a2 False ** apply hneq ** case refine'_3 \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 h : List.drop (\u2191n) (List.take (\u2191m) x) = [] hlen' : \u2191m \u2264 \u2191n \u22a2 n = m ** apply le_antisymm ** case refine'_3.a \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 h : List.drop (\u2191n) (List.take (\u2191m) x) = [] hlen' : \u2191m \u2264 \u2191n \u22a2 n \u2264 m  case refine'_3.a \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 h : List.drop (\u2191n) (List.take (\u2191m) x) = [] hlen' : \u2191m \u2264 \u2191n \u22a2 m \u2264 n ** assumption' ** \u03b1\u271d : Type u \u03c3\u271d : Type v M\u271d : DFA \u03b1\u271d \u03c3\u271d \u03b1 : Type u \u03c3 : Type v M : DFA \u03b1 \u03c3 inst\u271d : Fintype \u03c3 x : List \u03b1 s t : \u03c3 hlen : Fintype.card \u03c3 \u2264 List.length x hx : evalFrom M s x = t n m : Fin (Fintype.card \u03c3 + 1) hneq : n \u2260 m heq : evalFrom M s (List.take (\u2191n) x) = evalFrom M s (List.take (\u2191m) x) hle : \u2191n \u2264 \u2191m hm : \u2191m \u2264 Fintype.card \u03c3 \u22a2 evalFrom M (evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x))) (List.drop (\u2191n) (List.take (\u2191m) x)) =     evalFrom M s (List.take (\u2191n) (List.take (\u2191m) x)) ** rw [List.take_take, min_eq_left hle, \u2190 evalFrom_of_append, heq, \u2190 min_eq_left hle, \u2190\n  List.take_take, min_eq_left hle, List.take_append_drop] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.ofFn_get ** n : \u2115 \u03b1 : Type u_1 v : Vector \u03b1 n \u22a2 ofFn (get v) = v ** rcases v with \u27e8l, rfl\u27e9 ** case mk \u03b1 : Type u_1 l : List \u03b1 \u22a2 ofFn (get { val := l, property := (_ : List.length l = List.length l) }) =     { val := l, property := (_ : List.length l = List.length l) } ** apply toList_injective ** case mk.a \u03b1 : Type u_1 l : List \u03b1 \u22a2 toList (ofFn (get { val := l, property := (_ : List.length l = List.length l) })) =     toList { val := l, property := (_ : List.length l = List.length l) } ** dsimp ** case mk.a \u03b1 : Type u_1 l : List \u03b1 \u22a2 toList (ofFn (get { val := l, property := (_ : List.length l = List.length l) })) = l ** simpa only [toList_ofFn] using List.ofFn_get _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.gcd_gcd_self_right_right ** m n : Nat \u22a2 gcd m (gcd n m) = gcd n m ** rw [gcd_comm n m, gcd_gcd_self_right_left] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_not_mem_null_integral_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t N : Set \u03b1 f : \u03b1 \u2192 \u211d inst\u271d : IsProbabilityMeasure \u03bc hf : Integrable f hN : \u2191\u2191\u03bc N = 0 \u22a2 \u2203 x, \u00acx \u2208 N \u2227 \u222b (a : \u03b1), f a \u2202\u03bc \u2264 f x ** simpa only [average_eq_integral] using\n  exists_not_mem_null_average_le (IsProbabilityMeasure.ne_zero \u03bc) hf hN ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.reverse_get_zero ** n : \u2115 \u03b1 : Type u_1 v : Vector \u03b1 (n + 1) \u22a2 head (reverse v) = last v ** rw [\u2190 get_zero, last_def, get_eq_get, get_eq_get] ** n : \u2115 \u03b1 : Type u_1 v : Vector \u03b1 (n + 1) \u22a2 List.get (toList (reverse v)) (Fin.cast (_ : Nat.succ n = List.length (toList (reverse v))) 0) =     List.get (toList v) (Fin.cast (_ : n + 1 = List.length (toList v)) (Fin.last n)) ** simp_rw [toList_reverse] ** n : \u2115 \u03b1 : Type u_1 v : Vector \u03b1 (n + 1) \u22a2 List.get (List.reverse (toList v)) (Fin.cast (_ : Nat.succ n = List.length (toList (reverse v))) 0) =     List.get (toList v) (Fin.cast (_ : n + 1 = List.length (toList v)) (Fin.last n)) ** rw [\u2190 Option.some_inj, Fin.cast, Fin.cast, \u2190 List.get?_eq_get, \u2190 List.get?_eq_get,\n  List.get?_reverse] ** n : \u2115 \u03b1 : Type u_1 v : Vector \u03b1 (n + 1) \u22a2 List.get? (toList v) (List.length (toList v) - 1 - \u21910) = List.get? (toList v) \u2191(Fin.last n) ** congr ** case e_a n : \u2115 \u03b1 : Type u_1 v : Vector \u03b1 (n + 1) \u22a2 List.length (toList v) - 1 - \u21910 = \u2191(Fin.last n) ** simp ** case h n : \u2115 \u03b1 : Type u_1 v : Vector \u03b1 (n + 1) \u22a2 \u21910 < List.length (toList v) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.bind_pure ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 f : \u03b1 \u2192 PMF \u03b2 g : \u03b2 \u2192 PMF \u03b3 x y : \u03b1 hy : y \u2260 x \u22a2 \u2191p y * \u2191(pure y) x = 0 ** rw [pure_apply_of_ne _ _ hy.symm, mul_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 f : \u03b1 \u2192 PMF \u03b2 g : \u03b2 \u2192 PMF \u03b3 x : \u03b1 \u22a2 \u2191p x * \u2191(pure x) x = \u2191p x ** rw [pure_apply_self, mul_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.sum_variance_truncation_le ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 \u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a ** set Y := fun n : \u2115 => truncation X n ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a ** let \u03c1 : Measure \u211d := Measure.map X \u2119 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 \u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation X \u2191j ^ 2) a \u2264 2 * \u222b (a : \u03a9), X a ** have Y2 : \u2200 n, \ud835\udd3c[Y n ^ 2] = \u222b x in (0)..n, x ^ 2 \u2202\u03c1 := by\n  intro n\n  change \ud835\udd3c[fun x => Y n x ^ 2] = _\n  rw [moment_truncation_eq_intervalIntegral_of_nonneg hint.1 two_ne_zero hnonneg] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 \u22a2 \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 ** intro n ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 n : \u2115 \u22a2 \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 ** change \ud835\udd3c[fun x => Y n x ^ 2] = _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 n : \u2115 \u22a2 \u222b (a : \u03a9), (fun x => Y n x ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 ** rw [moment_truncation_eq_intervalIntegral_of_nonneg hint.1 two_ne_zero hnonneg] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (Y j ^ 2) a = \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 ** simp_rw [Y2] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 =     \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 ** congr 1 with j ** case e_f.h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 j : \u2115 \u22a2 (\u2191j ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 ** congr 1 ** case e_f.h.e_a \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 j : \u2115 \u22a2 \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 ** rw [intervalIntegral.sum_integral_adjacent_intervals] ** case e_f.h.e_a \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 j : \u2115 \u22a2 \u2200 (k : \u2115), k < j \u2192 IntervalIntegrable (fun x => x ^ 2) \u03c1 \u2191k \u2191(k + 1) ** intro k _ ** case e_f.h.e_a \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 j k : \u2115 a\u271d : k < j \u22a2 IntervalIntegrable (fun x => x ^ 2) \u03c1 \u2191k \u2191(k + 1) ** exact (continuous_id.pow _).intervalIntegrable _ _ ** case e_f.h.e_a \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 j : \u2115 \u22a2 \u222b (x : \u211d) in 0 ..\u2191j, x ^ 2 \u2202\u03c1 = \u222b (x : \u211d) in \u21910 ..\u2191j, x ^ 2 \u2202\u03c1 ** norm_cast ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u2211 k in range j, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 =     \u2211 k in range K, (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 ** simp_rw [mul_sum, sum_mul, sum_sigma'] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2211 x in Finset.sigma (range K) fun a => range a,       (\u2191x.fst ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in \u2191x.snd..\u2191(x.snd + 1), x ^ 2 \u2202Measure.map X \u2119 =     \u2211 x in Finset.sigma (range K) fun a => Ioo a K,       (\u2191x.snd ^ 2)\u207b\u00b9 * \u222b (x : \u211d) in \u2191x.fst..\u2191(x.fst + 1), x ^ 2 \u2202Measure.map X \u2119 ** refine' sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n  (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _ ** case refine'_1 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => range a),     (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ioo a K ** rintro \u27e8i, j\u27e9 hij ** case refine'_1.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => Ioo a K ** simp only [mem_sigma, mem_range, mem_filter] at hij ** case refine'_1.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 i j : \u2115 hij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a hij : i < K \u2227 j < i \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208 Finset.sigma (range K) fun a => Ioo a K ** simp [hij, mem_sigma, mem_range, and_self_iff, hij.2.trans hij.1] ** case refine'_2 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ioo a K),     (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => range a ** rintro \u27e8i, j\u27e9 hij ** case refine'_2.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208 Finset.sigma (range K) fun a => range a ** simp only [mem_sigma, mem_range, mem_Ioo] at hij ** case refine'_2.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 i j : \u2115 hij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K hij : i < K \u2227 i < j \u2227 j < K \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208 Finset.sigma (range K) fun a => range a ** simp only [hij, mem_sigma, mem_range, and_self_iff] ** case refine'_3 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => range a),     (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)         (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => Ioo a K) =       a ** rintro \u27e8i, j\u27e9 hij ** case refine'_3.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => range a \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)       (_ :         (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208           Finset.sigma (range K) fun a => Ioo a K) =     { fst := i, snd := j } ** rfl ** case refine'_4 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range K) fun a => Ioo a K),     (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)         (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range K) fun a => range a) =       a ** rintro \u27e8i, j\u27e9 hij ** case refine'_4.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range K) fun a => Ioo a K \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)       (_ :         (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208           Finset.sigma (range K) fun a => range a) =     { fst := i, snd := j } ** rfl ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2211 k in range K, (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264     \u2211 k in range K, 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 ** apply sum_le_sum fun k _ => ?_ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K \u22a2 (\u2211 j in Ioo k K, (\u2191j ^ 2)\u207b\u00b9) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264     2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 ** refine' mul_le_mul_of_nonneg_right (sum_Ioo_inv_sq_le _ _) _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K \u22a2 0 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 ** refine' intervalIntegral.integral_nonneg_of_forall _ fun u => sq_nonneg _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K \u22a2 \u2191k \u2264 \u2191(k + 1) ** simp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2211 k in range K, 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264     \u2211 k in range K, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1 ** apply sum_le_sum fun k _ => ?_ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K \u22a2 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1 ** have Ik : (k : \u211d) \u2264 (k + 1 : \u2115) := by simp ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) \u22a2 2 / (\u2191k + 1) * \u222b (x : \u211d) in \u2191k..\u2191(k + 1), x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1 ** rw [\u2190 intervalIntegral.integral_const_mul, intervalIntegral.integral_of_le Ik,\n  intervalIntegral.integral_of_le Ik] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) \u22a2 \u222b (x : \u211d) in Set.Ioc \u2191k \u2191(k + 1), 2 / (\u2191k + 1) * x ^ 2 \u2202\u03c1 \u2264 \u222b (x : \u211d) in Set.Ioc \u2191k \u2191(k + 1), 2 * x \u2202\u03c1 ** refine' set_integral_mono_on _ _ measurableSet_Ioc fun x hx => _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K \u22a2 \u2191k \u2264 \u2191(k + 1) ** simp ** case refine'_1 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) \u22a2 IntegrableOn (fun x => 2 / (\u2191k + 1) * x ^ 2) (Set.Ioc \u2191k \u2191(k + 1)) ** apply Continuous.integrableOn_Ioc ** case refine'_1.hf \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) \u22a2 Continuous fun x => 2 / (\u2191k + 1) * x ^ 2 ** exact continuous_const.mul (continuous_pow 2) ** case refine'_2 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) \u22a2 IntegrableOn (fun x => 2 * x) (Set.Ioc \u2191k \u2191(k + 1)) ** apply Continuous.integrableOn_Ioc ** case refine'_2.hf \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) \u22a2 Continuous fun x => 2 * x ** exact continuous_const.mul continuous_id' ** case refine'_3 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) x : \u211d hx : x \u2208 Set.Ioc \u2191k \u2191(k + 1) \u22a2 2 / (\u2191k + 1) * x ^ 2 \u2264 2 * x ** calc\n  \u21912 / (\u2191k + \u21911) * x ^ 2 = x / (k + 1) * (2 * x) := by ring\n  _ \u2264 1 * (2 * x) :=\n    (mul_le_mul_of_nonneg_right (by\n      convert (div_le_one _).2 hx.2; norm_cast\n      simp only [Nat.cast_add, Nat.cast_one]\n      linarith only [show (0 : \u211d) \u2264 k from Nat.cast_nonneg k])\n      (mul_nonneg zero_le_two ((Nat.cast_nonneg k).trans hx.1.le)))\n  _ = 2 * x := by rw [one_mul] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) x : \u211d hx : x \u2208 Set.Ioc \u2191k \u2191(k + 1) \u22a2 2 / (\u2191k + 1) * x ^ 2 = x / (\u2191k + 1) * (2 * x) ** ring ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) x : \u211d hx : x \u2208 Set.Ioc \u2191k \u2191(k + 1) \u22a2 x / (\u2191k + 1) \u2264 1 ** convert (div_le_one _).2 hx.2 ** case h.e'_3.h.e'_6 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) x : \u211d hx : x \u2208 Set.Ioc \u2191k \u2191(k + 1) \u22a2 \u2191k + 1 = \u2191(k + 1)  \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) x : \u211d hx : x \u2208 Set.Ioc \u2191k \u2191(k + 1) \u22a2 0 < \u2191(k + 1) ** norm_cast ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) x : \u211d hx : x \u2208 Set.Ioc \u2191k \u2191(k + 1) \u22a2 0 < \u2191(k + 1) ** simp only [Nat.cast_add, Nat.cast_one] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) x : \u211d hx : x \u2208 Set.Ioc \u2191k \u2191(k + 1) \u22a2 0 < \u2191k + 1 ** linarith only [show (0 : \u211d) \u2264 k from Nat.cast_nonneg k] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k \u2208 range K Ik : \u2191k \u2264 \u2191(k + 1) x : \u211d hx : x \u2208 Set.Ioc \u2191k \u2191(k + 1) \u22a2 1 * (2 * x) = 2 * x ** rw [one_mul] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u2211 k in range K, \u222b (x : \u211d) in \u2191k..\u2191(k + 1), 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1 ** rw [intervalIntegral.sum_integral_adjacent_intervals fun k _ => ?_] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u222b (x : \u211d) in \u21910 ..\u2191K, 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1  \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k < K \u22a2 IntervalIntegrable (fun x => 2 * x) \u03c1 \u2191k \u2191(k + 1) ** swap ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u222b (x : \u211d) in \u21910 ..\u2191K, 2 * x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1 ** rw [intervalIntegral.integral_const_mul] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 2 * \u222b (x : \u211d) in \u21910 ..\u2191K, x \u2202\u03c1 = 2 * \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1 ** norm_cast ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 k : \u2115 x\u271d : k < K \u22a2 IntervalIntegrable (fun x => 2 * x) \u03c1 \u2191k \u2191(k + 1) ** exact (continuous_const.mul continuous_id').intervalIntegrable _ _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u222b (x : \u211d) in 0 ..\u2191K, x \u2202\u03c1 \u2264 \u222b (a : \u03a9), X a ** rw [\u2190 integral_truncation_eq_intervalIntegral_of_nonneg hint.1 hnonneg] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation X \u2191n \u03c1 : Measure \u211d := Measure.map X \u2119 Y2 : \u2200 (n : \u2115), \u222b (a : \u03a9), (Y n ^ 2) a = \u222b (x : \u211d) in 0 ..\u2191n, x ^ 2 \u2202\u03c1 \u22a2 \u222b (x : \u03a9), truncation X (\u2191K) x \u2264 \u222b (a : \u03a9), X a ** exact integral_truncation_le_integral_of_nonneg hint hnonneg ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.centralMoment_one ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc \u22a2 centralMoment X 1 \u03bc = 0 ** by_cases h_int : Integrable X \u03bc ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : Integrable X \u22a2 centralMoment X 1 \u03bc = 0 ** rw [centralMoment_one' h_int] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : Integrable X \u22a2 (1 - ENNReal.toReal (\u2191\u2191\u03bc Set.univ)) * \u222b (x : \u03a9), X x \u2202\u03bc = 0 ** simp only [measure_univ, ENNReal.one_toReal, sub_self, zero_mul] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X \u22a2 centralMoment X 1 \u03bc = 0 ** simp only [centralMoment, Pi.sub_apply, pow_one] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X \u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0 ** have : \u00acIntegrable (fun x => X x - integral \u03bc X) \u03bc := by\n  refine' fun h_sub => h_int _\n  have h_add : X = (fun x => X x - integral \u03bc X) + fun _ => integral \u03bc X := by ext1 x; simp\n  rw [h_add]\n  exact h_sub.add (integrable_const _) ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X this : \u00acIntegrable fun x => X x - integral \u03bc X \u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0 ** rw [integral_undef this] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X \u22a2 \u00acIntegrable fun x => X x - integral \u03bc X ** refine' fun h_sub => h_int _ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X \u22a2 Integrable X ** have h_add : X = (fun x => X x - integral \u03bc X) + fun _ => integral \u03bc X := by ext1 x; simp ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X h_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X \u22a2 Integrable X ** rw [h_add] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X h_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X \u22a2 Integrable ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X) ** exact h_sub.add (integrable_const _) ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X \u22a2 X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X ** ext1 x ** case h \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc h_int : \u00acIntegrable X h_sub : Integrable fun x => X x - integral \u03bc X x : \u03a9 \u22a2 X x = ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X) x ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.Mem\u2112p.uniformIntegrable_of_identDistrib ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 E : Type u_5 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc \u03b9 : Type u_6 f : \u03b9 \u2192 \u03b1 \u2192 E j : \u03b9 p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 h\u2112p : Mem\u2112p (f j) p hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) \u22a2 UniformIntegrable f p \u03bc ** have hfmeas : \u2200 i, AEStronglyMeasurable (f i) \u03bc := fun i =>\n  (hf i).aestronglyMeasurable_iff.2 h\u2112p.1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 E : Type u_5 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc \u03b9 : Type u_6 f : \u03b9 \u2192 \u03b1 \u2192 E j : \u03b9 p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 h\u2112p : Mem\u2112p (f j) p hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) hfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc \u22a2 UniformIntegrable f p \u03bc ** set g : \u03b9 \u2192 \u03b1 \u2192 E := fun i => (hfmeas i).choose ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 E : Type u_5 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc \u03b9 : Type u_6 f : \u03b9 \u2192 \u03b1 \u2192 E j : \u03b9 p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 h\u2112p : Mem\u2112p (f j) p hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) hfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc g : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u22a2 UniformIntegrable f p \u03bc ** have hgmeas : \u2200 i, StronglyMeasurable (g i) := fun i => (Exists.choose_spec <| hfmeas i).1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 E : Type u_5 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc \u03b9 : Type u_6 f : \u03b9 \u2192 \u03b1 \u2192 E j : \u03b9 p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 h\u2112p : Mem\u2112p (f j) p hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) hfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc g : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) hgmeas : \u2200 (i : \u03b9), StronglyMeasurable (g i) \u22a2 UniformIntegrable f p \u03bc ** have hgeq : \u2200 i, g i =\u1d50[\u03bc] f i := fun i => (Exists.choose_spec <| hfmeas i).2.symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 E : Type u_5 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc \u03b9 : Type u_6 f : \u03b9 \u2192 \u03b1 \u2192 E j : \u03b9 p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 h\u2112p : Mem\u2112p (f j) p hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) hfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc g : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) hgmeas : \u2200 (i : \u03b9), StronglyMeasurable (g i) hgeq : \u2200 (i : \u03b9), g i =\u1d50[\u03bc] f i \u22a2 UniformIntegrable f p \u03bc ** have hg\u2112p : Mem\u2112p (g j) p \u03bc := h\u2112p.ae_eq (hgeq j).symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 E : Type u_5 inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc \u03b9 : Type u_6 f : \u03b9 \u2192 \u03b1 \u2192 E j : \u03b9 p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 h\u2112p : Mem\u2112p (f j) p hf : \u2200 (i : \u03b9), IdentDistrib (f i) (f j) hfmeas : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc g : \u03b9 \u2192 \u03b1 \u2192 E := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) hgmeas : \u2200 (i : \u03b9), StronglyMeasurable (g i) hgeq : \u2200 (i : \u03b9), g i =\u1d50[\u03bc] f i hg\u2112p : Mem\u2112p (g j) p \u22a2 UniformIntegrable f p \u03bc ** exact UniformIntegrable.ae_eq\n  (Mem\u2112p.uniformIntegrable_of_identDistrib_aux hp hp' hg\u2112p hgmeas fun i =>\n    (IdentDistrib.of_ae_eq (hgmeas i).aemeasurable (hgeq i)).trans\n      ((hf i).trans <| IdentDistrib.of_ae_eq (hfmeas j).aemeasurable (hgeq j).symm)) hgeq ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Ordered.lowerBound?_exists ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut h : Ordered cmp t \u22a2 (\u2203 x, lowerBound? cut t none = some x) \u2194 \u2203 x, x \u2208 t \u2227 cut x \u2260 Ordering.lt ** refine \u27e8fun \u27e8x, hx\u27e9 => \u27e8_, lowerBound?_mem hx, lowerBound?_le hx\u27e9, fun H => ?_\u27e9 ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut h : Ordered cmp t H : \u2203 x, x \u2208 t \u2227 cut x \u2260 Ordering.lt \u22a2 \u2203 x, lowerBound? cut t none = some x ** obtain \u27e8x, hx, e\u27e9 := H ** case intro.intro.nil \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut h : Ordered cmp nil x : \u03b1 hx : x \u2208 nil e : cut x \u2260 Ordering.lt \u22a2 \u2203 x, lowerBound? cut nil none = some x ** cases hx ** case intro.intro.node \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 ihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x r_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x h : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d) x : \u03b1 hx : x \u2208 node c\u271d l\u271d v\u271d r\u271d e : cut x \u2260 Ordering.lt \u22a2 \u2203 x, lowerBound? cut (node c\u271d l\u271d v\u271d r\u271d) none = some x ** simp [lowerBound?] ** case intro.intro.node \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 ihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x r_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x h : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d) x : \u03b1 hx : x \u2208 node c\u271d l\u271d v\u271d r\u271d e : cut x \u2260 Ordering.lt \u22a2 \u2203 x,     (match cut v\u271d with       | Ordering.lt => lowerBound? cut l\u271d none       | Ordering.gt => lowerBound? cut r\u271d (some v\u271d)       | Ordering.eq => some v\u271d) =       some x ** split ** case intro.intro.node.h_1 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 ihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x r_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x h : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d) x : \u03b1 hx : x \u2208 node c\u271d l\u271d v\u271d r\u271d e : cut x \u2260 Ordering.lt x\u271d : Ordering heq\u271d : cut v\u271d = Ordering.lt \u22a2 \u2203 x, lowerBound? cut l\u271d none = some x ** rcases hx with rfl | hx | hx ** case intro.intro.node.h_1.inl \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut c\u271d : RBColor l\u271d r\u271d : RBNode \u03b1 ihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x r_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x x : \u03b1 e : cut x \u2260 Ordering.lt x\u271d : Ordering h : Ordered cmp (node c\u271d l\u271d x r\u271d) heq\u271d : cut x = Ordering.lt \u22a2 \u2203 x, lowerBound? cut l\u271d none = some x ** contradiction ** case intro.intro.node.h_1.inr.inl \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 ihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x r_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x h : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d) x : \u03b1 e : cut x \u2260 Ordering.lt x\u271d : Ordering heq\u271d : cut v\u271d = Ordering.lt hx : Any (fun x_1 => x = x_1) l\u271d \u22a2 \u2203 x, lowerBound? cut l\u271d none = some x ** exact ihl h.2.2.1 _ hx e ** case intro.intro.node.h_1.inr.inr \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 ihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x r_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x h : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d) x : \u03b1 e : cut x \u2260 Ordering.lt x\u271d : Ordering heq\u271d : cut v\u271d = Ordering.lt hx : Any (fun x_1 => x = x_1) r\u271d \u22a2 \u2203 x, lowerBound? cut l\u271d none = some x ** next hv => cases e <| IsCut.lt_trans (All_def.1 h.2.1 _ hx).1 hv ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 ihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x r_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x h : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d) x : \u03b1 e : cut x \u2260 Ordering.lt x\u271d : Ordering hv : cut v\u271d = Ordering.lt hx : Any (fun x_1 => x = x_1) r\u271d \u22a2 \u2203 x, lowerBound? cut l\u271d none = some x ** cases e <| IsCut.lt_trans (All_def.1 h.2.1 _ hx).1 hv ** case intro.intro.node.h_2 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 ihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x r_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x h : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d) x : \u03b1 hx : x \u2208 node c\u271d l\u271d v\u271d r\u271d e : cut x \u2260 Ordering.lt x\u271d : Ordering heq\u271d : cut v\u271d = Ordering.gt \u22a2 \u2203 x, lowerBound? cut r\u271d (some v\u271d) = some x ** exact lowerBound?_of_some ** case intro.intro.node.h_3 \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut c\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 ihl : Ordered cmp l\u271d \u2192 \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut l\u271d none = some x r_ih\u271d : Ordered cmp r\u271d \u2192 \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 cut x \u2260 Ordering.lt \u2192 \u2203 x, lowerBound? cut r\u271d none = some x h : Ordered cmp (node c\u271d l\u271d v\u271d r\u271d) x : \u03b1 hx : x \u2208 node c\u271d l\u271d v\u271d r\u271d e : cut x \u2260 Ordering.lt x\u271d : Ordering heq\u271d : cut v\u271d = Ordering.eq \u22a2 \u2203 x, some v\u271d = some x ** exact \u27e8_, rfl\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.noncommProd_union_of_disjoint ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 h : Disjoint s t f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise {x | x \u2208 s \u222a t} fun a b => Commute (f a) (f b) \u22a2 noncommProd (s \u222a t) f comm =     noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) *       noncommProd t f (_ : Set.Pairwise \u2191t fun a b => Commute (f a) (f b)) ** obtain \u27e8sl, sl', rfl\u27e9 := exists_list_nodup_eq s ** case intro.intro F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 sl : List \u03b1 sl' : List.Nodup sl h : Disjoint (List.toFinset sl) t comm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a t} fun a b => Commute (f a) (f b) \u22a2 noncommProd (List.toFinset sl \u222a t) f comm =     noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *       noncommProd t f (_ : Set.Pairwise \u2191t fun a b => Commute (f a) (f b)) ** obtain \u27e8tl, tl', rfl\u27e9 := exists_list_nodup_eq t ** case intro.intro.intro.intro F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 f : \u03b1 \u2192 \u03b2 sl : List \u03b1 sl' : List.Nodup sl tl : List \u03b1 tl' : List.Nodup tl h : Disjoint (List.toFinset sl) (List.toFinset tl) comm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b) \u22a2 noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm =     noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *       noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b)) ** rw [List.disjoint_toFinset_iff_disjoint] at h ** case intro.intro.intro.intro F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 f : \u03b1 \u2192 \u03b2 sl : List \u03b1 sl' : List.Nodup sl tl : List \u03b1 tl' : List.Nodup tl h : List.Disjoint sl tl comm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b) \u22a2 noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm =     noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *       noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b)) ** calc noncommProd (List.toFinset sl \u222a List.toFinset tl) f comm\n   = noncommProd \u27e8\u2191(sl ++ tl), Multiset.coe_nodup.2 (sl'.append tl' h)\u27e9 f\n       (by convert comm; simp [Set.ext_iff]) := noncommProd_congr (by ext; simp) (by simp) _\n _ = noncommProd (List.toFinset sl) f (comm.mono <| coe_subset.2 <| subset_union_left _ _) *\n       noncommProd (List.toFinset tl) f (comm.mono <| coe_subset.2 <| subset_union_right _ _) :=\n  by simp [noncommProd, List.dedup_eq_self.2 sl', List.dedup_eq_self.2 tl', h] ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 f : \u03b1 \u2192 \u03b2 sl : List \u03b1 sl' : List.Nodup sl tl : List \u03b1 tl' : List.Nodup tl h : List.Disjoint sl tl comm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b) \u22a2 Set.Pairwise \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } fun a b => Commute (f a) (f b) ** convert comm ** case h.e'_2 F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 f : \u03b1 \u2192 \u03b2 sl : List \u03b1 sl' : List.Nodup sl tl : List \u03b1 tl' : List.Nodup tl h : List.Disjoint sl tl comm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b) \u22a2 \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } = {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} ** simp [Set.ext_iff] ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 f : \u03b1 \u2192 \u03b2 sl : List \u03b1 sl' : List.Nodup sl tl : List \u03b1 tl' : List.Nodup tl h : List.Disjoint sl tl comm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b) \u22a2 List.toFinset sl \u222a List.toFinset tl = { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } ** ext ** case a F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 f : \u03b1 \u2192 \u03b2 sl : List \u03b1 sl' : List.Nodup sl tl : List \u03b1 tl' : List.Nodup tl h : List.Disjoint sl tl comm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b) a\u271d : \u03b1 \u22a2 a\u271d \u2208 List.toFinset sl \u222a List.toFinset tl \u2194 a\u271d \u2208 { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } ** simp ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 f : \u03b1 \u2192 \u03b2 sl : List \u03b1 sl' : List.Nodup sl tl : List \u03b1 tl' : List.Nodup tl h : List.Disjoint sl tl comm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b) \u22a2 \u2200 (x : \u03b1), x \u2208 { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } \u2192 f x = f x ** simp ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 f : \u03b1 \u2192 \u03b2 sl : List \u03b1 sl' : List.Nodup sl tl : List \u03b1 tl' : List.Nodup tl h : List.Disjoint sl tl comm : Set.Pairwise {x | x \u2208 List.toFinset sl \u222a List.toFinset tl} fun a b => Commute (f a) (f b) \u22a2 noncommProd { val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } f       (_ :         Set.Pairwise \u2191{ val := \u2191(sl ++ tl), nodup := (_ : Multiset.Nodup \u2191(sl ++ tl)) } fun a b =>           Commute (f a) (f b)) =     noncommProd (List.toFinset sl) f (_ : Set.Pairwise \u2191(List.toFinset sl) fun a b => Commute (f a) (f b)) *       noncommProd (List.toFinset tl) f (_ : Set.Pairwise \u2191(List.toFinset tl) fun a b => Commute (f a) (f b)) ** simp [noncommProd, List.dedup_eq_self.2 sl', List.dedup_eq_self.2 tl', h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.mem_finset_prod ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 t : Finset \u03b9 f : \u03b9 \u2192 Set \u03b1 a : \u03b1 \u22a2 a \u2208 \u220f i in t, f i \u2194 \u2203 g x, \u220f i in t, g i = a ** induction' t using Finset.induction_on with i is hi ih generalizing a ** case insert \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a\u271d : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a a : \u03b1 \u22a2 a \u2208 \u220f i in insert i is, f i \u2194 \u2203 g x, \u220f i in insert i is, g i = a ** rw [Finset.prod_insert hi, Set.mem_mul] ** case insert \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a\u271d : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a a : \u03b1 \u22a2 (\u2203 x y, x \u2208 f i \u2227 y \u2208 \u220f x in is, f x \u2227 x * y = a) \u2194 \u2203 g x, \u220f i in insert i is, g i = a ** simp_rw [Finset.prod_insert hi] ** case insert \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a\u271d : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a a : \u03b1 \u22a2 (\u2203 x y, x \u2208 f i \u2227 y \u2208 \u220f x in is, f x \u2227 x * y = a) \u2194 \u2203 g h, g i * \u220f i in is, g i = a ** simp_rw [ih] ** case insert \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a\u271d : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a a : \u03b1 \u22a2 (\u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = a) \u2194 \u2203 g h, g i * \u220f i in is, g i = a ** constructor ** case empty \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a\u271d a : \u03b1 \u22a2 a \u2208 \u220f i in \u2205, f i \u2194 \u2203 g x, \u220f i in \u2205, g i = a ** simp_rw [Finset.prod_empty, Set.mem_one] ** case empty \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a\u271d a : \u03b1 \u22a2 a = 1 \u2194 \u2203 g h, 1 = a ** exact \u27e8fun h \u21a6 \u27e8fun _ \u21a6 a, fun hi \u21a6 False.elim (Finset.not_mem_empty _ hi), h.symm\u27e9,\n  fun \u27e8_, _, hf\u27e9 \u21a6 hf.symm\u27e9 ** case insert.mp \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a\u271d : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a a : \u03b1 \u22a2 (\u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = a) \u2192 \u2203 g h, g i * \u220f i in is, g i = a ** rintro \u27e8x, y, hx, \u27e8g, hg, rfl\u27e9, rfl\u27e9 ** case insert.mp.intro.intro.intro.intro.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a x : \u03b1 hx : x \u2208 f i g : \u03b9 \u2192 \u03b1 hg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i \u22a2 \u2203 g_1 h, g_1 i * \u220f i in is, g_1 i = x * \u220f i in is, g i ** refine \u27e8Function.update g i x, ?_, ?_\u27e9 ** case insert.mp.intro.intro.intro.intro.intro.intro.refine_1 \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a x : \u03b1 hx : x \u2208 f i g : \u03b9 \u2192 \u03b1 hg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i \u22a2 \u2200 {i_1 : \u03b9}, i_1 \u2208 insert i is \u2192 update g i x i_1 \u2208 f i_1 ** intro j hj ** case insert.mp.intro.intro.intro.intro.intro.intro.refine_1 \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a x : \u03b1 hx : x \u2208 f i g : \u03b9 \u2192 \u03b1 hg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i j : \u03b9 hj : j \u2208 insert i is \u22a2 update g i x j \u2208 f j ** obtain rfl | hj := Finset.mem_insert.mp hj ** case insert.mp.intro.intro.intro.intro.intro.intro.refine_1.inl \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a : \u03b1 is : Finset \u03b9 ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a x : \u03b1 g : \u03b9 \u2192 \u03b1 hg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i j : \u03b9 hi : \u00acj \u2208 is hx : x \u2208 f j hj : j \u2208 insert j is \u22a2 update g j x j \u2208 f j ** rwa [Function.update_same] ** case insert.mp.intro.intro.intro.intro.intro.intro.refine_1.inr \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a x : \u03b1 hx : x \u2208 f i g : \u03b9 \u2192 \u03b1 hg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i j : \u03b9 hj\u271d : j \u2208 insert i is hj : j \u2208 is \u22a2 update g i x j \u2208 f j ** rw [update_noteq (ne_of_mem_of_not_mem hj hi)] ** case insert.mp.intro.intro.intro.intro.intro.intro.refine_1.inr \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a x : \u03b1 hx : x \u2208 f i g : \u03b9 \u2192 \u03b1 hg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i j : \u03b9 hj\u271d : j \u2208 insert i is hj : j \u2208 is \u22a2 g j \u2208 f j ** exact hg hj ** case insert.mp.intro.intro.intro.intro.intro.intro.refine_2 \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a x : \u03b1 hx : x \u2208 f i g : \u03b9 \u2192 \u03b1 hg : \u2200 {i : \u03b9}, i \u2208 is \u2192 g i \u2208 f i \u22a2 update g i x i * \u220f i_1 in is, update g i x i_1 = x * \u220f i in is, g i ** rw [Finset.prod_update_of_not_mem hi, Function.update_same] ** case insert.mpr \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a\u271d : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a a : \u03b1 \u22a2 (\u2203 g h, g i * \u220f i in is, g i = a) \u2192 \u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = a ** rintro \u27e8g, hg, rfl\u27e9 ** case insert.mpr.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 F : Type u_4 inst\u271d\u00b2 : CommMonoid \u03b1 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : \u03b9 \u2192 Set \u03b1 a : \u03b1 i : \u03b9 is : Finset \u03b9 hi : \u00aci \u2208 is ih : \u2200 (a : \u03b1), a \u2208 \u220f i in is, f i \u2194 \u2203 g x, \u220f i in is, g i = a g : \u03b9 \u2192 \u03b1 hg : \u2200 {i_1 : \u03b9}, i_1 \u2208 insert i is \u2192 g i_1 \u2208 f i_1 \u22a2 \u2203 x y, x \u2208 f i \u2227 (\u2203 g x, \u220f i in is, g i = y) \u2227 x * y = g i * \u220f i in is, g i ** exact \u27e8g i, is.prod g, hg (is.mem_insert_self _),\n  \u27e8\u27e8g, fun hi \u21a6 hg (Finset.mem_insert_of_mem hi), rfl\u27e9, rfl\u27e9\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.BorelCantelli.martingalePart_process_ae_eq ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 \u2131\u271d : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 s\u271d : \u2115 \u2192 Set \u03a9 \u2131 : Filtration \u2115 m0 \u03bc : Measure \u03a9 s : \u2115 \u2192 Set \u03a9 n : \u2115 \u22a2 martingalePart (process s) \u2131 \u03bc n =     \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) ** simp only [martingalePart_eq_sum, process_zero, zero_add] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 \u2131\u271d : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 s\u271d : \u2115 \u2192 Set \u03a9 \u2131 : Filtration \u2115 m0 \u03bc : Measure \u03a9 s : \u2115 \u2192 Set \u03a9 n : \u2115 \u22a2 \u2211 i in Finset.range n, (process s (i + 1) - process s i - \u03bc[process s (i + 1) - process s i|\u2191\u2131 i]) =     \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) ** refine' Finset.sum_congr rfl fun k _ => _ ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc\u271d : Measure \u03a9 \u2131\u271d : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 s\u271d : \u2115 \u2192 Set \u03a9 \u2131 : Filtration \u2115 m0 \u03bc : Measure \u03a9 s : \u2115 \u2192 Set \u03a9 n k : \u2115 x\u271d : k \u2208 Finset.range n \u22a2 process s (k + 1) - process s k - \u03bc[process s (k + 1) - process s k|\u2191\u2131 k] =     Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k] ** simp only [process, Finset.sum_range_succ_sub_sum] ** Qed",
        "informal": ""
    },
    {
        "formal": "PosNum.size_to_nat ** \u03b1 : Type u_1 n : PosNum \u22a2 \u2191(size (bit0 n)) = Nat.size \u2191(bit0 n) ** rw [size, succ_to_nat, size_to_nat n, cast_bit0, Nat.size_bit0 <| ne_of_gt <| to_nat_pos n] ** \u03b1 : Type u_1 n : PosNum \u22a2 \u2191(size (bit1 n)) = Nat.size \u2191(bit1 n) ** rw [size, succ_to_nat, size_to_nat n, cast_bit1, Nat.size_bit1] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2075 : SMul R \u211d\u22650 inst\u271d\u2074 : SMul R \u211d\u22650\u221e inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_3 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 \u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f)) ** rw [tendsto_iff_weak_star_tendsto, tendsto_iff_forall_eval_tendsto_topDualPairing] ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2075 : SMul R \u211d\u22650 inst\u271d\u2074 : SMul R \u211d\u22650\u221e inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_3 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bc : FiniteMeasure \u03a9 \u22a2 (\u2200 (y : \u03a9 \u2192\u1d47 \u211d\u22650),       Tendsto (fun i => \u2191(\u2191(topDualPairing \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650)) (toWeakDualBCNN (\u03bcs i))) y) F         (\ud835\udcdd (\u2191(\u2191(topDualPairing \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650)) (toWeakDualBCNN \u03bc)) y))) \u2194     \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f)) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.lt_div_add_one_mul_self ** a b : Int H : 0 < b \u22a2 a < (div a b + 1) * b ** rw [Int.add_mul, Int.one_mul, Int.mul_comm] ** a b : Int H : 0 < b \u22a2 a < b * div a b + b ** exact Int.lt_add_of_sub_left_lt <| Int.mod_def .. \u25b8 mod_lt_of_pos _ H ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec'.vec_iff ** m n : \u2115 f : Vector \u2115 m \u2192 Vector \u2115 n h : Vec f \u22a2 Computable f ** simpa only [ofFn_get] using vector_ofFn fun i => to_part (h i) ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.dvd_to_int ** \u03b1 : Type u_1 m n : ZNum x\u271d : \u2191m \u2223 \u2191n k : \u2124 e : \u2191n = \u2191m * k \u22a2 n = m * \u2191k ** rw [\u2190 of_to_int n, e] ** \u03b1 : Type u_1 m n : ZNum x\u271d : \u2191m \u2223 \u2191n k : \u2124 e : \u2191n = \u2191m * k \u22a2 \u2191(\u2191m * k) = m * \u2191k ** simp ** \u03b1 : Type u_1 m n : ZNum x\u271d : m \u2223 n k : ZNum e : n = m * k \u22a2 \u2191n = \u2191m * \u2191k ** simp [e] ** Qed",
        "informal": ""
    },
    {
        "formal": "Embedding.aestronglyMeasurable_comp_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g \u22a2 AEStronglyMeasurable (fun x => g (f x)) \u03bc \u2194 AEStronglyMeasurable f \u03bc ** letI := pseudoMetrizableSpacePseudoMetric \u03b3 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 \u22a2 AEStronglyMeasurable (fun x => g (f x)) \u03bc \u2194 AEStronglyMeasurable f \u03bc ** borelize \u03b2 \u03b3 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 \u22a2 AEStronglyMeasurable (fun x => g (f x)) \u03bc \u2194 AEStronglyMeasurable f \u03bc ** refine'\n  \u27e8fun H => aestronglyMeasurable_iff_aemeasurable_separable.2 \u27e8_, _\u27e9, fun H =>\n    hg.continuous.comp_aestronglyMeasurable H\u27e9 ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc \u22a2 AEMeasurable f ** let G : \u03b2 \u2192 range g := codRestrict g (range g) mem_range_self ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc G : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x) \u22a2 AEMeasurable f ** have hG : ClosedEmbedding G :=\n  { hg.codRestrict _ _ with\n    closed_range := by\n      convert isClosed_univ (\u03b1 := \u21a5(range g))\n      apply eq_univ_of_forall\n      rintro \u27e8-, \u27e8x, rfl\u27e9\u27e9\n      exact mem_range_self x } ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc G : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x) hG : ClosedEmbedding G \u22a2 AEMeasurable f ** have : AEMeasurable (G \u2218 f) \u03bc := AEMeasurable.subtype_mk H.aemeasurable ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this\u271d\u2074 : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc G : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x) hG : ClosedEmbedding G this : AEMeasurable (G \u2218 f) \u22a2 AEMeasurable f ** exact hG.measurableEmbedding.aemeasurable_comp_iff.1 this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc G : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x) src\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=   Embedding.codRestrict hg (range g) mem_range_self \u22a2 IsClosed (range G) ** convert isClosed_univ (\u03b1 := \u21a5(range g)) ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc G : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x) src\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=   Embedding.codRestrict hg (range g) mem_range_self \u22a2 range G = univ ** apply eq_univ_of_forall ** case h.e'_3.a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc G : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x) src\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=   Embedding.codRestrict hg (range g) mem_range_self \u22a2 \u2200 (x : \u2191(range g)), x \u2208 range G ** rintro \u27e8-, \u27e8x, rfl\u27e9\u27e9 ** case h.e'_3.a.mk.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc G : \u03b2 \u2192 \u2191(range g) := codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x) src\u271d : _root_.Embedding (codRestrict g (range g) (_ : \u2200 (i : \u03b2), g i \u2208 range fun x => g x)) :=   Embedding.codRestrict hg (range g) mem_range_self x : \u03b2 \u22a2 { val := g x, property := (_ : \u2203 y, g y = g x) } \u2208 range G ** exact mem_range_self x ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc \u22a2 \u2203 t, IsSeparable t \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t ** rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 H).2 with \u27e8t, ht, h't\u27e9 ** case refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : PseudoMetrizableSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b3 g : \u03b2 \u2192 \u03b3 f : \u03b1 \u2192 \u03b2 hg : _root_.Embedding g this : PseudoMetricSpace \u03b3 := pseudoMetrizableSpacePseudoMetric \u03b3 this\u271d\u00b3 : MeasurableSpace \u03b2 := borel \u03b2 this\u271d\u00b2 : BorelSpace \u03b2 this\u271d\u00b9 : MeasurableSpace \u03b3 := borel \u03b3 this\u271d : BorelSpace \u03b3 H : AEStronglyMeasurable (fun x => g (f x)) \u03bc t : Set \u03b3 ht : IsSeparable t h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g (f x) \u2208 t \u22a2 \u2203 t, IsSeparable t \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2208 t ** exact \u27e8g \u207b\u00b9' t, hg.isSeparable_preimage ht, h't\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_coe_le_of_lintegral_coe_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d\u22650 b : \u211d\u22650 h : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b \u22a2 \u222b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b ** by_cases hf : Integrable (fun a => (f a : \u211d)) \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d\u22650 b : \u211d\u22650 h : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b hf : Integrable fun a => \u2191(f a) \u22a2 \u222b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b ** exact (lintegral_coe_le_coe_iff_integral_le hf).1 h ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d\u22650 b : \u211d\u22650 h : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b hf : \u00acIntegrable fun a => \u2191(f a) \u22a2 \u222b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b ** rw [integral_undef hf] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d\u22650 b : \u211d\u22650 h : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2264 \u2191b hf : \u00acIntegrable fun a => \u2191(f a) \u22a2 0 \u2264 \u2191b ** exact b.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.dvd_to_nat ** m n : Num x\u271d : \u2191m \u2223 \u2191n k : \u2115 e : \u2191n = \u2191m * k \u22a2 n = m * \u2191k ** rw [\u2190 of_to_nat n, e] ** m n : Num x\u271d : \u2191m \u2223 \u2191n k : \u2115 e : \u2191n = \u2191m * k \u22a2 \u2191(\u2191m * k) = m * \u2191k ** simp ** m n : Num x\u271d : m \u2223 n k : Num e : n = m * k \u22a2 \u2191n = \u2191m * \u2191k ** simp [e, mul_to_nat] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.WF.out ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t : RBNode \u03b1 h : WF cmp t \u22a2 Ordered cmp t \u2227 \u2203 c n, Balanced t c n ** induction h with\n| mk o h => exact \u27e8o, _, _, h\u27e9\n| insert _ ih => have \u27e8o, _, _, h\u27e9 := ih; exact \u27e8o.insert, h.insert\u27e9\n| erase _ ih => have \u27e8o, _, _, h\u27e9 := ih; exact \u27e8o.erase, _, h.erase\u27e9 ** case mk \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t t\u271d : RBNode \u03b1 c\u271d : RBColor n\u271d : Nat o : Ordered cmp t\u271d h : Balanced t\u271d c\u271d n\u271d \u22a2 Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n ** exact \u27e8o, _, _, h\u27e9 ** case insert \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t t\u271d : RBNode \u03b1 a\u271d\u00b9 : \u03b1 a\u271d : WF cmp t\u271d ih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n \u22a2 Ordered cmp (RBNode.insert cmp t\u271d a\u271d\u00b9) \u2227 \u2203 c n, Balanced (RBNode.insert cmp t\u271d a\u271d\u00b9) c n ** have \u27e8o, _, _, h\u27e9 := ih ** case insert \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t t\u271d : RBNode \u03b1 a\u271d\u00b9 : \u03b1 a\u271d : WF cmp t\u271d ih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n o : Ordered cmp t\u271d w\u271d\u00b9 : RBColor w\u271d : Nat h : Balanced t\u271d w\u271d\u00b9 w\u271d \u22a2 Ordered cmp (RBNode.insert cmp t\u271d a\u271d\u00b9) \u2227 \u2203 c n, Balanced (RBNode.insert cmp t\u271d a\u271d\u00b9) c n ** exact \u27e8o.insert, h.insert\u27e9 ** case erase \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t t\u271d : RBNode \u03b1 cut\u271d : \u03b1 \u2192 Ordering a\u271d : WF cmp t\u271d ih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n \u22a2 Ordered cmp (RBNode.erase cut\u271d t\u271d) \u2227 \u2203 c n, Balanced (RBNode.erase cut\u271d t\u271d) c n ** have \u27e8o, _, _, h\u27e9 := ih ** case erase \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t t\u271d : RBNode \u03b1 cut\u271d : \u03b1 \u2192 Ordering a\u271d : WF cmp t\u271d ih : Ordered cmp t\u271d \u2227 \u2203 c n, Balanced t\u271d c n o : Ordered cmp t\u271d w\u271d\u00b9 : RBColor w\u271d : Nat h : Balanced t\u271d w\u271d\u00b9 w\u271d \u22a2 Ordered cmp (RBNode.erase cut\u271d t\u271d) \u2227 \u2203 c n, Balanced (RBNode.erase cut\u271d t\u271d) c n ** exact \u27e8o.erase, _, h.erase\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "measurableSet_integrable ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) \u22a2 MeasurableSet {x | Integrable (f x)} ** simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u03b2 \u2192 E hf : StronglyMeasurable (uncurry f) \u22a2 MeasurableSet {x | HasFiniteIntegral (f x)} ** exact measurableSet_lt (Measurable.lintegral_prod_right hf.ennnorm) measurable_const ** Qed",
        "informal": ""
    },
    {
        "formal": "circleIntegral.integral_sub_inv_of_mem_ball ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I ** have hR : 0 < R := dist_nonneg.trans_lt hw ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I ** suffices H : HasSum (fun n : \u2115 => \u222e z in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u03c0 * I) ** case H E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R \u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I) ** have H : \u2200 n : \u2115, n \u2260 0 \u2192 (\u222e z in C(c, R), (z - c) ^ (-n - 1 : \u2124)) = 0 := by\n  refine' fun n hn => integral_sub_zpow_of_ne _ _ _ _; simpa ** case H E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0 \u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I) ** have : (\u222e z in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u03c0 * I := by simp [hR.ne'] ** case H E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0 this : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I \u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I) ** refine' this \u25b8 hasSum_single _ fun n hn => _ ** case H E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0 this : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I n : \u2115 hn : n \u2260 0 \u22a2 (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) = 0 ** simp only [div_eq_mul_inv, mul_pow, integral_const_mul, mul_assoc] ** case H E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0 this : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I n : \u2115 hn : n \u2260 0 \u22a2 ((w - c) ^ n * \u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 ^ n * (z - c)\u207b\u00b9) = 0 ** rw [(integral_congr hR.le fun z hz => _).trans (H n hn), mul_zero] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0 this : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I n : \u2115 hn : n \u2260 0 \u22a2 \u2200 (z : \u2102), z \u2208 sphere c R \u2192 (z - c)\u207b\u00b9 ^ n * (z - c)\u207b\u00b9 = (z - c) ^ (-\u2191n - 1) ** intro z _ ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0 this : (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I n : \u2115 hn : n \u2260 0 z : \u2102 hz\u271d : z \u2208 sphere c R \u22a2 (z - c)\u207b\u00b9 ^ n * (z - c)\u207b\u00b9 = (z - c) ^ (-\u2191n - 1) ** rw [\u2190 pow_succ', \u2190 zpow_ofNat, inv_zpow, \u2190 zpow_neg, Int.ofNat_succ, neg_add,\n  sub_eq_add_neg _ (1 : \u2124)] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I) \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I ** have A : CircleIntegrable (fun _ => (1 : \u2102)) c R := continuousOn_const.circleIntegrable' ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I) A : CircleIntegrable (fun x => 1) c R \u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) = 2 * \u2191\u03c0 * I ** refine' (H.unique _).symm ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (2 * \u2191\u03c0 * I) A : CircleIntegrable (fun x => 1) c R \u22a2 HasSum (fun n => \u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)\u207b\u00b9) (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9) ** simpa only [smul_eq_mul, mul_one, add_sub_cancel'_right] using\n  hasSum_two_pi_I_cauchyPowerSeries_integral A hw ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R \u22a2 \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0 ** refine' fun n hn => integral_sub_zpow_of_ne _ _ _ _ ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R n : \u2115 hn : n \u2260 0 \u22a2 -\u2191n - 1 \u2260 -1 ** simpa ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E c w : \u2102 R : \u211d hw : w \u2208 ball c R hR : 0 < R H : \u2200 (n : \u2115), n \u2260 0 \u2192 (\u222e (z : \u2102) in C(c, R), (z - c) ^ (-\u2191n - 1)) = 0 \u22a2 (\u222e (z : \u2102) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)\u207b\u00b9) = 2 * \u2191\u03c0 * I ** simp [hR.ne'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.univ_pi_Ioi_ae_eq_Ici ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u2074 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i) f : (i : \u03b9) \u2192 \u03b1 i \u22a2 (Set.pi univ fun i => Ioi (f i)) =\u1da0[ae (Measure.pi \u03bc)] Ici f ** rw [\u2190 pi_univ_Ici] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u2074 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i) f : (i : \u03b9) \u2192 \u03b1 i \u22a2 (Set.pi univ fun i => Ioi (f i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Ici (f i) ** exact pi_Ioi_ae_eq_pi_Ici ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_nnnorm_le_of_forall_fin_meas_integral_eq ** \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' s : Set \u03b1 hm : m \u2264 m0 f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hfi : IntegrableOn f s hg : StronglyMeasurable g hgi : IntegrableOn g s hgf : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016g x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016f x\u2016\u208a \u2202\u03bc ** rw [\u2190 ofReal_integral_norm_eq_lintegral_nnnorm hfi, \u2190\n  ofReal_integral_norm_eq_lintegral_nnnorm hgi, ENNReal.ofReal_le_ofReal_iff] ** \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' s : Set \u03b1 hm : m \u2264 m0 f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hfi : IntegrableOn f s hg : StronglyMeasurable g hgi : IntegrableOn g s hgf : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2016g x\u2016 \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, \u2016f x\u2016 \u2202\u03bc ** exact integral_norm_le_of_forall_fin_meas_integral_eq hm hf hfi hg hgi hgf hs h\u03bcs ** \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' s : Set \u03b1 hm : m \u2264 m0 f g : \u03b1 \u2192 \u211d hf : StronglyMeasurable f hfi : IntegrableOn f s hg : StronglyMeasurable g hgi : IntegrableOn g s hgf : \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 0 \u2264 \u222b (x : \u03b1) in s, \u2016f x\u2016 \u2202\u03bc ** exact integral_nonneg fun x => norm_nonneg _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM0.Machine.map_respects ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081\u271d : \u039b \u2192 \u039b' g\u2082\u271d : \u039b' \u2192 \u039b g\u2081 : PointedMap \u039b \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b ss : Supports M S f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q \u22a2 Respects (step M) (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b ** intro c _ \u27e8cs, rfl\u27e9 ** \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081\u271d : \u039b \u2192 \u039b' g\u2082\u271d : \u039b' \u2192 \u039b g\u2081 : PointedMap \u039b \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b ss : Supports M S f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q c : Cfg \u0393 \u039b a\u2082\u271d : Cfg \u0393' \u039b' cs : c.q \u2208 S \u22a2 match step M c with   | some b\u2081 =>     \u2203 b\u2082,       (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082   | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none ** cases e : step M c ** case none \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081\u271d : \u039b \u2192 \u039b' g\u2082\u271d : \u039b' \u2192 \u039b g\u2081 : PointedMap \u039b \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b ss : Supports M S f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q c : Cfg \u0393 \u039b a\u2082\u271d : Cfg \u0393' \u039b' cs : c.q \u2208 S e : step M c = none \u22a2 match none with   | some b\u2081 =>     \u2203 b\u2082,       (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082   | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none ** rw [\u2190 M.map_step f\u2081 f\u2082 g\u2081 g\u2082 f\u2082\u2081 g\u2082\u2081 _ cs, e] ** case none \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081\u271d : \u039b \u2192 \u039b' g\u2082\u271d : \u039b' \u2192 \u039b g\u2081 : PointedMap \u039b \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b ss : Supports M S f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q c : Cfg \u0393 \u039b a\u2082\u271d : Cfg \u0393' \u039b' cs : c.q \u2208 S e : step M c = none \u22a2 match none with   | some b\u2081 =>     \u2203 b\u2082,       (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082   | none => Option.map (Cfg.map f\u2081 g\u2081.f) none = none ** rfl ** case some \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081\u271d : \u039b \u2192 \u039b' g\u2082\u271d : \u039b' \u2192 \u039b g\u2081 : PointedMap \u039b \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b ss : Supports M S f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q c : Cfg \u0393 \u039b a\u2082\u271d : Cfg \u0393' \u039b' cs : c.q \u2208 S val\u271d : Cfg \u0393 \u039b e : step M c = some val\u271d \u22a2 match some val\u271d with   | some b\u2081 =>     \u2203 b\u2082,       (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082   | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none ** refine' \u27e8_, \u27e8step_supports M ss e cs, rfl\u27e9, TransGen.single _\u27e9 ** case some \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081\u271d : \u039b \u2192 \u039b' g\u2082\u271d : \u039b' \u2192 \u039b g\u2081 : PointedMap \u039b \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b ss : Supports M S f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q c : Cfg \u0393 \u039b a\u2082\u271d : Cfg \u0393' \u039b' cs : c.q \u2208 S val\u271d : Cfg \u0393 \u039b e : step M c = some val\u271d \u22a2 Cfg.map f\u2081 g\u2081.f val\u271d \u2208 step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) ** rw [\u2190 M.map_step f\u2081 f\u2082 g\u2081 g\u2082 f\u2082\u2081 g\u2082\u2081 _ cs, e] ** case some \u0393 : Type u_1 inst\u271d\u00b3 : Inhabited \u0393 \u0393' : Type u_2 inst\u271d\u00b2 : Inhabited \u0393' \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u039b' : Type u_4 inst\u271d : Inhabited \u039b' M : Machine \u0393 \u039b f\u2081 : PointedMap \u0393 \u0393' f\u2082 : PointedMap \u0393' \u0393 g\u2081\u271d : \u039b \u2192 \u039b' g\u2082\u271d : \u039b' \u2192 \u039b g\u2081 : PointedMap \u039b \u039b' g\u2082 : \u039b' \u2192 \u039b S : Set \u039b ss : Supports M S f\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f g\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q c : Cfg \u0393 \u039b a\u2082\u271d : Cfg \u0393' \u039b' cs : c.q \u2208 S val\u271d : Cfg \u0393 \u039b e : step M c = some val\u271d \u22a2 Cfg.map f\u2081 g\u2081.f val\u271d \u2208 Option.map (Cfg.map f\u2081 g\u2081.f) (some val\u271d) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1to0.tr_eval ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 l : List \u0393 \u22a2 Part.map (fun c => Tape.right\u2080 c.Tape)       ((fun a => trCfg M a) <$> eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk\u2081 l }) =     TM1.eval M l ** rw [Part.map_eq_map, Part.map_map, TM1.eval] ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2081 l : List \u0393 \u22a2 Part.map ((fun c => Tape.right\u2080 c.Tape) \u2218 fun a => trCfg M a)       (eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk\u2081 l }) =     Part.map (fun c => Tape.right\u2080 c.Tape) (eval (TM1.step M) (TM1.init l)) ** congr with \u27e8\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "IsCountablySpanning.pi ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Finite \u03b9 inst\u271d : Finite \u03b9' C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), IsCountablySpanning (C i) \u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C) ** choose s h1s h2s using hC ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Finite \u03b9 inst\u271d : Finite \u03b9' C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) s : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i h2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ \u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C) ** cases nonempty_encodable (\u03b9 \u2192 \u2115) ** case intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Finite \u03b9 inst\u271d : Finite \u03b9' C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) s : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i h2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ val\u271d : Encodable (\u03b9 \u2192 \u2115) \u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C) ** let e : \u2115 \u2192 \u03b9 \u2192 \u2115 := fun n => (@decode (\u03b9 \u2192 \u2115) _ n).iget ** case intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Finite \u03b9 inst\u271d : Finite \u03b9' C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) s : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i h2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ val\u271d : Encodable (\u03b9 \u2192 \u2115) e : \u2115 \u2192 \u03b9 \u2192 \u2115 := fun n => Option.iget (decode n) \u22a2 IsCountablySpanning (Set.pi univ '' Set.pi univ C) ** refine' \u27e8fun n => Set.pi univ fun i => s i (e n i), fun n =>\n  mem_image_of_mem _ fun i _ => h1s i _, _\u27e9 ** case intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Finite \u03b9 inst\u271d : Finite \u03b9' C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) s : (i : \u03b9) \u2192 \u2115 \u2192 Set (\u03b1 i) h1s : \u2200 (i : \u03b9) (n : \u2115), s i n \u2208 C i h2s : \u2200 (i : \u03b9), \u22c3 n, s i n = univ val\u271d : Encodable (\u03b9 \u2192 \u2115) e : \u2115 \u2192 \u03b9 \u2192 \u2115 := fun n => Option.iget (decode n) \u22a2 \u22c3 n, (fun n => Set.pi univ fun i => s i (e n i)) n = univ ** simp_rw [(surjective_decode_iget (\u03b9 \u2192 \u2115)).iUnion_comp fun x => Set.pi univ fun i => s i (x i),\n  iUnion_univ_pi s, h2s, pi_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_const_of_cdf ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E a b c\u271d d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : IsFiniteMeasure \u03bc c : E \u22a2 \u222b (x : \u211d) in a..b, c \u2202\u03bc = (ENNReal.toReal (\u2191\u2191\u03bc (Iic b)) - ENNReal.toReal (\u2191\u2191\u03bc (Iic a))) \u2022 c ** simp only [sub_smul, \u2190 set_integral_const] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E a b c\u271d d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d inst\u271d : IsFiniteMeasure \u03bc c : E \u22a2 \u222b (x : \u211d) in a..b, c \u2202\u03bc = \u222b (x : \u211d) in Iic b, c \u2202\u03bc - \u222b (x : \u211d) in Iic a, c \u2202\u03bc ** refine' (integral_Iic_sub_Iic _ _).symm <;>\n  simp only [integrableOn_const, measure_lt_top, or_true_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_set_integral_of_monotone ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s hfi : IntegrableOn f (\u22c3 n, s n) \u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, s n, f a \u2202\u03bc)) ** have hfi' : (\u222b\u207b x in \u22c3 n, s n, \u2016f x\u2016\u208a \u2202\u03bc) < \u221e := hfi.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s hfi : IntegrableOn f (\u22c3 n, s n) hfi' : \u222b\u207b (x : \u03b1) in \u22c3 n, s n, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4 \u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, s n, f a \u2202\u03bc)) ** set S := \u22c3 i, s i ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hfi' : \u222b\u207b (x : \u03b1) in S, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4 \u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc)) ** have hSm : MeasurableSet S := MeasurableSet.iUnion hsm ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hfi' : \u222b\u207b (x : \u03b1) in S, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4 hSm : MeasurableSet S \u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc)) ** have hsub : \u2200 {i}, s i \u2286 S := @(subset_iUnion s) ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hfi' : \u222b\u207b (x : \u03b1) in S, \u2191\u2016f x\u2016\u208a \u2202\u03bc < \u22a4 hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc)) ** rw [\u2190 withDensity_apply _ hSm] at hfi' ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hfi' : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a) S < \u22a4 hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc)) ** set \u03bd := \u03bc.withDensity fun x => \u2016f x\u2016\u208a with h\u03bd ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a hfi' : \u2191\u2191\u03bd S < \u22a4 h\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a \u22a2 Tendsto (fun i => \u222b (a : \u03b1) in s i, f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in S, f a \u2202\u03bc)) ** refine' Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun \u03b5 \u03b50 => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a hfi' : \u2191\u2191\u03bd S < \u22a4 h\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a \u03b5 : \u211d \u03b50 : 0 < \u03b5 \u22a2 \u2200\u1da0 (x : \u03b9) in atTop, \u222b (a : \u03b1) in s x, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u03b5 ** lift \u03b5 to \u211d\u22650 using \u03b50.le ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a hfi' : \u2191\u2191\u03bd S < \u22a4 h\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 \u22a2 \u2200\u1da0 (x : \u03b9) in atTop, \u222b (a : \u03b1) in s x, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u2191\u03b5 ** have : \u2200\u1da0 i in atTop, \u03bd (s i) \u2208 Icc (\u03bd S - \u03b5) (\u03bd S + \u03b5) :=\n  tendsto_measure_iUnion h_mono (ENNReal.Icc_mem_nhds hfi'.ne (ENNReal.coe_pos.2 \u03b50).ne') ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a hfi' : \u2191\u2191\u03bd S < \u22a4 h\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 this : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) \u22a2 \u2200\u1da0 (x : \u03b9) in atTop, \u222b (a : \u03b1) in s x, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u2191\u03b5 ** refine' this.mono fun i hi => _ ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a hfi' : \u2191\u2191\u03bd S < \u22a4 h\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 this : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) i : \u03b9 hi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) \u22a2 \u222b (a : \u03b1) in s i, f a \u2202\u03bc \u2208 Metric.closedBall (\u222b (a : \u03b1) in S, f a \u2202\u03bc) \u2191\u03b5 ** rw [mem_closedBall_iff_norm', \u2190 integral_diff (hsm i) hfi hsub, \u2190 coe_nnnorm, NNReal.coe_le_coe, \u2190\n  ENNReal.coe_le_coe] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a hfi' : \u2191\u2191\u03bd S < \u22a4 h\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 this : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) i : \u03b9 hi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) \u22a2 \u2191\u2016\u222b (x : \u03b1) in S \\ s i, f x \u2202\u03bc\u2016\u208a \u2264 \u2191\u03b5 ** refine' (ennnorm_integral_le_lintegral_ennnorm _).trans _ ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a hfi' : \u2191\u2191\u03bd S < \u22a4 h\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 this : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) i : \u03b9 hi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) \u22a2 \u222b\u207b (a : \u03b1) in S \\ s i, \u2191\u2016f a\u2016\u208a \u2202\u03bc \u2264 \u2191\u03b5 ** rw [\u2190 withDensity_apply _ (hSm.diff (hsm _)), \u2190 h\u03bd, measure_diff hsub (hsm _)] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a hfi' : \u2191\u2191\u03bd S < \u22a4 h\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 this : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) i : \u03b9 hi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) \u22a2 \u2191\u2191\u03bd S - \u2191\u2191\u03bd (s i) \u2264 \u2191\u03b5  case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b2 : NormedSpace \u211d E \u03b9 : Type u_5 inst\u271d\u00b9 : Countable \u03b9 inst\u271d : SemilatticeSup \u03b9 s : \u03b9 \u2192 Set \u03b1 hsm : \u2200 (i : \u03b9), MeasurableSet (s i) h_mono : Monotone s S : Set \u03b1 := \u22c3 i, s i hfi : IntegrableOn f S hSm : MeasurableSet S hsub : \u2200 {i : \u03b9}, s i \u2286 S \u03bd : Measure \u03b1 := Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a hfi' : \u2191\u2191\u03bd S < \u22a4 h\u03bd : \u03bd = Measure.withDensity \u03bc fun x => \u2191\u2016f x\u2016\u208a \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 this : \u2200\u1da0 (i : \u03b9) in atTop, \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) i : \u03b9 hi : \u2191\u2191\u03bd (s i) \u2208 Icc (\u2191\u2191\u03bd S - \u2191\u03b5) (\u2191\u2191\u03bd S + \u2191\u03b5) \u22a2 \u2191\u2191\u03bd (s i) \u2260 \u22a4 ** exacts [tsub_le_iff_tsub_le.mp hi.1,\n  (hi.2.trans_lt <| ENNReal.add_lt_top.2 \u27e8hfi', ENNReal.coe_lt_top\u27e9).ne] ** Qed",
        "informal": ""
    },
    {
        "formal": "QPF.Cofix.dest_corec ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : \u03b1 \u2192 F \u03b1 x : \u03b1 \u22a2 dest (corec g x) = corec g <$> g x ** conv =>\n  lhs\n  rw [Cofix.dest, Cofix.corec]; ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : \u03b1 \u2192 F \u03b1 x : \u03b1 \u22a2 Quot.lift (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x))       (_ :         \u2200 (x y : PFunctor.M (P F)),           Mcongr x y \u2192             (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) x =               (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) y)       (Quot.mk Mcongr (corecF g x)) =     corec g <$> g x ** dsimp ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : \u03b1 \u2192 F \u03b1 x : \u03b1 \u22a2 Quot.mk Mcongr <$> abs (PFunctor.M.dest (corecF g x)) = corec g <$> g x ** rw [corecF_eq, abs_map, abs_repr, \u2190 comp_map] ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : \u03b1 \u2192 F \u03b1 x : \u03b1 \u22a2 (Quot.mk Mcongr \u2218 corecF g) <$> g x = corec g <$> g x ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.ofFunction_caratheodory ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 h\u2080 : m \u2205 = 0 hs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t \u22a2 MeasurableSet s ** apply (isCaratheodory_iff_le _).mpr ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 h\u2080 : m \u2205 = 0 hs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t \u22a2 \u2200 (t : Set \u03b1),     \u2191(OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + \u2191(OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264       \u2191(OuterMeasure.ofFunction m h\u2080) t ** refine' fun t => le_iInf fun f => le_iInf fun hf => _ ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 h\u2080 : m \u2205 = 0 hs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t t : Set \u03b1 f : \u2115 \u2192 Set \u03b1 hf : t \u2286 \u22c3 i, f i \u22a2 \u2191(OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + \u2191(OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264 \u2211' (i : \u2115), m (f i) ** refine'\n  le_trans\n    (add_le_add ((iInf_le_of_le fun i => f i \u2229 s) <| iInf_le _ _)\n      ((iInf_le_of_le fun i => f i \\ s) <| iInf_le _ _))\n    _ ** case refine'_1 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 h\u2080 : m \u2205 = 0 hs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t t : Set \u03b1 f : \u2115 \u2192 Set \u03b1 hf : t \u2286 \u22c3 i, f i \u22a2 t \u2229 s \u2286 \u22c3 i, (fun i => f i \u2229 s) i ** rw [\u2190 iUnion_inter] ** case refine'_1 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 h\u2080 : m \u2205 = 0 hs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t t : Set \u03b1 f : \u2115 \u2192 Set \u03b1 hf : t \u2286 \u22c3 i, f i \u22a2 t \u2229 s \u2286 (\u22c3 i, f i) \u2229 s ** exact inter_subset_inter_left _ hf ** case refine'_2 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 h\u2080 : m \u2205 = 0 hs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t t : Set \u03b1 f : \u2115 \u2192 Set \u03b1 hf : t \u2286 \u22c3 i, f i \u22a2 t \\ s \u2286 \u22c3 i, (fun i => f i \\ s) i ** rw [\u2190 iUnion_diff] ** case refine'_2 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 h\u2080 : m \u2205 = 0 hs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t t : Set \u03b1 f : \u2115 \u2192 Set \u03b1 hf : t \u2286 \u22c3 i, f i \u22a2 t \\ s \u2286 (\u22c3 i, f i) \\ s ** exact diff_subset_diff_left hf ** case refine'_3 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 h\u2080 : m \u2205 = 0 hs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t t : Set \u03b1 f : \u2115 \u2192 Set \u03b1 hf : t \u2286 \u22c3 i, f i \u22a2 \u2211' (i : \u2115), m ((fun i => f i \u2229 s) i) + \u2211' (i : \u2115), m ((fun i => f i \\ s) i) \u2264 \u2211' (i : \u2115), m (f i) ** rw [\u2190 ENNReal.tsum_add] ** case refine'_3 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 h\u2080 : m \u2205 = 0 hs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t t : Set \u03b1 f : \u2115 \u2192 Set \u03b1 hf : t \u2286 \u22c3 i, f i \u22a2 \u2211' (a : \u2115), (m ((fun i => f i \u2229 s) a) + m ((fun i => f i \\ s) a)) \u2264 \u2211' (i : \u2115), m (f i) ** exact ENNReal.tsum_le_tsum fun i => hs _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.liftOn_toFinmap ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v \u03b3 : Type u_1 s : AList \u03b2 f : AList \u03b2 \u2192 \u03b3 H : \u2200 (a b : AList \u03b2), a.entries ~ b.entries \u2192 f a = f b \u22a2 liftOn \u27e6s\u27e7 f H = f s ** cases s ** case mk \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v \u03b3 : Type u_1 f : AList \u03b2 \u2192 \u03b3 H : \u2200 (a b : AList \u03b2), a.entries ~ b.entries \u2192 f a = f b entries\u271d : List (Sigma \u03b2) nodupKeys\u271d : NodupKeys entries\u271d \u22a2 liftOn \u27e6{ entries := entries\u271d, nodupKeys := nodupKeys\u271d }\u27e7 f H = f { entries := entries\u271d, nodupKeys := nodupKeys\u271d } ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.mem_support_bindOnSupport_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 f : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 b : \u03b2 \u22a2 b \u2208 support (bindOnSupport p f) \u2194 \u2203 a h, b \u2208 support (f a h) ** simp only [support_bindOnSupport, Set.mem_setOf_eq, Set.mem_iUnion] ** Qed",
        "informal": ""
    },
    {
        "formal": "WithBot.image_coe_Iic ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Iic a = Ioc \u22a5 \u2191a ** rw [\u2190 preimage_coe_Iic, image_preimage_eq_inter_range, range_coe, inter_comm, Ioi_inter_Iic] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.gcd_eq_right ** i j : \u2124 H : j \u2223 i \u22a2 gcd i j = natAbs j ** rw [gcd_comm, gcd_eq_left H] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hausdorffMeasure_pi_real ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 \u22a2 \u03bcH[\u2191(Fintype.card \u03b9)] = volume ** refine'\n  (pi_eq_generateFrom (fun _ => Real.borel_eq_generateFrom_Ioo_rat.symm)\n      (fun _ => Real.isPiSystem_Ioo_rat) (fun _ => Real.finiteSpanningSetsInIooRat _) _).symm ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 \u22a2 \u2200 (s : \u03b9 \u2192 Set \u211d),     (\u2200 (i : \u03b9), s i \u2208 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b}) \u2192       \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i) ** simp only [mem_iUnion, mem_singleton_iff] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 \u22a2 \u2200 (s : \u03b9 \u2192 Set \u211d),     (\u2200 (i : \u03b9), \u2203 i_1 i_2 h, s i = Ioo \u2191i_1 \u2191i_2) \u2192 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i) ** intro s hs ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 s : \u03b9 \u2192 Set \u211d hs : \u2200 (i : \u03b9), \u2203 i_1 i_2 h, s i = Ioo \u2191i_1 \u2191i_2 \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i) ** choose a b H using hs ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 s : \u03b9 \u2192 Set \u211d a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), \u2203 h, s i = Ioo \u2191(a i) \u2191(b i) \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i) ** obtain rfl : s = fun i => Ioo (\u03b1 := \u211d) (a i) (b i) ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 s : \u03b9 \u2192 Set \u211d a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), \u2203 h, s i = Ioo \u2191(a i) \u2191(b i) \u22a2 s = fun i => Ioo \u2191(a i) \u2191(b i)  \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), \u2203 h, (fun i => Ioo \u2191(a i) \u2191(b i)) i = Ioo \u2191(a i) \u2191(b i) \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) ** exact funext fun i => (H i).2 ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), \u2203 h, (fun i => Ioo \u2191(a i) \u2191(b i)) i = Ioo \u2191(a i) \u2191(b i) \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) ** replace H := fun i => (H i).1 ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) ** apply le_antisymm _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) ** have I : \u2200 i, 0 \u2264 (b i : \u211d) - a i := fun i => by\n  simpa only [sub_nonneg, Rat.cast_le] using (H i).le ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) ** let \u03b3 := fun n : \u2115 => \u2200 i : \u03b9, Fin \u2308((b i : \u211d) - a i) * n\u2309\u208a ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) ** let t : \u2200 n : \u2115, \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) := fun n f =>\n  Set.pi univ fun i => Icc (a i + f i / n) (a i + (f i + 1) / n) ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) ** have A : Tendsto (fun n : \u2115 => 1 / (n : \u211d\u22650\u221e)) atTop (\ud835\udcdd 0) := by\n  simp only [one_div, ENNReal.tendsto_inv_nat_nhds_zero] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) \u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) ** have B : \u2200\u1da0 n in atTop, \u2200 i : \u03b3 n, diam (t n i) \u2264 1 / n := by\n  refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n  intro f\n  refine' diam_pi_le_of_le fun b => _\n  simp only [Real.ediam_Icc, add_div, ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), le_refl,\n    add_sub_add_left_eq_sub, add_sub_cancel', ENNReal.ofReal_one, ENNReal.ofReal_coe_nat] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i \u22a2 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) ** have Hle : volume \u2264 (\u03bcH[Fintype.card \u03b9] : Measure (\u03b9 \u2192 \u211d)) := by\n  refine' le_hausdorffMeasure _ _ \u221e ENNReal.coe_lt_top fun s _ => _\n  rw [ENNReal.rpow_nat_cast]\n  exact Real.volume_pi_le_diam_pow s ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i Hle : volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)] \u22a2 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) ** rw [\u2190 volume_pi_pi fun i => Ioo (a i : \u211d) (b i)] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i Hle : volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)] \u22a2 \u2191\u2191volume (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) ** exact Measure.le_iff'.1 Hle _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i \u22a2 volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)] ** refine' le_hausdorffMeasure _ _ \u221e ENNReal.coe_lt_top fun s _ => _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i s : Set (\u03b9 \u2192 \u211d) x\u271d : diam s \u2264 \u22a4 \u22a2 \u2191\u2191volume s \u2264 diam s ^ \u2191(Fintype.card \u03b9) ** rw [ENNReal.rpow_nat_cast] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i s : Set (\u03b9 \u2192 \u211d) x\u271d : diam s \u2264 \u22a4 \u22a2 \u2191\u2191volume s \u2264 diam s ^ Fintype.card \u03b9 ** exact Real.volume_pi_le_diam_pow s ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i i : \u03b9 \u22a2 0 \u2264 \u2191(b i) - \u2191(a i) ** simpa only [sub_nonneg, Rat.cast_le] using (H i).le ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) \u22a2 Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) ** simp only [one_div, ENNReal.tendsto_inv_nat_nhds_zero] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) \u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n ** refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9 ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) n : \u2115 hn : n \u2265 1 \u22a2 \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n ** intro f ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) n : \u2115 hn : n \u2265 1 f : \u03b3 n \u22a2 diam (t n f) \u2264 1 / \u2191n ** refine' diam_pi_le_of_le fun b => _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b\u271d : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b\u271d i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b\u271d i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b\u271d i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) n : \u2115 hn : n \u2265 1 f : \u03b3 n b : \u03b9 \u22a2 diam (Icc (\u2191(a b) + \u2191\u2191(f b) / \u2191n) (\u2191(a b) + (\u2191\u2191(f b) + 1) / \u2191n)) \u2264 1 / \u2191n ** simp only [Real.ediam_Icc, add_div, ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), le_refl,\n  add_sub_add_left_eq_sub, add_sub_cancel', ENNReal.ofReal_one, ENNReal.ofReal_coe_nat] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n \u22a2 \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i ** refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9 ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 \u22a2 (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i ** have npos : (0 : \u211d) < n := Nat.cast_pos.2 hn ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n \u22a2 (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i ** intro x hx ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : x \u2208 Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i) \u22a2 x \u2208 \u22c3 i, t n i ** simp only [mem_Ioo, mem_univ_pi] at hx ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) \u22a2 x \u2208 \u22c3 i, t n i ** simp only [mem_iUnion, mem_Ioo, mem_univ_pi] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) f : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) } \u22a2 \u2203 i, \u2200 (i_1 : \u03b9), x i_1 \u2208 Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191n) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191n) ** refine' \u27e8f, fun i => \u27e8_, _\u27e9\u27e9 ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) i : \u03b9 \u22a2 \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a ** apply Nat.floor_lt_ceil_of_lt_of_pos ** case h \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) i : \u03b9 \u22a2 (x i - \u2191(a i)) * \u2191n < (\u2191(b i) - \u2191(a i)) * \u2191n ** refine' (mul_lt_mul_right npos).2 _ ** case h \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) i : \u03b9 \u22a2 x i - \u2191(a i) < \u2191(b i) - \u2191(a i) ** simp only [(hx i).right, sub_lt_sub_iff_right] ** case h' \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) i : \u03b9 \u22a2 0 < (\u2191(b i) - \u2191(a i)) * \u2191n ** refine' mul_pos _ npos ** case h' \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) i : \u03b9 \u22a2 0 < \u2191(b i) - \u2191(a i) ** simpa only [Rat.cast_lt, sub_pos] using H i ** case refine'_1 \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) f : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) } i : \u03b9 \u22a2 \u2191(a i) + \u2191\u2191(f i) / \u2191n \u2264 x i ** calc\n  (a i : \u211d) + \u230a(x i - a i) * n\u230b\u208a / n \u2264 (a i : \u211d) + (x i - a i) * n / n := by\n    refine' add_le_add le_rfl ((div_le_div_right npos).2 _)\n    exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le)\n  _ = x i := by field_simp [npos.ne'] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) f : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) } i : \u03b9 \u22a2 \u2191(a i) + \u2191\u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a / \u2191n \u2264 \u2191(a i) + (x i - \u2191(a i)) * \u2191n / \u2191n ** refine' add_le_add le_rfl ((div_le_div_right npos).2 _) ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) f : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) } i : \u03b9 \u22a2 \u2191\u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a \u2264 (x i - \u2191(a i)) * \u2191n ** exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le) ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) f : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) } i : \u03b9 \u22a2 \u2191(a i) + (x i - \u2191(a i)) * \u2191n / \u2191n = x i ** field_simp [npos.ne'] ** case refine'_2 \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) f : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) } i : \u03b9 \u22a2 x i \u2264 \u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n ** calc\n  x i = (a i : \u211d) + (x i - a i) * n / n := by field_simp [npos.ne']\n  _ \u2264 (a i : \u211d) + (\u230a(x i - a i) * n\u230b\u208a + 1) / n :=\n    add_le_add le_rfl ((div_le_div_right npos).2 (Nat.lt_floor_add_one _).le) ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n n : \u2115 hn : n \u2265 1 npos : 0 < \u2191n x : \u03b9 \u2192 \u211d hx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i) f : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) } i : \u03b9 \u22a2 x i = \u2191(a i) + (x i - \u2191(a i)) * \u2191n / \u2191n ** field_simp [npos.ne'] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i \u22a2 liminf (fun n => \u2211 i : \u03b3 n, diam (t n i) ^ \u2191(Fintype.card \u03b9)) atTop \u2264     liminf (fun n => \u2211 i : \u03b3 n, (1 / \u2191n) ^ Fintype.card \u03b9) atTop ** refine' liminf_le_liminf _ _ ** case refine'_1 \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i \u22a2 \u2200\u1da0 (a_1 : \u2115) in atTop, \u2211 i : \u03b3 a_1, diam (t a_1 i) ^ \u2191(Fintype.card \u03b9) \u2264 \u2211 i : \u03b3 a_1, (1 / \u2191a_1) ^ Fintype.card \u03b9 ** filter_upwards [B] with _ hn ** case h \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i a\u271d : \u2115 hn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d \u22a2 \u2211 i : \u03b3 a\u271d, diam (t a\u271d i) ^ \u2191(Fintype.card \u03b9) \u2264 \u2211 i : \u03b3 a\u271d, (1 / \u2191a\u271d) ^ Fintype.card \u03b9 ** apply Finset.sum_le_sum fun i _ => _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i a\u271d : \u2115 hn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d \u22a2 \u2200 (i : \u03b3 a\u271d), i \u2208 Finset.univ \u2192 diam (t a\u271d i) ^ \u2191(Fintype.card \u03b9) \u2264 (1 / \u2191a\u271d) ^ Fintype.card \u03b9 ** simp only [ENNReal.rpow_nat_cast] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i a\u271d : \u2115 hn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d \u22a2 \u2200 (i : \u03b3 a\u271d),     i \u2208 Finset.univ \u2192       diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^           Fintype.card \u03b9 \u2264         (1 / \u2191a\u271d) ^ Fintype.card \u03b9 ** intros i _ ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i a\u271d : \u2115 hn : \u2200 (i : \u03b3 a\u271d), diam (t a\u271d i) \u2264 1 / \u2191a\u271d i : \u03b3 a\u271d x\u271d : i \u2208 Finset.univ \u22a2 diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^ Fintype.card \u03b9 \u2264     (1 / \u2191a\u271d) ^ Fintype.card \u03b9 ** exact pow_le_pow_of_le_left' (hn i) _ ** case refine'_2 \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i \u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => \u2211 i : \u03b3 n, diam (t n i) ^ \u2191(Fintype.card \u03b9) ** isBoundedDefault ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i \u22a2 liminf (fun n => \u2211 i : \u03b3 n, (1 / \u2191n) ^ Fintype.card \u03b9) atTop =     liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop ** simp only [Finset.card_univ, Nat.cast_prod, one_mul, Fintype.card_fin, Finset.sum_const,\n  nsmul_eq_mul, Fintype.card_pi, div_eq_mul_inv, Finset.prod_mul_distrib, Finset.prod_const] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i \u22a2 liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop = \u220f i : \u03b9, \u2191\u2191volume (Ioo \u2191(a i) \u2191(b i)) ** simp only [Real.volume_Ioo] ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i \u22a2 liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop = \u220f x : \u03b9, ENNReal.ofReal (\u2191(b x) - \u2191(a x)) ** apply Tendsto.liminf_eq ** case h \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i \u22a2 Tendsto (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop (\ud835\udcdd (\u220f x : \u03b9, ENNReal.ofReal (\u2191(b x) - \u2191(a x)))) ** refine' ENNReal.tendsto_finset_prod_of_ne_top _ (fun i _ => _) fun i _ => _ ** case h.refine'_1 \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i i : \u03b9 x\u271d : i \u2208 Finset.univ \u22a2 Tendsto (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop (\ud835\udcdd (ENNReal.ofReal (\u2191(b i) - \u2191(a i)))) ** apply\n  Tendsto.congr' _\n    ((ENNReal.continuous_ofReal.tendsto _).comp\n      ((tendsto_nat_ceil_mul_div_atTop (I i)).comp tendsto_nat_cast_atTop_atTop)) ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i i : \u03b9 x\u271d : i \u2208 Finset.univ \u22a2 ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast =\u1da0[atTop] fun n =>     \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n ** apply eventually_atTop.2 \u27e81, fun n hn => _\u27e9 ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i i : \u03b9 x\u271d : i \u2208 Finset.univ \u22a2 \u2200 (n : \u2115),     n \u2265 1 \u2192       (ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast) n =         (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) n ** intros n hn ** \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i i : \u03b9 x\u271d : i \u2208 Finset.univ n : \u2115 hn : n \u2265 1 \u22a2 (ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast) n =     (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) n ** simp only [ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), comp_apply,\n  ENNReal.ofReal_coe_nat] ** case h.refine'_2 \u03b9\u271d : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2076 : EMetricSpace X inst\u271d\u2075 : EMetricSpace Y inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : BorelSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : BorelSpace Y \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 a b : \u03b9 \u2192 \u211a H : \u2200 (i : \u03b9), a i < b i I : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i) \u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a t : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=   fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n) A : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0) B : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n C : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 i, t n i i : \u03b9 x\u271d : i \u2208 Finset.univ \u22a2 ENNReal.ofReal (\u2191(b i) - \u2191(a i)) \u2260 \u22a4 ** simp only [ENNReal.ofReal_ne_top, Ne.def, not_false_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_map ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g\u271d : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : AEMeasurable \u03c6 f : \u03b2 \u2192 G hfm : AEStronglyMeasurable f (Measure.map \u03c6 \u03bc) g : \u03b2 \u2192 G := AEStronglyMeasurable.mk f hfm \u22a2 \u222b (y : \u03b2), g y \u2202Measure.map \u03c6 \u03bc = \u222b (y : \u03b2), g y \u2202Measure.map (AEMeasurable.mk \u03c6 h\u03c6) \u03bc ** congr 1 ** case e_\u03bc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g\u271d : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : MeasurableSpace \u03b2 \u03c6 : \u03b1 \u2192 \u03b2 h\u03c6 : AEMeasurable \u03c6 f : \u03b2 \u2192 G hfm : AEStronglyMeasurable f (Measure.map \u03c6 \u03bc) g : \u03b2 \u2192 G := AEStronglyMeasurable.mk f hfm \u22a2 Measure.map \u03c6 \u03bc = Measure.map (AEMeasurable.mk \u03c6 h\u03c6) \u03bc ** exact Measure.map_congr h\u03c6.ae_eq_mk ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hausdorffMeasure_lineMap_image ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u00b9 : EMetricSpace X inst\u271d\u00b9\u2070 : EMetricSpace Y inst\u271d\u2079 : MeasurableSpace X inst\u271d\u2078 : BorelSpace X inst\u271d\u2077 : MeasurableSpace Y inst\u271d\u2076 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 P : Type u_6 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace P inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor E P inst\u271d : BorelSpace P x y : P s : Set \u211d \u22a2 \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s ** borelize E ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u00b9 : EMetricSpace X inst\u271d\u00b9\u2070 : EMetricSpace Y inst\u271d\u2079 : MeasurableSpace X inst\u271d\u2078 : BorelSpace X inst\u271d\u2077 : MeasurableSpace Y inst\u271d\u2076 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 P : Type u_6 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace P inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor E P inst\u271d : BorelSpace P x y : P s : Set \u211d this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E \u22a2 \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s ** rw [IsometryEquiv.hausdorffMeasure_image, hausdorffMeasure_smul_right_image,\n  nndist_eq_nnnorm_vsub' E] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u00b9 : EMetricSpace X inst\u271d\u00b9\u2070 : EMetricSpace Y inst\u271d\u2079 : MeasurableSpace X inst\u271d\u2078 : BorelSpace X inst\u271d\u2077 : MeasurableSpace Y inst\u271d\u2076 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 P : Type u_6 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace P inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor E P inst\u271d : BorelSpace P x y : P s : Set \u211d this : \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s \u22a2 \u2191\u2191\u03bcH[1] (\u2191(AffineMap.lineMap x y) '' s) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s ** simpa only [Set.image_image] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.TProd.elim_mk ** \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i\u271d j\u271d : \u03b9 l : List \u03b9 f\u271d : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 i : \u03b9 is : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i j : \u03b9 hj : j \u2208 i :: is \u22a2 TProd.elim (TProd.mk (i :: is) f) hj = f j ** by_cases hji : j = i ** case pos \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i\u271d j\u271d : \u03b9 l : List \u03b9 f\u271d : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 i : \u03b9 is : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i j : \u03b9 hj : j \u2208 i :: is hji : j = i \u22a2 TProd.elim (TProd.mk (i :: is) f) hj = f j ** subst hji ** case pos \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i j\u271d : \u03b9 l : List \u03b9 f\u271d : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 is : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i j : \u03b9 hj : j \u2208 j :: is \u22a2 TProd.elim (TProd.mk (j :: is) f) hj = f j ** simp ** case neg \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i\u271d j\u271d : \u03b9 l : List \u03b9 f\u271d : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 i : \u03b9 is : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i j : \u03b9 hj : j \u2208 i :: is hji : \u00acj = i \u22a2 TProd.elim (TProd.mk (i :: is) f) hj = f j ** rw [TProd.elim_of_ne _ hji, snd_mk, elim_mk is] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.atEnd_of_valid ** cs cs' : List Char \u22a2 atEnd { data := cs ++ cs' } { byteIdx := utf8Len cs } = true \u2194 cs' = [] ** rw [atEnd_iff] ** cs cs' : List Char \u22a2 endPos { data := cs ++ cs' } \u2264 { byteIdx := utf8Len cs } \u2194 cs' = [] ** cases cs' <;> simp [Nat.lt_add_of_pos_right add_csize_pos] ** Qed",
        "informal": ""
    },
    {
        "formal": "LazyList.append_nil ** \u03b1 : Type u_1 xs : LazyList \u03b1 \u22a2 append xs (Thunk.pure nil) = xs ** induction' xs using LazyList.rec with _ _ _ _ ih ** case nil \u03b1 : Type u_1 \u22a2 append nil (Thunk.pure nil) = nil ** rfl ** case cons \u03b1 : Type u_1 xs : LazyList \u03b1 hd\u271d : \u03b1 tl\u271d : Thunk (LazyList \u03b1) tl_ih\u271d : ?m.62213 tl\u271d \u22a2 append (cons hd\u271d tl\u271d) (Thunk.pure nil) = cons hd\u271d tl\u271d ** simpa only [append, cons.injEq, true_and] ** case mk \u03b1 : Type u_1 xs : LazyList \u03b1 fn\u271d : Unit \u2192 LazyList \u03b1 ih : \u2200 (a : Unit), append (fn\u271d a) (Thunk.pure nil) = fn\u271d a \u22a2 { fn := fun x => append (Thunk.get { fn := fn\u271d }) (Thunk.pure nil) } = { fn := fn\u271d } ** ext ** case mk.eq \u03b1 : Type u_1 xs : LazyList \u03b1 fn\u271d : Unit \u2192 LazyList \u03b1 ih : \u2200 (a : Unit), append (fn\u271d a) (Thunk.pure nil) = fn\u271d a \u22a2 Thunk.get { fn := fun x => append (Thunk.get { fn := fn\u271d }) (Thunk.pure nil) } = Thunk.get { fn := fn\u271d } ** apply ih ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul ** \u03b1 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =     ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1)) ** rw [lintegral_rpow_eq_lintegral_meas_le_mul \u03bc f_nn f_mble p_pos] ** \u03b1 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p \u22a2 ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) =     ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1)) ** apply congr_arg fun z => ENNReal.ofReal p * z ** \u03b1 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1)) ** apply lintegral_congr_ae ** case h \u03b1 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p \u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (a ^ (p - 1))) =\u1da0[ae (Measure.restrict volume (Ioi 0))] fun a =>     \u2191\u2191\u03bc {a_1 | a < f a_1} * ENNReal.ofReal (a ^ (p - 1)) ** filter_upwards [meas_le_ae_eq_meas_lt \u03bc (volume.restrict (Ioi 0)) f]\n  with t ht ** case h \u03b1 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSingletonClass \u03b2 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p t : \u211d ht : \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a} \u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) = \u2191\u2191\u03bc {a | t < f a} * ENNReal.ofReal (t ^ (p - 1)) ** rw [ht] ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.lift_graph ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 f : \u03b1 \u2192 \u03b2 a : \u03b1 b : \u03b2 \u22a2 (\u2203 x, f a = b) \u2194 f a = b ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.integral_of_fun_eq_integral ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hf : Integrable f \u22a2 \u222b (a : \u03b1), \u2191\u2191(Integrable.toL1 f hf) a \u2202\u03bc = \u222b (a : \u03b1), f a \u2202\u03bc ** by_cases hG : CompleteSpace G ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hf : Integrable f hG : CompleteSpace G \u22a2 \u222b (a : \u03b1), \u2191\u2191(Integrable.toL1 f hf) a \u2202\u03bc = \u222b (a : \u03b1), f a \u2202\u03bc ** simp only [MeasureTheory.integral, hG, L1.integral] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hf : Integrable f hG : CompleteSpace G \u22a2 (if h : True then       if hf_1 : Integrable fun a => \u2191\u2191(Integrable.toL1 f hf) a then         \u2191integralCLM (Integrable.toL1 (fun a => \u2191\u2191(Integrable.toL1 f hf) a) hf_1)       else 0     else 0) =     if h : True then if hf : Integrable fun a => f a then \u2191integralCLM (Integrable.toL1 (fun a => f a) hf) else 0 else 0 ** exact setToFun_toL1 (dominatedFinMeasAdditive_weightedSMul \u03bc) hf ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 G hf : Integrable f hG : \u00acCompleteSpace G \u22a2 \u222b (a : \u03b1), \u2191\u2191(Integrable.toL1 f hf) a \u2202\u03bc = \u222b (a : \u03b1), f a \u2202\u03bc ** simp [MeasureTheory.integral, hG] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_add_measure ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f \u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd ** by_cases hG : CompleteSpace G ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : CompleteSpace G \u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd  case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : \u00acCompleteSpace G \u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd ** swap ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : CompleteSpace G \u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd ** have hfi := h\u03bc.add_measure h\u03bd ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : CompleteSpace G hfi : Integrable f \u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd ** simp_rw [integral_eq_setToFun] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : CompleteSpace G hfi : Integrable f \u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>       f a) =     (setToFun \u03bc (weightedSMul \u03bc) (_ : DominatedFinMeasAdditive \u03bc (weightedSMul \u03bc) 1) fun a => f a) +       setToFun \u03bd (weightedSMul \u03bd) (_ : DominatedFinMeasAdditive \u03bd (weightedSMul \u03bd) 1) fun a => f a ** have h\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc : Set \u03b1 \u2192 G \u2192L[\u211d] G) 1 :=\n  DominatedFinMeasAdditive.add_measure_right \u03bc \u03bd (dominatedFinMeasAdditive_weightedSMul \u03bc)\n    zero_le_one ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : CompleteSpace G hfi : Integrable f h\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1 \u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>       f a) =     (setToFun \u03bc (weightedSMul \u03bc) (_ : DominatedFinMeasAdditive \u03bc (weightedSMul \u03bc) 1) fun a => f a) +       setToFun \u03bd (weightedSMul \u03bd) (_ : DominatedFinMeasAdditive \u03bd (weightedSMul \u03bd) 1) fun a => f a ** have h\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd : Set \u03b1 \u2192 G \u2192L[\u211d] G) 1 :=\n  DominatedFinMeasAdditive.add_measure_left \u03bc \u03bd (dominatedFinMeasAdditive_weightedSMul \u03bd)\n    zero_le_one ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : CompleteSpace G hfi : Integrable f h\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1 h\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1 \u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>       f a) =     (setToFun \u03bc (weightedSMul \u03bc) (_ : DominatedFinMeasAdditive \u03bc (weightedSMul \u03bc) 1) fun a => f a) +       setToFun \u03bd (weightedSMul \u03bd) (_ : DominatedFinMeasAdditive \u03bd (weightedSMul \u03bd) 1) fun a => f a ** rw [\u2190 setToFun_congr_measure_of_add_right h\u03bc_dfma\n      (dominatedFinMeasAdditive_weightedSMul \u03bc) f hfi,\n  \u2190 setToFun_congr_measure_of_add_left h\u03bd_dfma (dominatedFinMeasAdditive_weightedSMul \u03bd) f hfi] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : CompleteSpace G hfi : Integrable f h\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1 h\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1 \u22a2 (setToFun (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) (_ : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul (\u03bc + \u03bd)) 1) fun a =>       f a) =     setToFun (\u03bc + \u03bd) (weightedSMul \u03bc) h\u03bc_dfma f + setToFun (\u03bc + \u03bd) (weightedSMul \u03bd) h\u03bd_dfma f ** refine' setToFun_add_left' _ _ _ (fun s _ h\u03bc\u03bds => _) f ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : CompleteSpace G hfi : Integrable f h\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1 h\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1 s : Set \u03b1 x\u271d : MeasurableSet s h\u03bc\u03bds : \u2191\u2191(\u03bc + \u03bd) s < \u22a4 \u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s ** rw [Measure.coe_add, Pi.add_apply, add_lt_top] at h\u03bc\u03bds ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : CompleteSpace G hfi : Integrable f h\u03bc_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bc) 1 h\u03bd_dfma : DominatedFinMeasAdditive (\u03bc + \u03bd) (weightedSMul \u03bd) 1 s : Set \u03b1 x\u271d : MeasurableSet s h\u03bc\u03bds : \u2191\u2191\u03bc s < \u22a4 \u2227 \u2191\u2191\u03bd s < \u22a4 \u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s ** rw [weightedSMul, weightedSMul, weightedSMul, \u2190 add_smul, Measure.coe_add, Pi.add_apply,\ntoReal_add h\u03bc\u03bds.1.ne h\u03bc\u03bds.2.ne] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 f : \u03b1 \u2192 G h\u03bc : Integrable f h\u03bd : Integrable f hG : \u00acCompleteSpace G \u22a2 \u222b (x : \u03b1), f x \u2202(\u03bc + \u03bd) = \u222b (x : \u03b1), f x \u2202\u03bc + \u222b (x : \u03b1), f x \u2202\u03bd ** simp [integral, hG] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.JordanDecomposition.toSignedMeasure_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j : JordanDecomposition \u03b1 \u22a2 toSignedMeasure 0 = 0 ** ext1 i hi ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j : JordanDecomposition \u03b1 i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(toSignedMeasure 0) i = \u21910 i ** rw [toSignedMeasure, toSignedMeasure_sub_apply hi, zero_posPart, zero_negPart, sub_self,\n  VectorMeasure.coe_zero, Pi.zero_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.extract.go\u2081_cons_addChar ** c : Char cs : List Char b e : Pos \u22a2 go\u2081 (c :: cs) 0 (b + c) (e + c) = go\u2081 cs 0 b e ** simp [go\u2081, Pos.ext_iff, Nat.ne_of_lt add_csize_pos] ** c : Char cs : List Char b e : Pos \u22a2 go\u2081 cs (0 + c) (b + c) (e + c) = go\u2081 cs 0 b e ** apply go\u2081_add_right_cancel ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.addBottom_modifyNth ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 f : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n : \u2115 \u22a2 ListBlank.modifyNth (fun a => (a.1, f a.2)) n (addBottom L) = addBottom (ListBlank.modifyNth f n L) ** cases n <;>\n  simp only [addBottom, ListBlank.head_cons, ListBlank.modifyNth, ListBlank.tail_cons] ** case succ K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 f : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n\u271d : \u2115 \u22a2 ListBlank.cons (true, ListBlank.head L)       (ListBlank.modifyNth (fun a => (a.1, f a.2)) n\u271d         (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }           (ListBlank.tail L))) =     ListBlank.cons (true, ListBlank.head L)       (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }         (ListBlank.modifyNth f n\u271d (ListBlank.tail L))) ** congr ** case succ.e_l K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 f : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n\u271d : \u2115 \u22a2 ListBlank.modifyNth (fun a => (a.1, f a.2)) n\u271d       (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) =     ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }       (ListBlank.modifyNth f n\u271d (ListBlank.tail L)) ** symm ** case succ.e_l K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 f : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n\u271d : \u2115 \u22a2 ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }       (ListBlank.modifyNth f n\u271d (ListBlank.tail L)) =     ListBlank.modifyNth (fun a => (a.1, f a.2)) n\u271d       (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) ** apply ListBlank.map_modifyNth ** case succ.e_l.H K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 f : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n\u271d : \u2115 \u22a2 \u2200 (x : (k : K) \u2192 Option (\u0393 k)),     PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (f x) =       ((PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x).1,         f (PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x).2) ** intro ** case succ.e_l.H K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 f : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k) L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n\u271d : \u2115 x\u271d : (k : K) \u2192 Option (\u0393 k) \u22a2 PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (f x\u271d) =     ((PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x\u271d).1,       f (PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x\u271d).2) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.ne_of_lt ** a b : Int h : a < b e : a = b \u22a2 False ** cases e ** case refl a : Int h : a < a \u22a2 False ** exact Int.lt_irrefl _ h ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.bit_val ** b : Bool n : \u2124 \u22a2 bit b n = 2 * n + bif b then 1 else 0 ** cases b ** case false n : \u2124 \u22a2 bit false n = 2 * n + bif false then 1 else 0  case true n : \u2124 \u22a2 bit true n = 2 * n + bif true then 1 else 0 ** apply (bit0_val n).trans (add_zero _).symm ** case true n : \u2124 \u22a2 bit true n = 2 * n + bif true then 1 else 0 ** apply bit1_val ** Qed",
        "informal": ""
    },
    {
        "formal": "List.toFinset_cons ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 l l' : List \u03b1 a : \u03b1 \u22a2 (toFinset (a :: l)).val = (insert a (toFinset l)).val ** by_cases h : a \u2208 l <;> simp [Finset.insert_val', Multiset.dedup_cons, h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hausdorffMeasure_affineSegment ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u00b9 : EMetricSpace X inst\u271d\u00b9\u2070 : EMetricSpace Y inst\u271d\u2079 : MeasurableSpace X inst\u271d\u2078 : BorelSpace X inst\u271d\u2077 : MeasurableSpace Y inst\u271d\u2076 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 P : Type u_6 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace P inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor E P inst\u271d : BorelSpace P x y : P \u22a2 \u2191\u2191\u03bcH[1] (affineSegment \u211d x y) = edist x y ** rw [affineSegment, hausdorffMeasure_lineMap_image, hausdorffMeasure_real, Real.volume_Icc,\n  sub_zero, ENNReal.ofReal_one, \u2190 Algebra.algebraMap_eq_smul_one] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9\u00b9 : EMetricSpace X inst\u271d\u00b9\u2070 : EMetricSpace Y inst\u271d\u2079 : MeasurableSpace X inst\u271d\u2078 : BorelSpace X inst\u271d\u2077 : MeasurableSpace Y inst\u271d\u2076 : BorelSpace Y \ud835\udd5c : Type u_4 E : Type u_5 P : Type u_6 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace P inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor E P inst\u271d : BorelSpace P x y : P \u22a2 \u2191(algebraMap \u211d\u22650 \u211d\u22650\u221e) (nndist x y) = edist x y ** exact (edist_nndist _ _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.set_integral_condCdf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d s : Set \u03b1 hs : MeasurableSet s \u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)) ** have h := set_lintegral_condCdf \u03c1 x hs ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d s : Set \u03b1 hs : MeasurableSet s h : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) \u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)) ** rw [\u2190 ofReal_integral_eq_lintegral_ofReal] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d s : Set \u03b1 hs : MeasurableSet s h : ENNReal.ofReal (\u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) \u22a2 \u222b (a : \u03b1) in s, \u2191(condCdf \u03c1 a) x \u2202Measure.fst \u03c1 = ENNReal.toReal (\u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x)) ** rw [\u2190 h, ENNReal.toReal_ofReal] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d s : Set \u03b1 hs : MeasurableSet s h : ENNReal.ofReal (\u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1) = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) \u22a2 0 \u2264 \u222b (x_1 : \u03b1) in s, \u2191(condCdf \u03c1 x_1) x \u2202Measure.fst \u03c1 ** exact integral_nonneg fun _ => condCdf_nonneg _ _ _ ** case hfi \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d s : Set \u03b1 hs : MeasurableSet s h : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) \u22a2 Integrable fun a => \u2191(condCdf \u03c1 a) x ** exact (integrable_condCdf _ _).integrableOn ** case f_nn \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d s : Set \u03b1 hs : MeasurableSet s h : \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) \u22a2 0 \u2264\u1d50[Measure.restrict (Measure.fst \u03c1) s] fun a => \u2191(condCdf \u03c1 a) x ** exact eventually_of_forall fun _ => condCdf_nonneg _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "List.Sublist.reverse ** \u03b1\u271d : Type u_1 l\u2081\u271d l\u2082 l\u2081 l\u2082\u271d : List \u03b1\u271d a\u271d : \u03b1\u271d h : l\u2081 <+ l\u2082\u271d \u22a2 List.reverse l\u2081 <+ List.reverse (a\u271d :: l\u2082\u271d) ** rw [reverse_cons] ** \u03b1\u271d : Type u_1 l\u2081\u271d l\u2082 l\u2081 l\u2082\u271d : List \u03b1\u271d a\u271d : \u03b1\u271d h : l\u2081 <+ l\u2082\u271d \u22a2 List.reverse l\u2081 <+ List.reverse l\u2082\u271d ++ [a\u271d] ** exact sublist_append_of_sublist_left h.reverse ** \u03b1\u271d : Type u_1 l\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List \u03b1\u271d a\u271d : \u03b1\u271d h : l\u2081\u271d <+ l\u2082\u271d \u22a2 List.reverse (a\u271d :: l\u2081\u271d) <+ List.reverse (a\u271d :: l\u2082\u271d) ** rw [reverse_cons, reverse_cons] ** \u03b1\u271d : Type u_1 l\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List \u03b1\u271d a\u271d : \u03b1\u271d h : l\u2081\u271d <+ l\u2082\u271d \u22a2 List.reverse l\u2081\u271d ++ [a\u271d] <+ List.reverse l\u2082\u271d ++ [a\u271d] ** exact h.reverse.append_right _ ** Qed",
        "informal": ""
    },
    {
        "formal": "integral_re_add_im ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e f : \u03b1 \u2192 \ud835\udd5c hf : Integrable f \u22a2 \u2191(\u222b (x : \u03b1), \u2191IsROrC.re (f x) \u2202\u03bc) + \u2191(\u222b (x : \u03b1), \u2191IsROrC.im (f x) \u2202\u03bc) * IsROrC.I = \u222b (x : \u03b1), f x \u2202\u03bc ** rw [\u2190 integral_ofReal, \u2190 integral_ofReal, integral_coe_re_add_coe_im hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "mem_generatePiSystem_iUnion_elim' ** \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) t : Set \u03b1 h_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : t \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) \u22a2 \u2203 T f, \u2191T \u2286 s \u2227 t = \u22c2 b \u2208 T, f b \u2227 \u2200 (b : \u03b2), b \u2208 T \u2192 f b \u2208 g b ** rcases @mem_generatePiSystem_iUnion_elim \u03b1 (Subtype s) (g \u2218 Subtype.val)\n    (fun b => h_pi b.val b.property) t this with\n  \u27e8T, \u27e8f, \u27e8rfl, h_t'\u27e9\u27e9\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) \u22a2 \u2203 T_1 f_1, \u2191T_1 \u2286 s \u2227 \u22c2 b \u2208 T, f b = \u22c2 b \u2208 T_1, f_1 b \u2227 \u2200 (b : \u03b2), b \u2208 T_1 \u2192 f_1 b \u2208 g b ** refine'\n  \u27e8T.image (fun x : s => (x : \u03b2)),\n    Function.extend (fun x : s => (x : \u03b2)) f fun _ : \u03b2 => (\u2205 : Set \u03b1), by simp, _, _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) t : Set \u03b1 h_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) \u22a2 t \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) ** suffices h1 : \u22c3 b : Subtype s, (g \u2218 Subtype.val) b = \u22c3 b \u2208 s, g b ** case h1 \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) t : Set \u03b1 h_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) \u22a2 \u22c3 b, (g \u2218 Subtype.val) b = \u22c3 b \u2208 s, g b ** ext x ** case h1.h \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) t : Set \u03b1 h_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) x : Set \u03b1 \u22a2 x \u2208 \u22c3 b, (g \u2218 Subtype.val) b \u2194 x \u2208 \u22c3 b \u2208 s, g b ** simp only [exists_prop, Set.mem_iUnion, Function.comp_apply, Subtype.exists, Subtype.coe_mk] ** case h1.h \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) t : Set \u03b1 h_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) x : Set \u03b1 \u22a2 (\u2203 a, s a \u2227 x \u2208 g a) \u2194 \u2203 i, i \u2208 s \u2227 x \u2208 g i ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) t : Set \u03b1 h_t : t \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) h1 : \u22c3 b, (g \u2218 Subtype.val) b = \u22c3 b \u2208 s, g b \u22a2 t \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) ** rwa [h1] ** \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) \u22a2 \u2191(Finset.image (fun x => \u2191x) T) \u2286 s ** simp ** case intro.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) \u22a2 \u22c2 b \u2208 T, f b = \u22c2 b \u2208 Finset.image (fun x => \u2191x) T, Function.extend (fun x => \u2191x) f (fun x => \u2205) b ** ext a ** case intro.intro.intro.refine'_1.h.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) a : \u03b1 \u22a2 a \u2208 \u22c2 b \u2208 Finset.image (fun x => \u2191x) T, Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2192 a \u2208 \u22c2 b \u2208 T, f b ** simp (config := { proj := false }) only\n  [Set.mem_iInter, Subtype.forall, Finset.set_biInter_finset_image] ** case intro.intro.intro.refine'_1.h.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) a : \u03b1 \u22a2 (\u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),       { val := a_1, property := b } \u2208 T \u2192         a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b }) \u2192     \u2200 (a_2 : \u03b2) (b : s a_2), { val := a_2, property := b } \u2208 T \u2192 a \u2208 f { val := a_2, property := b } ** intro h1 b h_b h_b_in_T ** case intro.intro.intro.refine'_1.h.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) a : \u03b1 h1 :   \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),     { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b } b : \u03b2 h_b : s b h_b_in_T : { val := b, property := h_b } \u2208 T \u22a2 a \u2208 f { val := b, property := h_b } ** have h2 := h1 b h_b h_b_in_T ** case intro.intro.intro.refine'_1.h.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) a : \u03b1 h1 :   \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),     { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b } b : \u03b2 h_b : s b h_b_in_T : { val := b, property := h_b } \u2208 T h2 : a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := b, property := h_b } \u22a2 a \u2208 f { val := b, property := h_b } ** revert h2 ** case intro.intro.intro.refine'_1.h.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) a : \u03b1 h1 :   \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),     { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b } b : \u03b2 h_b : s b h_b_in_T : { val := b, property := h_b } \u2208 T \u22a2 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := b, property := h_b } \u2192 a \u2208 f { val := b, property := h_b } ** rw [Subtype.val_injective.extend_apply] ** case intro.intro.intro.refine'_1.h.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) a : \u03b1 h1 :   \u2200 (a_1 : \u03b2) (b : a_1 \u2208 s),     { val := a_1, property := b } \u2208 T \u2192 a \u2208 Function.extend (fun x => \u2191x) f (fun x => \u2205) \u2191{ val := a_1, property := b } b : \u03b2 h_b : s b h_b_in_T : { val := b, property := h_b } \u2208 T \u22a2 a \u2208 f { val := b, property := h_b } \u2192 a \u2208 f { val := b, property := h_b } ** apply id ** case intro.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) \u22a2 \u2200 (b : \u03b2), b \u2208 Finset.image (fun x => \u2191x) T \u2192 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b ** intros b h_b ** case intro.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) b : \u03b2 h_b : b \u2208 Finset.image (fun x => \u2191x) T \u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b ** simp_rw [Finset.mem_image, Subtype.exists, exists_and_right, exists_eq_right]\n  at h_b ** case intro.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) b : \u03b2 h_b : \u2203 x, { val := b, property := (_ : b \u2208 s) } \u2208 T \u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b ** cases' h_b with h_b_w h_b_h ** case intro.intro.intro.refine'_2.intro \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) b : \u03b2 h_b_w : b \u2208 s h_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T \u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b ** have h_b_alt : b = (Subtype.mk b h_b_w).val := rfl ** case intro.intro.intro.refine'_2.intro \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) b : \u03b2 h_b_w : b \u2208 s h_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T h_b_alt : b = \u2191{ val := b, property := h_b_w } \u22a2 Function.extend (fun x => \u2191x) f (fun x => \u2205) b \u2208 g b ** rw [h_b_alt, Subtype.val_injective.extend_apply] ** case intro.intro.intro.refine'_2.intro \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) b : \u03b2 h_b_w : b \u2208 s h_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T h_b_alt : b = \u2191{ val := b, property := h_b_w } \u22a2 f { val := b, property := h_b_w } \u2208 g \u2191{ val := b, property := h_b_w } ** apply h_t' ** case intro.intro.intro.refine'_2.intro.a \u03b1 : Type u_1 \u03b2 : Type u_2 g : \u03b2 \u2192 Set (Set \u03b1) s : Set \u03b2 h_pi : \u2200 (b : \u03b2), b \u2208 s \u2192 IsPiSystem (g b) T : Finset (Subtype s) f : Subtype s \u2192 Set \u03b1 h_t' : \u2200 (b : Subtype s), b \u2208 T \u2192 f b \u2208 (g \u2218 Subtype.val) b h_t : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b \u2208 s, g b) this : \u22c2 b \u2208 T, f b \u2208 generatePiSystem (\u22c3 b, (g \u2218 Subtype.val) b) b : \u03b2 h_b_w : b \u2208 s h_b_h : { val := b, property := (_ : b \u2208 s) } \u2208 T h_b_alt : b = \u2191{ val := b, property := h_b_w } \u22a2 { val := b, property := h_b_w } \u2208 T ** apply h_b_h ** Qed",
        "informal": ""
    },
    {
        "formal": "StieltjesFunction.length_Ioc ** f : StieltjesFunction a b : \u211d \u22a2 length f (Ioc a b) = ofReal (\u2191f b - \u2191f a) ** refine'\n  le_antisymm (iInf_le_of_le a <| iInf\u2082_le b Subset.rfl)\n    (le_iInf fun a' => le_iInf fun b' => le_iInf fun h => ENNReal.coe_le_coe.2 _) ** f : StieltjesFunction a b a' b' : \u211d h : Ioc a b \u2286 Ioc a' b' \u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a') ** cases' le_or_lt b a with ab ab ** case inr f : StieltjesFunction a b a' b' : \u211d h : Ioc a b \u2286 Ioc a' b' ab : a < b \u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a') ** cases' (Ioc_subset_Ioc_iff ab).1 h with h\u2081 h\u2082 ** case inr.intro f : StieltjesFunction a b a' b' : \u211d h : Ioc a b \u2286 Ioc a' b' ab : a < b h\u2081 : b \u2264 b' h\u2082 : a' \u2264 a \u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a') ** exact Real.toNNReal_le_toNNReal (sub_le_sub (f.mono h\u2081) (f.mono h\u2082)) ** case inl f : StieltjesFunction a b a' b' : \u211d h : Ioc a b \u2286 Ioc a' b' ab : b \u2264 a \u22a2 Real.toNNReal (\u2191f b - \u2191f a) \u2264 Real.toNNReal (\u2191f b' - \u2191f a') ** rw [Real.toNNReal_of_nonpos (sub_nonpos.2 (f.mono ab))] ** case inl f : StieltjesFunction a b a' b' : \u211d h : Ioc a b \u2286 Ioc a' b' ab : b \u2264 a \u22a2 0 \u2264 Real.toNNReal (\u2191f b' - \u2191f a') ** apply zero_le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_ae_eq_condexpL1 ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc f) ** rw [condexp_of_sigmaFinite hm] ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' \u22a2 (if Integrable f then       if StronglyMeasurable f then f       else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)     else 0) =\u1d50[\u03bc]     \u2191\u2191(condexpL1 hm \u03bc f) ** by_cases hfi : Integrable f \u03bc ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' hfi : \u00acIntegrable f \u22a2 (if Integrable f then       if StronglyMeasurable f then f       else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)     else 0) =\u1d50[\u03bc]     \u2191\u2191(condexpL1 hm \u03bc f) ** rw [if_neg hfi, condexpL1_undef hfi] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' hfi : \u00acIntegrable f \u22a2 0 =\u1d50[\u03bc] \u2191\u21910 ** exact (coeFn_zero _ _ _).symm ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' hfi : Integrable f \u22a2 (if Integrable f then       if StronglyMeasurable f then f       else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)     else 0) =\u1d50[\u03bc]     \u2191\u2191(condexpL1 hm \u03bc f) ** rw [if_pos hfi] ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' hfi : Integrable f \u22a2 (if StronglyMeasurable f then f     else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) =\u1d50[\u03bc]     \u2191\u2191(condexpL1 hm \u03bc f) ** by_cases hfm : StronglyMeasurable[m] f ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' hfi : Integrable f hfm : StronglyMeasurable f \u22a2 (if StronglyMeasurable f then f     else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) =\u1d50[\u03bc]     \u2191\u2191(condexpL1 hm \u03bc f) ** rw [if_pos hfm] ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' hfi : Integrable f hfm : StronglyMeasurable f \u22a2 f =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc f) ** exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)\n  hfi).symm ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' hfi : Integrable f hfm : \u00acStronglyMeasurable f \u22a2 (if StronglyMeasurable f then f     else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) =\u1d50[\u03bc]     \u2191\u2191(condexpL1 hm \u03bc f) ** rw [if_neg hfm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) f : \u03b1 \u2192 F' hfi : Integrable f hfm : \u00acStronglyMeasurable f \u22a2 AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc) =\u1d50[\u03bc]     \u2191\u2191(condexpL1 hm \u03bc f) ** exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.mem_ite_empty_right ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u : Set \u03b1 p : Prop inst\u271d : Decidable p t : Set \u03b1 x : \u03b1 \u22a2 (\u2203 h, x \u2208 t) \u2194 p \u2227 x \u2208 t ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable\u2080 ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e hs : MeasurableSet s h'f : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x < \u22a4 \u22a2 \u222b\u207b (a : \u03b1) in s, g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1) in s, (f * g) a \u2202\u03bc ** rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul_non_measurable\u2080 _ hf h'f] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_pos_preimage_ball ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d : PseudoMetricSpace \u03b4 f : \u03b1 \u2192 \u03b4 x : \u03b4 h\u03bc : \u03bc \u2260 0 \u22a2 \u22c3 i, f \u207b\u00b9' Metric.ball x \u2191i = univ ** rw [\u2190 preimage_iUnion, Metric.iUnion_ball_nat, preimage_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toOuterMeasure_apply_eq_zero_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 \u22a2 \u2191(toOuterMeasure p) s = 0 \u2194 Disjoint (support p) s ** rw [toOuterMeasure_apply, ENNReal.tsum_eq_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 \u22a2 (\u2200 (i : \u03b1), Set.indicator s (\u2191p) i = 0) \u2194 Disjoint (support p) s ** exact Function.funext_iff.symm.trans Set.indicator_eq_zero' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_le_snorm_mul_snorm_of_nnnorm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** by_cases hp_zero : p = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** have hq_ne_zero : q \u2260 0 := by\n  intro hq_zero\n  simp only [hq_zero, hp_zero, one_div, ENNReal.inv_zero, top_add, ENNReal.inv_eq_top] at hpqr ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** have hr_ne_zero : r \u2260 0 := by\n  intro hr_zero\n  simp only [hr_zero, hp_zero, one_div, ENNReal.inv_zero, add_top, ENNReal.inv_eq_top] at hpqr ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** by_cases hq_top : q = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** by_cases hr_top : r = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 hpq : p < q \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** rw [snorm_eq_snorm' hp_zero (hpq.trans_le le_top).ne, snorm_eq_snorm' hq_ne_zero hq_top,\n  snorm_eq_snorm' hr_ne_zero hr_top] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 hpq : p < q \u22a2 snorm' (fun x => b (f x) (g x)) (ENNReal.toReal p) \u03bc \u2264 snorm' f (ENNReal.toReal q) \u03bc * snorm' g (ENNReal.toReal r) \u03bc ** refine' snorm'_le_snorm'_mul_snorm' hf hg _ h _ _ _ ** case neg.refine'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 hpq : p < q \u22a2 1 / ENNReal.toReal p = 1 / ENNReal.toReal q + 1 / ENNReal.toReal r ** rw [\u2190 ENNReal.one_toReal, \u2190 ENNReal.toReal_div, \u2190 ENNReal.toReal_div, \u2190 ENNReal.toReal_div, hpqr,\n  ENNReal.toReal_add] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : p = 0 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** simp [hp_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 \u22a2 q \u2260 0 ** intro hq_zero ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_zero : q = 0 \u22a2 False ** simp only [hq_zero, hp_zero, one_div, ENNReal.inv_zero, top_add, ENNReal.inv_eq_top] at hpqr ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 \u22a2 r \u2260 0 ** intro hr_zero ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_zero : r = 0 \u22a2 False ** simp only [hr_zero, hp_zero, one_div, ENNReal.inv_zero, add_top, ENNReal.inv_eq_top] at hpqr ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : q = \u22a4 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** have hpr : p = r := by\n  simpa only [hq_top, one_div, ENNReal.inv_top, zero_add, inv_inj] using hpqr ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : q = \u22a4 hpr : p = r \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** rw [\u2190 hpr, hq_top] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : q = \u22a4 hpr : p = r \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f \u22a4 \u03bc * snorm g p \u03bc ** exact snorm_le_snorm_top_mul_snorm p f hg b h ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : q = \u22a4 \u22a2 p = r ** simpa only [hq_top, one_div, ENNReal.inv_top, zero_add, inv_inj] using hpqr ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : r = \u22a4 \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** have hpq : p = q := by\n  simpa only [hr_top, one_div, ENNReal.inv_top, add_zero, inv_inj] using hpqr ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : r = \u22a4 hpq : p = q \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f q \u03bc * snorm g r \u03bc ** rw [\u2190 hpq, hr_top] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : r = \u22a4 hpq : p = q \u22a2 snorm (fun x => b (f x) (g x)) p \u03bc \u2264 snorm f p \u03bc * snorm g \u22a4 \u03bc ** exact snorm_le_snorm_mul_snorm_top p hf g b h ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : r = \u22a4 \u22a2 p = q ** simpa only [hr_top, one_div, ENNReal.inv_top, add_zero, inv_inj] using hpqr ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 \u22a2 p < q ** suffices 1 / q < 1 / p by rwa [one_div, one_div, ENNReal.inv_lt_inv] at this ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 \u22a2 1 / q < 1 / p ** rw [hpqr] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 \u22a2 1 / q < 1 / q + 1 / r ** refine' ENNReal.lt_add_right _ _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 this : 1 / q < 1 / p \u22a2 p < q ** rwa [one_div, one_div, ENNReal.inv_lt_inv] at this ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 \u22a2 1 / q \u2260 \u22a4 ** simp only [hq_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff] ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 \u22a2 1 / r \u2260 0 ** simp only [hr_top, one_div, Ne.def, ENNReal.inv_eq_zero, not_false_iff] ** case neg.refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 hpq : p < q \u22a2 0 < ENNReal.toReal p ** exact ENNReal.toReal_pos hp_zero (hpq.trans_le le_top).ne ** case neg.refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 hpq : p < q \u22a2 ENNReal.toReal p < ENNReal.toReal q ** exact ENNReal.toReal_strict_mono hq_top hpq ** case neg.refine'_3.ha \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 hpq : p < q \u22a2 1 / q \u2260 \u22a4 ** simp only [hq_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff] ** case neg.refine'_3.hb \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q\u271d : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G p q r : \u211d\u22650\u221e f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc g : \u03b1 \u2192 F hg : AEStronglyMeasurable g \u03bc b : E \u2192 F \u2192 G h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a hpqr : 1 / p = 1 / q + 1 / r hp_zero : \u00acp = 0 hq_ne_zero : q \u2260 0 hr_ne_zero : r \u2260 0 hq_top : \u00acq = \u22a4 hr_top : \u00acr = \u22a4 hpq : p < q \u22a2 1 / r \u2260 \u22a4 ** simp only [hr_ne_zero, one_div, Ne.def, ENNReal.inv_eq_top, not_false_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_emetric_closedBall ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d r : \u211d\u22650\u221e \u22a2 \u2191\u2191volume (EMetric.closedBall a r) = 2 * r ** rcases eq_or_ne r \u221e with (rfl | hr) ** case inl \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d \u22a2 \u2191\u2191volume (EMetric.closedBall a \u22a4) = 2 * \u22a4 ** rw [EMetric.closedBall_top, volume_univ, two_mul, _root_.top_add] ** case inr \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d r : \u211d\u22650\u221e hr : r \u2260 \u22a4 \u22a2 \u2191\u2191volume (EMetric.closedBall a r) = 2 * r ** lift r to \u211d\u22650 using hr ** case inr.intro \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d r : \u211d\u22650 \u22a2 \u2191\u2191volume (EMetric.closedBall a \u2191r) = 2 * \u2191r ** rw [Metric.emetric_closedBall_nnreal, volume_closedBall, two_mul, \u2190 NNReal.coe_add,\n  ENNReal.ofReal_coe_nnreal, ENNReal.coe_add, two_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.castNum_shiftRight ** \u03b1 : Type u_1 m : Num n : \u2115 \u22a2 \u2191(m >>> n) = \u2191m >>> n ** cases' m with m <;> dsimp only [\u2190shiftr_eq_shiftRight, shiftr] ** case pos \u03b1 : Type u_1 n : \u2115 m : PosNum \u22a2 \u2191(m >>> n) = \u2191(pos m) >>> n ** induction' n with n IH generalizing m ** case pos.succ \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2191(m >>> Nat.succ n) = \u2191(pos m) >>> Nat.succ n ** cases' m with m m <;> dsimp only [PosNum.shiftr, \u2190PosNum.shiftr_eq_shiftRight] ** case zero \u03b1 : Type u_1 n : \u2115 \u22a2 \u21910 = \u2191zero >>> n ** symm ** case zero \u03b1 : Type u_1 n : \u2115 \u22a2 \u2191zero >>> n = \u21910 ** apply Nat.zero_shiftRight ** case pos.zero \u03b1 : Type u_1 m\u271d m : PosNum \u22a2 \u2191(m >>> Nat.zero) = \u2191(pos m) >>> Nat.zero ** cases m <;> rfl ** case pos.succ.one \u03b1 : Type u_1 m : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n \u22a2 \u21910 = \u2191(pos one) >>> Nat.succ n ** rw [Nat.shiftRight_eq_div_pow] ** case pos.succ.one \u03b1 : Type u_1 m : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n \u22a2 \u21910 = \u2191(pos one) / 2 ^ Nat.succ n ** symm ** case pos.succ.one \u03b1 : Type u_1 m : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n \u22a2 \u2191(pos one) / 2 ^ Nat.succ n = \u21910 ** apply Nat.div_eq_of_lt ** case pos.succ.one.h\u2080 \u03b1 : Type u_1 m : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n \u22a2 \u2191(pos one) < 2 ^ Nat.succ n ** simp [@Nat.pow_lt_pow_of_lt_right 2 (by decide) 0 (n + 1) (Nat.succ_pos _)] ** \u03b1 : Type u_1 m : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n \u22a2 1 < 2 ** decide ** case pos.succ.bit1 \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2191(PosNum.shiftr m n) = \u2191(pos (PosNum.bit1 m)) >>> Nat.succ n ** trans ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2191(PosNum.shiftr m n) = ?m.631939  \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 ?m.631939 = \u2191(pos (PosNum.bit1 m)) >>> Nat.succ n  \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2115 ** apply IH ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2191(pos m) >>> n = \u2191(pos (PosNum.bit1 m)) >>> Nat.succ n ** change Nat.shiftRight m n = Nat.shiftRight (_root_.bit1 m) (n + 1) ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 Nat.shiftRight (\u2191m) n = Nat.shiftRight (_root_.bit1 \u2191m) (n + 1) ** rw [add_comm n 1, @Nat.shiftRight_eq _ (1 + n), Nat.shiftRight_add] ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 Nat.shiftRight (\u2191m) n = _root_.bit1 \u2191m >>> 1 >>> n ** apply congr_arg fun x => Nat.shiftRight x n ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2191m = _root_.bit1 \u2191m >>> 1 ** simp [Nat.shiftRight_succ, Nat.shiftRight_zero, \u2190 Nat.div2_val] ** case pos.succ.bit0 \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2191(PosNum.shiftr m n) = \u2191(pos (PosNum.bit0 m)) >>> Nat.succ n ** trans ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2191(PosNum.shiftr m n) = ?m.633935  \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 ?m.633935 = \u2191(pos (PosNum.bit0 m)) >>> Nat.succ n  \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2115 ** apply IH ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2191(pos m) >>> n = \u2191(pos (PosNum.bit0 m)) >>> Nat.succ n ** change Nat.shiftRight m n = Nat.shiftRight (_root_.bit0 m) (n + 1) ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 Nat.shiftRight (\u2191m) n = Nat.shiftRight (_root_.bit0 \u2191m) (n + 1) ** rw [add_comm n 1,  @Nat.shiftRight_eq _ (1 + n), Nat.shiftRight_add] ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 Nat.shiftRight (\u2191m) n = _root_.bit0 \u2191m >>> 1 >>> n ** apply congr_arg fun x => Nat.shiftRight x n ** \u03b1 : Type u_1 m\u271d : PosNum n : \u2115 IH : \u2200 (m : PosNum), \u2191(m >>> n) = \u2191(pos m) >>> n m : PosNum \u22a2 \u2191m = _root_.bit0 \u2191m >>> 1 ** simp [Nat.shiftRight_succ, Nat.shiftRight_zero, \u2190 Nat.div2_val] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_biSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 s : Set \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hs : Set.Countable s hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i) \u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b ** haveI : Encodable s := hs.toEncodable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 s : Set \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hs : Set.Countable s hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i) this : Encodable \u2191s \u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b ** by_cases H : \u2200 i, i \u2208 s ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 s : Set \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hs : Set.Countable s hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i) this : Encodable \u2191s H : \u2200 (i : \u03b9), i \u2208 s \u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b ** have : \u2200 b, \u2a06 i \u2208 s, f i b = \u2a06 (i : s), f i b :=\n  fun b \u21a6 cbiSup_eq_of_forall (f := fun i \u21a6 f i b) H ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 s : Set \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hs : Set.Countable s hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i) this\u271d : Encodable \u2191s H : \u2200 (i : \u03b9), i \u2208 s this : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = \u2a06 i, f (\u2191i) b \u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b ** simp only [this] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 s : Set \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hs : Set.Countable s hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i) this\u271d : Encodable \u2191s H : \u2200 (i : \u03b9), i \u2208 s this : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = \u2a06 i, f (\u2191i) b \u22a2 Measurable fun b => \u2a06 i, f (\u2191i) b ** exact measurable_iSup (fun (i : s) \u21a6 hf i i.2) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 s : Set \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hs : Set.Countable s hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i) this : Encodable \u2191s H : \u00ac\u2200 (i : \u03b9), i \u2208 s \u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b ** have : \u2200 b, \u2a06 i \u2208 s, f i b = (\u2a06 (i : s), f i b) \u2294 sSup \u2205 :=\n  fun b \u21a6 cbiSup_eq_of_not_forall (f := fun i \u21a6 f i b) H ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 s : Set \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hs : Set.Countable s hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i) this\u271d : Encodable \u2191s H : \u00ac\u2200 (i : \u03b9), i \u2208 s this : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205 \u22a2 Measurable fun b => \u2a06 i \u2208 s, f i b ** simp only [this] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 s : Set \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hs : Set.Countable s hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i) this\u271d : Encodable \u2191s H : \u00ac\u2200 (i : \u03b9), i \u2208 s this : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205 \u22a2 Measurable fun b => (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205 ** apply Measurable.sup _ measurable_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 s : Set \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hs : Set.Countable s hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Measurable (f i) this\u271d : Encodable \u2191s H : \u00ac\u2200 (i : \u03b9), i \u2208 s this : \u2200 (b : \u03b4), \u2a06 i \u2208 s, f i b = (\u2a06 i, f (\u2191i) b) \u2294 sSup \u2205 \u22a2 Measurable fun a => \u2a06 i, f (\u2191i) a ** exact measurable_iSup (fun (i : s) \u21a6 hf i i.2) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.stoppedProcess_eq ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n : \u2115 \u22a2 stoppedProcess u \u03c4 n = Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.range n, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i) ** rw [stoppedProcess_eq'' n] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n : \u2115 \u22a2 Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.Iio n, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i) =     Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.range n, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i) ** congr with i ** case e_a.e_s.a \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 f : Filtration \u2115 m u : \u2115 \u2192 \u03a9 \u2192 \u03b2 \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : AddCommMonoid \u03b2 n i : \u2115 \u22a2 i \u2208 Finset.Iio n \u2194 i \u2208 Finset.range n ** rw [Finset.mem_Iio, Finset.mem_range] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vitali.exists_disjoint_subfamily_covering_enlargment ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) \u22a2 \u2203 u x, PairwiseDisjoint u B \u2227 \u2200 (a : \u03b9), a \u2208 t \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b ** let T : Set (Set \u03b9) := { u | u \u2286 t \u2227 u.PairwiseDisjoint B \u2227\n  \u2200 a \u2208 t, \u2200 b \u2208 u, (B a \u2229 B b).Nonempty \u2192 \u2203 c \u2208 u, (B a \u2229 B c).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c } ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} \u22a2 \u2203 u x, PairwiseDisjoint u B \u2227 \u2200 (a : \u03b9), a \u2208 t \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b ** obtain \u27e8u, uT, hu\u27e9 : \u2203 u \u2208 T, \u2200 v \u2208 T, u \u2286 v \u2192 v = u := by\n  refine' zorn_subset _ fun U UT hU => _\n  refine' \u27e8\u22c3\u2080 U, _, fun s hs => subset_sUnion_of_mem hs\u27e9\n  simp only [Set.sUnion_subset_iff, and_imp, exists_prop, forall_exists_index, mem_sUnion,\n    Set.mem_setOf_eq]\n  refine'\n    \u27e8fun u hu => (UT hu).1, (pairwiseDisjoint_sUnion hU.directedOn).2 fun u hu => (UT hu).2.1,\n      fun a hat b u uU hbu hab => _\u27e9\n  obtain \u27e8c, cu, ac, hc\u27e9 : \u2203 c, c \u2208 u \u2227 (B a \u2229 B c).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c :=\n    (UT uU).2.2 a hat b hbu hab\n  exact \u27e8c, \u27e8u, uU, cu\u27e9, ac, hc\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T hu : \u2200 (v : Set \u03b9), v \u2208 T \u2192 u \u2286 v \u2192 v = u \u22a2 \u2203 u x, PairwiseDisjoint u B \u2227 \u2200 (a : \u03b9), a \u2208 t \u2192 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b ** refine' \u27e8u, uT.1, uT.2.1, fun a hat => _\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T hu : \u2200 (v : Set \u03b9), v \u2208 T \u2192 u \u2286 v \u2192 v = u a : \u03b9 hat : a \u2208 t \u22a2 \u2203 b, b \u2208 u \u2227 Set.Nonempty (B a \u2229 B b) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 b ** contrapose! hu ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a \u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u ** have a_disj : \u2200 c \u2208 u, Disjoint (B a) (B c) := by\n  intro c hc\n  by_contra h\n  rw [not_disjoint_iff_nonempty_inter] at h\n  obtain \u27e8d, du, ad, hd\u27e9 : \u2203 d, d \u2208 u \u2227 (B a \u2229 B d).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 d :=\n    uT.2.2 a hat c hc h\n  exact lt_irrefl _ ((hu d du ad).trans_le hd) ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) \u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u ** let A := { a' | a' \u2208 t \u2227 \u2200 c \u2208 u, Disjoint (B a') (B c) } ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} \u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u ** have Anonempty : A.Nonempty := \u27e8a, hat, a_disj\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A \u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u ** let m := sSup (\u03b4 '' A) ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) \u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u ** have bddA : BddAbove (\u03b4 '' A) := by\n  refine' \u27e8R, fun x xA => _\u27e9\n  rcases (mem_image _ _ _).1 xA with \u27e8a', ha', rfl\u27e9\n  exact \u03b4le a' ha'.1 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' \u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u ** clear hat hu a_disj a ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' \u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u ** have a'_ne_u : a' \u2209 u := fun H => (hne _ a'A.1).ne_empty (disjoint_self.1 (a'A.2 _ H)) ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u \u22a2 \u2203 v, v \u2208 T \u2227 u \u2286 v \u2227 v \u2260 u ** refine' \u27e8insert a' u, \u27e8_, _, _\u27e9, subset_insert _ _, (ne_insert_of_not_mem _ a'_ne_u).symm\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} \u22a2 \u2203 u, u \u2208 T \u2227 \u2200 (v : Set \u03b9), v \u2208 T \u2192 u \u2286 v \u2192 v = u ** refine' zorn_subset _ fun U UT hU => _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} U : Set (Set \u03b9) UT : U \u2286 T hU : IsChain (fun x x_1 => x \u2286 x_1) U \u22a2 \u2203 ub, ub \u2208 T \u2227 \u2200 (s : Set \u03b9), s \u2208 U \u2192 s \u2286 ub ** refine' \u27e8\u22c3\u2080 U, _, fun s hs => subset_sUnion_of_mem hs\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} U : Set (Set \u03b9) UT : U \u2286 T hU : IsChain (fun x x_1 => x \u2286 x_1) U \u22a2 \u22c3\u2080 U \u2208 T ** simp only [Set.sUnion_subset_iff, and_imp, exists_prop, forall_exists_index, mem_sUnion,\n  Set.mem_setOf_eq] ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} U : Set (Set \u03b9) UT : U \u2286 T hU : IsChain (fun x x_1 => x \u2286 x_1) U \u22a2 (\u2200 (t' : Set \u03b9), t' \u2208 U \u2192 t' \u2286 t) \u2227     PairwiseDisjoint (\u22c3\u2080 U) B \u2227       \u2200 (a : \u03b9),         a \u2208 t \u2192           \u2200 (b : \u03b9) (x : Set \u03b9),             x \u2208 U \u2192               b \u2208 x \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, (\u2203 t, t \u2208 U \u2227 c \u2208 t) \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c ** refine'\n  \u27e8fun u hu => (UT hu).1, (pairwiseDisjoint_sUnion hU.directedOn).2 fun u hu => (UT hu).2.1,\n    fun a hat b u uU hbu hab => _\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} U : Set (Set \u03b9) UT : U \u2286 T hU : IsChain (fun x x_1 => x \u2286 x_1) U a : \u03b9 hat : a \u2208 t b : \u03b9 u : Set \u03b9 uU : u \u2208 U hbu : b \u2208 u hab : Set.Nonempty (B a \u2229 B b) \u22a2 \u2203 c, (\u2203 t, t \u2208 U \u2227 c \u2208 t) \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c ** obtain \u27e8c, cu, ac, hc\u27e9 : \u2203 c, c \u2208 u \u2227 (B a \u2229 B c).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c :=\n  (UT uU).2.2 a hat b hbu hab ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} U : Set (Set \u03b9) UT : U \u2286 T hU : IsChain (fun x x_1 => x \u2286 x_1) U a : \u03b9 hat : a \u2208 t b : \u03b9 u : Set \u03b9 uU : u \u2208 U hbu : b \u2208 u hab : Set.Nonempty (B a \u2229 B b) c : \u03b9 cu : c \u2208 u ac : Set.Nonempty (B a \u2229 B c) hc : \u03b4 a \u2264 \u03c4 * \u03b4 c \u22a2 \u2203 c, (\u2203 t, t \u2208 U \u2227 c \u2208 t) \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c ** exact \u27e8c, \u27e8u, uU, cu\u27e9, ac, hc\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a \u22a2 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) ** intro c hc ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a c : \u03b9 hc : c \u2208 u \u22a2 Disjoint (B a) (B c) ** by_contra h ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a c : \u03b9 hc : c \u2208 u h : \u00acDisjoint (B a) (B c) \u22a2 False ** rw [not_disjoint_iff_nonempty_inter] at h ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a c : \u03b9 hc : c \u2208 u h : Set.Nonempty (B a \u2229 B c) \u22a2 False ** obtain \u27e8d, du, ad, hd\u27e9 : \u2203 d, d \u2208 u \u2227 (B a \u2229 B d).Nonempty \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 d :=\n  uT.2.2 a hat c hc h ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a c : \u03b9 hc : c \u2208 u h : Set.Nonempty (B a \u2229 B c) d : \u03b9 du : d \u2208 u ad : Set.Nonempty (B a \u2229 B d) hd : \u03b4 a \u2264 \u03c4 * \u03b4 d \u22a2 False ** exact lt_irrefl _ ((hu d du ad).trans_le hd) ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) \u22a2 BddAbove (\u03b4 '' A) ** refine' \u27e8R, fun x xA => _\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) x : \u211d xA : x \u2208 \u03b4 '' A \u22a2 x \u2264 R ** rcases (mem_image _ _ _).1 xA with \u27e8a', ha', rfl\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) a' : \u03b9 ha' : a' \u2208 A xA : \u03b4 a' \u2208 \u03b4 '' A \u22a2 \u03b4 a' \u2264 R ** exact \u03b4le a' ha'.1 ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) \u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a' ** have : 0 \u2264 m := (\u03b4nonneg a hat).trans (le_csSup bddA (mem_image_of_mem _ \u27e8hat, a_disj\u27e9)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m \u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a' ** rcases eq_or_lt_of_le this with (mzero | mpos) ** case inl \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m mzero : 0 = m \u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a' ** refine' \u27e8a, \u27e8hat, a_disj\u27e9, _\u27e9 ** case inl \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m mzero : 0 = m \u22a2 m / \u03c4 \u2264 \u03b4 a ** simpa only [\u2190 mzero, zero_div] using \u03b4nonneg a hat ** case inr \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m mpos : 0 < m \u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a' ** have I : m / \u03c4 < m := by\n  rw [div_lt_iff (zero_lt_one.trans h\u03c4)]\n  conv_lhs => rw [\u2190 mul_one m]\n  exact (mul_lt_mul_left mpos).2 h\u03c4 ** case inr \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m mpos : 0 < m I : m / \u03c4 < m \u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a' ** rcases exists_lt_of_lt_csSup (nonempty_image_iff.2 Anonempty) I with \u27e8x, xA, hx\u27e9 ** case inr.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m mpos : 0 < m I : m / \u03c4 < m x : \u211d xA : x \u2208 \u03b4 '' A hx : m / \u03c4 < x \u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a' ** rcases (mem_image _ _ _).1 xA with \u27e8a', ha', rfl\u27e9 ** case inr.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m mpos : 0 < m I : m / \u03c4 < m a' : \u03b9 ha' : a' \u2208 A xA : \u03b4 a' \u2208 \u03b4 '' A hx : m / \u03c4 < \u03b4 a' \u22a2 \u2203 a', a' \u2208 A \u2227 m / \u03c4 \u2264 \u03b4 a' ** exact \u27e8a', ha', hx.le\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m mpos : 0 < m \u22a2 m / \u03c4 < m ** rw [div_lt_iff (zero_lt_one.trans h\u03c4)] ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m mpos : 0 < m \u22a2 m < m * \u03c4 ** conv_lhs => rw [\u2190 mul_one m] ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T a : \u03b9 hat : a \u2208 t hu : \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u03c4 * \u03b4 b < \u03b4 a a_disj : \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a) (B c) A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) this : 0 \u2264 m mpos : 0 < m \u22a2 m * 1 < m * \u03c4 ** exact (mul_lt_mul_left mpos).2 h\u03c4 ** case intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u \u22a2 insert a' u \u2286 t ** rw [insert_subset_iff] ** case intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u \u22a2 a' \u2208 t \u2227 u \u2286 t ** exact \u27e8a'A.1, uT.1\u27e9 ** case intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u \u22a2 PairwiseDisjoint (insert a' u) B ** exact uT.2.1.insert fun b bu _ => a'A.2 b bu ** case intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u \u22a2 \u2200 (a : \u03b9),     a \u2208 t \u2192       \u2200 (b : \u03b9),         b \u2208 insert a' u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 insert a' u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c ** intro c ct b ba'u hcb ** case intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u c : \u03b9 ct : c \u2208 t b : \u03b9 ba'u : b \u2208 insert a' u hcb : Set.Nonempty (B c \u2229 B b) \u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** by_cases H : \u2203 d \u2208 u, (B c \u2229 B d).Nonempty ** case pos \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u c : \u03b9 ct : c \u2208 t b : \u03b9 ba'u : b \u2208 insert a' u hcb : Set.Nonempty (B c \u2229 B b) H : \u2203 d, d \u2208 u \u2227 Set.Nonempty (B c \u2229 B d) \u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** rcases H with \u27e8d, du, hd\u27e9 ** case pos.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u c : \u03b9 ct : c \u2208 t b : \u03b9 ba'u : b \u2208 insert a' u hcb : Set.Nonempty (B c \u2229 B b) d : \u03b9 du : d \u2208 u hd : Set.Nonempty (B c \u2229 B d) \u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** rcases uT.2.2 c ct d du hd with \u27e8d', d'u, hd'\u27e9 ** case pos.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u c : \u03b9 ct : c \u2208 t b : \u03b9 ba'u : b \u2208 insert a' u hcb : Set.Nonempty (B c \u2229 B b) d : \u03b9 du : d \u2208 u hd : Set.Nonempty (B c \u2229 B d) d' : \u03b9 d'u : d' \u2208 u hd' : Set.Nonempty (B c \u2229 B d') \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 d' \u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** exact \u27e8d', mem_insert_of_mem _ d'u, hd'\u27e9 ** case neg \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u c : \u03b9 ct : c \u2208 t b : \u03b9 ba'u : b \u2208 insert a' u hcb : Set.Nonempty (B c \u2229 B b) H : \u00ac\u2203 d, d \u2208 u \u2227 Set.Nonempty (B c \u2229 B d) \u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** push_neg at H ** case neg \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u c : \u03b9 ct : c \u2208 t b : \u03b9 ba'u : b \u2208 insert a' u hcb : Set.Nonempty (B c \u2229 B b) H : \u2200 (d : \u03b9), d \u2208 u \u2192 \u00acSet.Nonempty (B c \u2229 B d) \u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** simp only [\u2190 not_disjoint_iff_nonempty_inter, Classical.not_not] at H ** case neg \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u c : \u03b9 ct : c \u2208 t b : \u03b9 ba'u : b \u2208 insert a' u hcb : Set.Nonempty (B c \u2229 B b) H : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d) \u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** rcases mem_insert_iff.1 ba'u with (rfl | H') ** case neg.inl \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) c : \u03b9 ct : c \u2208 t b : \u03b9 hcb : Set.Nonempty (B c \u2229 B b) H : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d) a'A : b \u2208 A ha' : m / \u03c4 \u2264 \u03b4 b a'_ne_u : \u00acb \u2208 u ba'u : b \u2208 insert b u \u22a2 \u2203 c_1, c_1 \u2208 insert b u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** refine' \u27e8b, mem_insert _ _, hcb, _\u27e9 ** case neg.inl \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) c : \u03b9 ct : c \u2208 t b : \u03b9 hcb : Set.Nonempty (B c \u2229 B b) H : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d) a'A : b \u2208 A ha' : m / \u03c4 \u2264 \u03b4 b a'_ne_u : \u00acb \u2208 u ba'u : b \u2208 insert b u \u22a2 \u03b4 c \u2264 \u03c4 * \u03b4 b ** calc\n  \u03b4 c \u2264 m := le_csSup bddA (mem_image_of_mem _ \u27e8ct, H\u27e9)\n  _ = \u03c4 * (m / \u03c4) := by\n    field_simp [(zero_lt_one.trans h\u03c4).ne']\n    ring\n  _ \u2264 \u03c4 * \u03b4 b := mul_le_mul_of_nonneg_left ha' (zero_le_one.trans h\u03c4.le) ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) c : \u03b9 ct : c \u2208 t b : \u03b9 hcb : Set.Nonempty (B c \u2229 B b) H : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d) a'A : b \u2208 A ha' : m / \u03c4 \u2264 \u03b4 b a'_ne_u : \u00acb \u2208 u ba'u : b \u2208 insert b u \u22a2 m = \u03c4 * (m / \u03c4) ** field_simp [(zero_lt_one.trans h\u03c4).ne'] ** \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) c : \u03b9 ct : c \u2208 t b : \u03b9 hcb : Set.Nonempty (B c \u2229 B b) H : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d) a'A : b \u2208 A ha' : m / \u03c4 \u2264 \u03b4 b a'_ne_u : \u00acb \u2208 u ba'u : b \u2208 insert b u \u22a2 sSup (\u03b4 '' {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}) * \u03c4 =     \u03c4 * sSup (\u03b4 '' {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)}) ** ring ** case neg.inr \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u c : \u03b9 ct : c \u2208 t b : \u03b9 ba'u : b \u2208 insert a' u hcb : Set.Nonempty (B c \u2229 B b) H : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d) H' : b \u2208 u \u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** rw [\u2190 not_disjoint_iff_nonempty_inter] at hcb ** case neg.inr \u03b1 : Type u_1 \u03b9 : Type u_2 B : \u03b9 \u2192 Set \u03b1 t : Set \u03b9 \u03b4 : \u03b9 \u2192 \u211d \u03c4 : \u211d h\u03c4 : 1 < \u03c4 \u03b4nonneg : \u2200 (a : \u03b9), a \u2208 t \u2192 0 \u2264 \u03b4 a R : \u211d \u03b4le : \u2200 (a : \u03b9), a \u2208 t \u2192 \u03b4 a \u2264 R hne : \u2200 (a : \u03b9), a \u2208 t \u2192 Set.Nonempty (B a) T : Set (Set \u03b9) :=   {u |     u \u2286 t \u2227       PairwiseDisjoint u B \u2227         \u2200 (a : \u03b9),           a \u2208 t \u2192 \u2200 (b : \u03b9), b \u2208 u \u2192 Set.Nonempty (B a \u2229 B b) \u2192 \u2203 c, c \u2208 u \u2227 Set.Nonempty (B a \u2229 B c) \u2227 \u03b4 a \u2264 \u03c4 * \u03b4 c} u : Set \u03b9 uT : u \u2208 T A : Set \u03b9 := {a' | a' \u2208 t \u2227 \u2200 (c : \u03b9), c \u2208 u \u2192 Disjoint (B a') (B c)} Anonempty : Set.Nonempty A m : \u211d := sSup (\u03b4 '' A) bddA : BddAbove (\u03b4 '' A) a' : \u03b9 a'A : a' \u2208 A ha' : m / \u03c4 \u2264 \u03b4 a' a'_ne_u : \u00aca' \u2208 u c : \u03b9 ct : c \u2208 t b : \u03b9 ba'u : b \u2208 insert a' u hcb : \u00acDisjoint (B c) (B b) H : \u2200 (d : \u03b9), d \u2208 u \u2192 Disjoint (B c) (B d) H' : b \u2208 u \u22a2 \u2203 c_1, c_1 \u2208 insert a' u \u2227 Set.Nonempty (B c \u2229 B c_1) \u2227 \u03b4 c \u2264 \u03c4 * \u03b4 c_1 ** exact (hcb (H _ H')).elim ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_add ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hf : Integrable f hg : Integrable g \u22a2 setToFun \u03bc T hT (f + g) = setToFun \u03bc T hT f + setToFun \u03bc T hT g ** rw [setToFun_eq hT (hf.add hg), setToFun_eq hT hf, setToFun_eq hT hg, Integrable.toL1_add,\n  (L1.setToL1 hT).map_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.fillNones_eq_fillNonesTR ** \u22a2 @fillNones = @fillNonesTR ** funext \u03b1 as as' ** case h.h.h \u03b1 : Type u_1 as : List (Option \u03b1) as' : List \u03b1 \u22a2 fillNones as as' = fillNonesTR as as' ** simp [fillNonesTR] ** case h.h.h \u03b1 : Type u_1 as : List (Option \u03b1) as' : List \u03b1 \u22a2 fillNones as as' = fillNonesTR.go as as' #[] ** simp [fillNonesTR, go] ** \u03b1 : Type u_1 as : List (Option \u03b1) as' : List \u03b1 acc : Array \u03b1 x\u271d : List \u03b1 \u22a2 fillNonesTR.go [] x\u271d acc = acc.data ++ fillNones [] x\u271d ** simp [fillNonesTR.go] ** \u03b1 : Type u_1 as\u271d : List (Option \u03b1) as'\u271d : List \u03b1 acc : Array \u03b1 a : \u03b1 as : List (Option \u03b1) as' : List \u03b1 \u22a2 fillNonesTR.go (some a :: as) as' acc = acc.data ++ fillNones (some a :: as) as' ** simp [fillNonesTR.go, go _ as as'] ** \u03b1 : Type u_1 as\u271d : List (Option \u03b1) as' : List \u03b1 acc : Array \u03b1 as : List (Option \u03b1) \u22a2 fillNonesTR.go (none :: as) [] acc = acc.data ++ fillNones (none :: as) [] ** simp [fillNonesTR.go, reduceOption, filterMap_eq_filterMapTR.go] ** \u03b1 : Type u_1 as\u271d : List (Option \u03b1) as'\u271d : List \u03b1 acc : Array \u03b1 as : List (Option \u03b1) a : \u03b1 as' : List \u03b1 \u22a2 fillNonesTR.go (none :: as) (a :: as') acc = acc.data ++ fillNones (none :: as) (a :: as') ** simp [fillNonesTR.go, go _ as as'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.nat_mod ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 m n : \u2115 \u22a2 (m, n).1 - (m, n).2 * (fun x x_1 => x / x_1) (m, n).1 (m, n).2 = m % n ** apply Nat.sub_eq_of_eq_add ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 m n : \u2115 \u22a2 (m, n).1 = m % n + (m, n).2 * (fun x x_1 => x / x_1) (m, n).1 (m, n).2 ** simp [add_comm (m % n), Nat.div_add_mod] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.restrict_toMeasure_support ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03b1 p\u271d : PMF \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 p : PMF \u03b1 \u22a2 Measure.restrict (toMeasure p) (support p) = toMeasure p ** ext s hs ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03b1 p\u271d : PMF \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 p : PMF \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.restrict (toMeasure p) (support p)) s = \u2191\u2191(toMeasure p) s ** apply (MeasureTheory.Measure.restrict_apply hs).trans ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03b1 p\u271d : PMF \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 p : PMF \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(toMeasure p) (s \u2229 support p) = \u2191\u2191(toMeasure p) s ** apply toMeasure_apply_inter_support p s hs p.support_countable.measurableSet ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.insert_inter_distrib ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t\u271d u v : Finset \u03b1 a\u271d b : \u03b1 s t : Finset \u03b1 a : \u03b1 \u22a2 insert a (s \u2229 t) = insert a s \u2229 insert a t ** simp_rw [insert_eq, union_distrib_left] ** Qed",
        "informal": ""
    },
    {
        "formal": "Basis.map_addHaar ** \u03b9 : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u2070 : Fintype \u03b9 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : MeasurableSpace E inst\u271d\u2074 : MeasurableSpace F inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : BorelSpace F inst\u271d\u00b9 : SecondCountableTopology F inst\u271d : SigmaCompactSpace F b : Basis \u03b9 \u211d E f : E \u2243L[\u211d] F \u22a2 Measure.map (\u2191f) (addHaar b) = addHaar (Basis.map b f.toLinearEquiv) ** have : IsAddHaarMeasure (map f b.addHaar) :=\n  AddEquiv.isAddHaarMeasure_map b.addHaar f.toAddEquiv f.continuous f.symm.continuous ** \u03b9 : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u2070 : Fintype \u03b9 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : MeasurableSpace E inst\u271d\u2074 : MeasurableSpace F inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : BorelSpace F inst\u271d\u00b9 : SecondCountableTopology F inst\u271d : SigmaCompactSpace F b : Basis \u03b9 \u211d E f : E \u2243L[\u211d] F this : IsAddHaarMeasure (Measure.map (\u2191f) (addHaar b)) \u22a2 Measure.map (\u2191f) (addHaar b) = addHaar (Basis.map b f.toLinearEquiv) ** rw [eq_comm, Basis.addHaar_eq_iff, Measure.map_apply f.continuous.measurable\n  (PositiveCompacts.isCompact _).measurableSet, Basis.coe_parallelepiped, Basis.coe_map] ** \u03b9 : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u2070 : Fintype \u03b9 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : MeasurableSpace E inst\u271d\u2074 : MeasurableSpace F inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : BorelSpace F inst\u271d\u00b9 : SecondCountableTopology F inst\u271d : SigmaCompactSpace F b : Basis \u03b9 \u211d E f : E \u2243L[\u211d] F this : IsAddHaarMeasure (Measure.map (\u2191f) (addHaar b)) \u22a2 \u2191\u2191(addHaar b) (\u2191f \u207b\u00b9' _root_.parallelepiped (\u2191f.toLinearEquiv \u2218 \u2191b)) = 1 ** erw [\u2190 image_parallelepiped, f.toEquiv.preimage_image, addHaar_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.ext_of_Ico_finite ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b2\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b2\u00b9 : MeasurableSpace \u03b1\u271d inst\u271d\u00b2\u2070 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2079 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2078 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2076 : TopologicalSpace \u03b3 inst\u271d\u00b9\u2075 : MeasurableSpace \u03b3 inst\u271d\u00b9\u2074 : BorelSpace \u03b3 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\u2082 inst\u271d\u00b9\u00b9 : BorelSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2079 : TopologicalSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b1' inst\u271d\u2077 : LinearOrder \u03b1\u271d inst\u271d\u2076 : OrderClosedTopology \u03b1\u271d a b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u2075 : TopologicalSpace \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc h\u03bc\u03bd : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ h : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b) \u22a2 \u03bc = \u03bd ** refine'\n  ext_of_generate_finite _ (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico \u03b1))\n    (isPiSystem_Ico (id : \u03b1 \u2192 \u03b1) id) _ h\u03bc\u03bd ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b2\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b2\u00b9 : MeasurableSpace \u03b1\u271d inst\u271d\u00b2\u2070 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2079 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2078 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2076 : TopologicalSpace \u03b3 inst\u271d\u00b9\u2075 : MeasurableSpace \u03b3 inst\u271d\u00b9\u2074 : BorelSpace \u03b3 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\u2082 inst\u271d\u00b9\u00b9 : BorelSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2079 : TopologicalSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b1' inst\u271d\u2077 : LinearOrder \u03b1\u271d inst\u271d\u2076 : OrderClosedTopology \u03b1\u271d a b x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u2075 : TopologicalSpace \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc h\u03bc\u03bd : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ h : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b) \u22a2 \u2200 (s : Set \u03b1), s \u2208 {S | \u2203 l u, l < u \u2227 Ico l u = S} \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s ** rintro - \u27e8a, b, hlt, rfl\u27e9 ** case intro.intro.intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b2\u00b2 : TopologicalSpace \u03b1\u271d inst\u271d\u00b2\u00b9 : MeasurableSpace \u03b1\u271d inst\u271d\u00b2\u2070 : OpensMeasurableSpace \u03b1\u271d inst\u271d\u00b9\u2079 : TopologicalSpace \u03b2 inst\u271d\u00b9\u2078 : MeasurableSpace \u03b2 inst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b2 inst\u271d\u00b9\u2076 : TopologicalSpace \u03b3 inst\u271d\u00b9\u2075 : MeasurableSpace \u03b3 inst\u271d\u00b9\u2074 : BorelSpace \u03b3 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\u2082 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\u2082 inst\u271d\u00b9\u00b9 : BorelSpace \u03b3\u2082 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b4 \u03b1' : Type u_6 inst\u271d\u2079 : TopologicalSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b1' inst\u271d\u2077 : LinearOrder \u03b1\u271d inst\u271d\u2076 : OrderClosedTopology \u03b1\u271d a\u271d b\u271d x : \u03b1\u271d \u03b1 : Type u_7 inst\u271d\u2075 : TopologicalSpace \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc h\u03bc\u03bd : \u2191\u2191\u03bc univ = \u2191\u2191\u03bd univ h : \u2200 \u2983a b : \u03b1\u2984, a < b \u2192 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b) a b : \u03b1 hlt : a < b \u22a2 \u2191\u2191\u03bc (Ico a b) = \u2191\u2191\u03bd (Ico a b) ** exact h hlt ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_le_coe_iff ** \u03b1 : Type u_1 s t : Set \u03b1 k : \u2115 h : encard s \u2264 \u2191k \u22a2 \u2203 n\u2080, encard s = \u2191n\u2080 \u2227 n\u2080 \u2264 k ** rwa [ENat.le_coe_iff] at h ** \u03b1 : Type u_1 s t : Set \u03b1 k : \u2115 x\u271d : Set.Finite s \u2227 \u2203 n\u2080, encard s = \u2191n\u2080 \u2227 n\u2080 \u2264 k left\u271d : Set.Finite s n\u2080 : \u2115 hs : encard s = \u2191n\u2080 hle : n\u2080 \u2264 k \u22a2 encard s \u2264 \u2191k ** rwa [hs, Nat.cast_le] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux2 ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) ** have :\n  Tendsto (fun \u03b5 : \u211d\u22650 => \u03bc (f '' s) + 2 * \u03b5 * \u03bc s) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (f '' s) + 2 * (0 : \u211d\u22650) * \u03bc s)) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  refine' tendsto_const_nhds.add _\n  refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)\n  exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s this : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s)) \u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) ** simp only [add_zero, zero_mul, mul_zero, ENNReal.coe_zero] at this ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s this : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s))) \u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) ** apply ge_of_tendsto this ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s this : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s))) \u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,     \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191c * \u2191\u2191\u03bc s ** filter_upwards [self_mem_nhdsWithin] ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s this : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s))) \u22a2 \u2200 (a : \u211d\u22650),     a \u2208 Ioi 0 \u2192 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191a * \u2191\u2191\u03bc s ** rintro \u03b5 (\u03b5pos : 0 < \u03b5) ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s this : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s))) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s ** exact lintegral_abs_det_fderiv_le_addHaar_image_aux1 \u03bc hs hf' hf \u03b5pos ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s)) ** apply Tendsto.mono_left _ nhdsWithin_le_nhds ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s)) ** refine' tendsto_const_nhds.add _ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910 * \u2191\u2191\u03bc s)) ** refine' ENNReal.Tendsto.mul_const _ (Or.inr h's) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910)) ** exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top) ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.appendList_data ** \u03b1 : Type u_1 arr : Array \u03b1 l : List \u03b1 \u22a2 (arr ++ l).data = arr.data ++ l ** rw [\u2190 appendList_eq_append] ** \u03b1 : Type u_1 arr : Array \u03b1 l : List \u03b1 \u22a2 (Array.appendList arr l).data = arr.data ++ l ** unfold Array.appendList ** \u03b1 : Type u_1 arr : Array \u03b1 l : List \u03b1 \u22a2 (List.foldl (fun r v => push r v) arr l).data = arr.data ++ l ** induction l generalizing arr <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.pow_subset_pow ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s t : Finset \u03b1 a : \u03b1 m n : \u2115 hst : s \u2286 t \u22a2 s ^ 0 \u2286 t ^ 0 ** simp [pow_zero] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s t : Finset \u03b1 a : \u03b1 m n\u271d : \u2115 hst : s \u2286 t n : \u2115 \u22a2 s ^ (n + 1) \u2286 t ^ (n + 1) ** rw [pow_succ] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s t : Finset \u03b1 a : \u03b1 m n\u271d : \u2115 hst : s \u2286 t n : \u2115 \u22a2 s * s ^ n \u2286 t ^ (n + 1) ** exact mul_subset_mul hst (pow_subset_pow hst n) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_add_left' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' hT'' : DominatedFinMeasAdditive \u03bc T'' C'' h_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s f : \u03b1 \u2192 E \u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' hT'' : DominatedFinMeasAdditive \u03bc T'' C'' h_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s f : \u03b1 \u2192 E hf : Integrable f \u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f ** simp_rw [setToFun_eq _ hf, L1.setToL1_add_left' hT hT' hT'' h_add] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' hT'' : DominatedFinMeasAdditive \u03bc T'' C'' h_add : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T'' s = T s + T' s f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 setToFun \u03bc T'' hT'' f = setToFun \u03bc T hT f + setToFun \u03bc T' hT' f ** simp_rw [setToFun_undef _ hf, add_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.LocallyIntegrable.exists_nat_integrableOn ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2075 : MeasurableSpace X inst\u271d\u2074 : TopologicalSpace X inst\u271d\u00b3 : MeasurableSpace Y inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d : SecondCountableTopology X hf : LocallyIntegrable f \u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 \u22c3 n, u n = univ \u2227 \u2200 (n : \u2115), IntegrableOn f (u n) ** rcases (hf.locallyIntegrableOn univ).exists_nat_integrableOn with \u27e8u, u_open, u_union, hu\u27e9 ** case intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2075 : MeasurableSpace X inst\u271d\u2074 : TopologicalSpace X inst\u271d\u00b3 : MeasurableSpace Y inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d : SecondCountableTopology X hf : LocallyIntegrable f u : \u2115 \u2192 Set X u_open : \u2200 (n : \u2115), IsOpen (u n) u_union : univ \u2286 \u22c3 n, u n hu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 univ) \u22a2 \u2203 u, (\u2200 (n : \u2115), IsOpen (u n)) \u2227 \u22c3 n, u n = univ \u2227 \u2200 (n : \u2115), IntegrableOn f (u n) ** refine' \u27e8u, u_open, eq_univ_of_univ_subset u_union, fun n \u21a6 _\u27e9 ** case intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2075 : MeasurableSpace X inst\u271d\u2074 : TopologicalSpace X inst\u271d\u00b3 : MeasurableSpace Y inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d : SecondCountableTopology X hf : LocallyIntegrable f u : \u2115 \u2192 Set X u_open : \u2200 (n : \u2115), IsOpen (u n) u_union : univ \u2286 \u22c3 n, u n hu : \u2200 (n : \u2115), IntegrableOn f (u n \u2229 univ) n : \u2115 \u22a2 IntegrableOn f (u n) ** simpa only [inter_univ] using hu n ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.isUnit_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : DivisionMonoid \u03b1 s t : Finset \u03b1 \u22a2 IsUnit s \u2194 \u2203 a, s = {a} \u2227 IsUnit a ** constructor ** case mp F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : DivisionMonoid \u03b1 s t : Finset \u03b1 \u22a2 IsUnit s \u2192 \u2203 a, s = {a} \u2227 IsUnit a ** rintro \u27e8u, rfl\u27e9 ** case mp.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : DivisionMonoid \u03b1 t : Finset \u03b1 u : (Finset \u03b1)\u02e3 \u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a ** obtain \u27e8a, b, ha, hb, h\u27e9 := Finset.mul_eq_one_iff.1 u.mul_inv ** case mp.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : DivisionMonoid \u03b1 t : Finset \u03b1 u : (Finset \u03b1)\u02e3 a b : \u03b1 ha : \u2191u = {a} hb : \u2191u\u207b\u00b9 = {b} h : a * b = 1 \u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a ** refine' \u27e8a, ha, \u27e8a, b, h, singleton_injective _\u27e9, rfl\u27e9 ** case mp.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : DivisionMonoid \u03b1 t : Finset \u03b1 u : (Finset \u03b1)\u02e3 a b : \u03b1 ha : \u2191u = {a} hb : \u2191u\u207b\u00b9 = {b} h : a * b = 1 \u22a2 {b * a} = {1} ** rw [\u2190 singleton_mul_singleton, \u2190 ha, \u2190 hb] ** case mp.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : DivisionMonoid \u03b1 t : Finset \u03b1 u : (Finset \u03b1)\u02e3 a b : \u03b1 ha : \u2191u = {a} hb : \u2191u\u207b\u00b9 = {b} h : a * b = 1 \u22a2 \u2191u\u207b\u00b9 * \u2191u = {1} ** exact u.inv_mul ** case mpr F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : DivisionMonoid \u03b1 s t : Finset \u03b1 \u22a2 (\u2203 a, s = {a} \u2227 IsUnit a) \u2192 IsUnit s ** rintro \u27e8a, rfl, ha\u27e9 ** case mpr.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : DivisionMonoid \u03b1 t : Finset \u03b1 a : \u03b1 ha : IsUnit a \u22a2 IsUnit {a} ** exact ha.finset ** Qed",
        "informal": ""
    },
    {
        "formal": "PEquiv.trans_eq_none ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 g : \u03b2 \u2243. \u03b3 a : \u03b1 \u22a2 \u2191(PEquiv.trans f g) a = none \u2194 \u2200 (b : \u03b2) (c : \u03b3), \u00acb \u2208 \u2191f a \u2228 \u00acc \u2208 \u2191g b ** simp only [eq_none_iff_forall_not_mem, mem_trans, imp_iff_not_or.symm] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 g : \u03b2 \u2243. \u03b3 a : \u03b1 \u22a2 (\u2200 (a_1 : \u03b3), \u00ac\u2203 b, b \u2208 \u2191f a \u2227 a_1 \u2208 \u2191g b) \u2194 \u2200 (b : \u03b2) (c : \u03b3), b \u2208 \u2191f a \u2192 \u00acc \u2208 \u2191g b ** push_neg ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x f : \u03b1 \u2243. \u03b2 g : \u03b2 \u2243. \u03b3 a : \u03b1 \u22a2 (\u2200 (a_1 : \u03b3) (b : \u03b2), b \u2208 \u2191f a \u2192 \u00aca_1 \u2208 \u2191g b) \u2194 \u2200 (b : \u03b2) (c : \u03b3), b \u2208 \u2191f a \u2192 \u00acc \u2208 \u2191g b ** exact forall_swap ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto ** \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F ** rcases L.eq_or_neBot with rfl | hne ** case inr \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F ** apply ENNReal.le_of_forall_pos_le_add ** case inr.h \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u2191\u03bc F < \u22a4 \u2192 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** intro \u03b5 \u03b5_pos _ ** case inr.h \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** let \u03b4s := fun n : \u2115 => (1 : \u211d) / (n + 1) ** case inr.h \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** have \u03b4s_pos : \u2200 n, 0 < \u03b4s n := fun n => Nat.one_div_pos_of_nat ** case inr.h \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** have \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) := tendsto_one_div_add_atTop_nhds_0_nat ** case inr.h \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** have key\u2081 :=\n  tendsto_lintegral_thickenedIndicator_of_isClosed (\u03bc : Measure \u03a9) F_closed \u03b4s_pos \u03b4s_lim ** case inr.h \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** have room\u2081 : (\u03bc : Measure \u03a9) F < (\u03bc : Measure \u03a9) F + \u03b5 / 2 := by\n  apply\n    ENNReal.lt_add_right (measure_lt_top (\u03bc : Measure \u03a9) F).ne\n      (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, ENNReal.two_ne_top\u27e9).ne.symm ** case inr.h \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** rcases eventually_atTop.mp (eventually_lt_of_tendsto_lt room\u2081 key\u2081) with \u27e8M, hM\u27e9 ** case inr.h.intro \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** have key\u2082 :=\n  FiniteMeasure.tendsto_iff_forall_lintegral_tendsto.mp \u03bcs_lim (thickenedIndicator (\u03b4s_pos M) F) ** case inr.h.intro \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 key\u2082 :   Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L     (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc)) \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** have room\u2082 :\n  (lintegral (\u03bc : Measure \u03a9) fun a => thickenedIndicator (\u03b4s_pos M) F a) <\n    (lintegral (\u03bc : Measure \u03a9) fun a => thickenedIndicator (\u03b4s_pos M) F a) + \u03b5 / 2 := by\n  apply ENNReal.lt_add_right (ne_of_lt ?_)\n      (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, ENNReal.two_ne_top\u27e9).ne.symm\n  apply BoundedContinuousFunction.lintegral_lt_top_of_nnreal ** case inr.h.intro \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 key\u2082 :   Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L     (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc)) room\u2082 :   \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <     \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** have ev_near := Eventually.mono (eventually_lt_of_tendsto_lt room\u2082 key\u2082) fun n => le_of_lt ** case inr.h.intro \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 key\u2082 :   Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L     (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc)) room\u2082 :   \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <     \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ev_near :   \u2200\u1da0 (x : \u03b9) in L,     \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264       \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** have ev_near' := Eventually.mono ev_near fun n => le_trans\n  (measure_le_lintegral_thickenedIndicator (\u03bcs n : Measure \u03a9) F_closed.measurableSet (\u03b4s_pos M)) ** case inr.h.intro \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 key\u2082 :   Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L     (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc)) room\u2082 :   \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <     \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ev_near :   \u2200\u1da0 (x : \u03b9) in L,     \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264       \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** apply (Filter.limsup_le_limsup ev_near').trans ** case inr.h.intro \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 key\u2082 :   Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L     (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc)) room\u2082 :   \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <     \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ev_near :   \u2200\u1da0 (x : \u03b9) in L,     \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264       \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 \u22a2 limsup (fun x => \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2) L \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** rw [limsup_const] ** case inr.h.intro \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 key\u2082 :   Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L     (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc)) room\u2082 :   \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <     \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ev_near :   \u2200\u1da0 (x : \u03b9) in L,     \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264       \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 \u22a2 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** apply le_trans (add_le_add (hM M rfl.le).le (le_refl (\u03b5 / 2 : \u211d\u22650\u221e))) ** case inr.h.intro \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 key\u2082 :   Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L     (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc)) room\u2082 :   \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <     \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ev_near :   \u2200\u1da0 (x : \u03b9) in L,     \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs x) \u2264       \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ev_near' : \u2200\u1da0 (x : \u03b9) in L, \u2191\u2191\u2191(\u03bcs x) F \u2264 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 \u22a2 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 + \u2191\u03b5 / 2 \u2264 \u2191\u2191\u2191\u03bc F + \u2191\u03b5 ** simp only [add_assoc, ENNReal.add_halves, le_refl] ** case inl \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 F : Set \u03a9 F_closed : IsClosed F \u03bcs_lim : Tendsto \u03bcs \u22a5 (\ud835\udcdd \u03bc) \u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) \u22a5 \u2264 \u2191\u2191\u2191\u03bc F ** simp only [limsup_bot, bot_le] ** \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) \u22a2 \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 ** apply\n  ENNReal.lt_add_right (measure_lt_top (\u03bc : Measure \u03a9) F).ne\n    (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, ENNReal.two_ne_top\u27e9).ne.symm ** \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 key\u2082 :   Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L     (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc)) \u22a2 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc <     \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc + \u2191\u03b5 / 2 ** apply ENNReal.lt_add_right (ne_of_lt ?_)\n    (ENNReal.div_pos_iff.mpr \u27e8(ENNReal.coe_pos.mpr \u03b5_pos).ne.symm, ENNReal.two_ne_top\u27e9).ne.symm ** \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bcs : \u03b9 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) F : Set \u03a9 F_closed : IsClosed F hne : NeBot L \u03b5 : \u211d\u22650 \u03b5_pos : 0 < \u03b5 a\u271d : \u2191\u2191\u2191\u03bc F < \u22a4 \u03b4s : \u2115 \u2192 \u211d := fun n => 1 / (\u2191n + 1) \u03b4s_pos : \u2200 (n : \u2115), 0 < \u03b4s n \u03b4s_lim : Tendsto \u03b4s atTop (\ud835\udcdd 0) key\u2081 : Tendsto (fun n => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s n) F) \u03c9) \u2202\u2191\u03bc) atTop (\ud835\udcdd (\u2191\u2191\u2191\u03bc F)) room\u2081 : \u2191\u2191\u2191\u03bc F < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 M : \u2115 hM : \u2200 (b : \u2115), b \u2265 M \u2192 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s b) F) \u03c9) \u2202\u2191\u03bc < \u2191\u2191\u2191\u03bc F + \u2191\u03b5 / 2 key\u2082 :   Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191(\u03bcs i)) L     (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) x) \u2202\u2191\u03bc)) \u22a2 \u222b\u207b (a : \u03a9), \u2191(\u2191(thickenedIndicator (_ : 0 < \u03b4s M) F) a) \u2202\u2191\u03bc < \u22a4 ** apply BoundedContinuousFunction.lintegral_lt_top_of_nnreal ** Qed",
        "informal": ""
    },
    {
        "formal": "Finsupp.card_pi ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f\u271d : \u03b9 \u2192\u2080 \u03b1 f : \u03b9 \u2192\u2080 Finset \u03b1 \u22a2 card (pi f) = prod f fun i => \u2191(card (\u2191f i)) ** rw [pi, card_finsupp] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f\u271d : \u03b9 \u2192\u2080 \u03b1 f : \u03b9 \u2192\u2080 Finset \u03b1 \u22a2 \u220f i in f.support, card (\u2191f i) = prod f fun i => \u2191(card (\u2191f i)) ** exact Finset.prod_congr rfl fun i _ => by simp only [Pi.nat_apply, Nat.cast_id] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f\u271d : \u03b9 \u2192\u2080 \u03b1 f : \u03b9 \u2192\u2080 Finset \u03b1 i : \u03b9 x\u271d : i \u2208 f.support \u22a2 card (\u2191f i) = (fun i => \u2191(card (\u2191f i))) i (\u2191f i) ** simp only [Pi.nat_apply, Nat.cast_id] ** Qed",
        "informal": ""
    },
    {
        "formal": "Setoid.sSup_def ** \u03b1 : Type u_1 \u03b2 : Type u_2 s : Set (Setoid \u03b1) \u22a2 sSup s = EqvGen.Setoid (sSup (Rel '' s)) ** rw [sSup_eq_eqvGen, sSup_image] ** \u03b1 : Type u_1 \u03b2 : Type u_2 s : Set (Setoid \u03b1) \u22a2 (EqvGen.Setoid fun x y => \u2203 r, r \u2208 s \u2227 Rel r x y) = EqvGen.Setoid (\u2a06 a \u2208 s, Rel a) ** congr with (x y) ** case e_r.h.h.a \u03b1 : Type u_1 \u03b2 : Type u_2 s : Set (Setoid \u03b1) x y : \u03b1 \u22a2 (\u2203 r, r \u2208 s \u2227 Rel r x y) \u2194 iSup (fun a => \u2a06 (_ : a \u2208 s), Rel a) x y ** simp only [iSup_apply, iSup_Prop_eq, exists_prop] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.snorm_lim_le_liminf_snorm ** \u03b1 : Type u_1 E\u271d : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G E : Type u_5 inst\u271d : NormedAddCommGroup E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc f_lim : \u03b1 \u2192 E h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) \u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop ** obtain rfl|hp0 := eq_or_ne p 0 ** case inr \u03b1 : Type u_1 E\u271d : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G E : Type u_5 inst\u271d : NormedAddCommGroup E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc f_lim : \u03b1 \u2192 E h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) hp0 : p \u2260 0 \u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop ** by_cases hp_top : p = \u221e ** case neg \u03b1 : Type u_1 E\u271d : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G E : Type u_5 inst\u271d : NormedAddCommGroup E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc f_lim : \u03b1 \u2192 E h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 \u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop ** simp_rw [snorm_eq_snorm' hp0 hp_top] ** case neg \u03b1 : Type u_1 E\u271d : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G E : Type u_5 inst\u271d : NormedAddCommGroup E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc f_lim : \u03b1 \u2192 E h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 \u22a2 snorm' f_lim (ENNReal.toReal p) \u03bc \u2264 liminf (fun n => snorm' (f n) (ENNReal.toReal p) \u03bc) atTop ** have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_top ** case neg \u03b1 : Type u_1 E\u271d : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G E : Type u_5 inst\u271d : NormedAddCommGroup E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc f_lim : \u03b1 \u2192 E h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) hp0 : p \u2260 0 hp_top : \u00acp = \u22a4 hp_pos : 0 < ENNReal.toReal p \u22a2 snorm' f_lim (ENNReal.toReal p) \u03bc \u2264 liminf (fun n => snorm' (f n) (ENNReal.toReal p) \u03bc) atTop ** exact snorm'_lim_le_liminf_snorm' hp_pos hf h_lim ** case inl \u03b1 : Type u_1 E\u271d : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G E : Type u_5 inst\u271d : NormedAddCommGroup E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc f_lim : \u03b1 \u2192 E h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) \u22a2 snorm f_lim 0 \u03bc \u2264 liminf (fun n => snorm (f n) 0 \u03bc) atTop ** simp ** case pos \u03b1 : Type u_1 E\u271d : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G E : Type u_5 inst\u271d : NormedAddCommGroup E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc f_lim : \u03b1 \u2192 E h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) hp0 : p \u2260 0 hp_top : p = \u22a4 \u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop ** simp_rw [hp_top] ** case pos \u03b1 : Type u_1 E\u271d : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E\u271d inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G E : Type u_5 inst\u271d : NormedAddCommGroup E f : \u2115 \u2192 \u03b1 \u2192 E hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc f_lim : \u03b1 \u2192 E h_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x)) hp0 : p \u2260 0 hp_top : p = \u22a4 \u22a2 snorm f_lim \u22a4 \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop ** exact snorm_exponent_top_lim_le_liminf_snorm_exponent_top h_lim ** Qed",
        "informal": ""
    },
    {
        "formal": "List.mapM_cons ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 a : \u03b1 l : List \u03b1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 m \u03b2 \u22a2 mapM f (a :: l) = do     let __do_lift \u2190 f a     let __do_lift_1 \u2190 mapM f l     pure (__do_lift :: __do_lift_1) ** simp [\u2190 mapM'_eq_mapM, mapM'] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.finset_sum_apply' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 I : Finset \u03b9 \u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } a : \u03b1 s : Set \u03b2 \u22a2 \u2191\u2191(\u2191(\u2211 i in I, \u03ba i) a) s = \u2211 i in I, \u2191\u2191(\u2191(\u03ba i) a) s ** rw [finset_sum_apply, Measure.finset_sum_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.mem_insert_self ** \u03b1 : Type u_1 c : RBColor n : Nat v : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t : RBNode \u03b1 ht : Balanced t c n \u22a2 v \u2208 insert cmp t v ** rw [\u2190 mem_toList, List.mem_iff_append] ** \u03b1 : Type u_1 c : RBColor n : Nat v : \u03b1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t : RBNode \u03b1 ht : Balanced t c n \u22a2 \u2203 s t_1, toList (insert cmp t v) = s ++ v :: t_1 ** exact match e : zoom (cmp v) t with\n| (nil, p) => let \u27e8_, _, _, h\u27e9 := exists_insert_toList_zoom_nil ht e; \u27e8_, _, h\u27e9\n| (node .., p) => let \u27e8_, _, _, h\u27e9 := exists_insert_toList_zoom_node ht e; \u27e8_, _, h\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "WithBot.image_coe_Ioi ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ioi a = Ioi \u2191a ** rw [\u2190 preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Ioi_subset_Ioi bot_le)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.mkRat_mul_mkRat ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** if z\u2081 : d\u2081 = 0 then simp [z\u2081] else if z\u2082 : d\u2082 = 0 then simp [z\u2082] else\nrw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_mul_normalize, normalize_eq_mkRat] ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat z\u2081 : d\u2081 = 0 \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** simp [z\u2081] ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat z\u2081 : \u00acd\u2081 = 0 \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** if z\u2082 : d\u2082 = 0 then simp [z\u2082] else\nrw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_mul_normalize, normalize_eq_mkRat] ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat z\u2081 : \u00acd\u2081 = 0 z\u2082 : d\u2082 = 0 \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** simp [z\u2082] ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat z\u2081 : \u00acd\u2081 = 0 z\u2082 : \u00acd\u2082 = 0 \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** rw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_mul_normalize, normalize_eq_mkRat] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_of_uncrossing_lt_top ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4 hf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4 h : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => |f n \u03c9| \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** rw [isBoundedUnder_le_abs] at h ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4 hf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4 h :   (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227     IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9 \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** refine' tendsto_of_no_upcrossings Rat.denseRange_cast _ h.1 h.2 ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4 hf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4 h :   (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227     IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9 \u22a2 \u2200 (a : \u211d),     a \u2208 Set.range Rat.cast \u2192       \u2200 (b : \u211d), b \u2208 Set.range Rat.cast \u2192 a < b \u2192 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9) ** intro a ha b hb hab ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4 hf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4 h :   (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227     IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9 a : \u211d ha : a \u2208 Set.range Rat.cast b : \u211d hb : b \u2208 Set.range Rat.cast hab : a < b \u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9) ** obtain \u27e8\u27e8a, rfl\u27e9, \u27e8b, rfl\u27e9\u27e9 := ha, hb ** case pos.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4 hf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4 h :   (IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => f n \u03c9) \u2227     IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => f n \u03c9 a b : \u211a hab : \u2191a < \u2191b \u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < \u2191a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, \u2191b < f n \u03c9) ** exact not_frequently_of_upcrossings_lt_top hab (hf\u2082 a b (Rat.cast_lt.1 hab)).ne ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4 hf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4 h : \u00acIsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => |f n \u03c9| \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** obtain \u27e8a, b, hab, h\u2081, h\u2082\u27e9 := ENNReal.exists_upcrossings_of_not_bounded_under hf\u2081.ne h ** case neg.intro.intro.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hf\u2081 : liminf (fun n => \u2191\u2016f n \u03c9\u2016\u208a) atTop < \u22a4 hf\u2082 : \u2200 (a b : \u211a), a < b \u2192 upcrossings (\u2191a) (\u2191b) f \u03c9 < \u22a4 h : \u00acIsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun n => |f n \u03c9| a b : \u211a hab : a < b h\u2081 : \u2203\u1da0 (i : \u2115) in atTop, f i \u03c9 < \u2191a h\u2082 : \u2203\u1da0 (i : \u2115) in atTop, \u2191b < f i \u03c9 \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** exact\n  False.elim ((hf\u2082 a b hab).ne (upcrossings_eq_top_of_frequently_lt (Rat.cast_lt.2 hab) h\u2081 h\u2082)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_indicator_const' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 0 hp : p \u2260 0 \u22a2 snorm (Set.indicator s fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** by_cases hp_top : p = \u221e ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 0 hp : p \u2260 0 hp_top : p = \u22a4 \u22a2 snorm (Set.indicator s fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** simp [hp_top, snormEssSup_indicator_const_eq s c h\u03bcs] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c\u271d : E f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc s : Set \u03b1 c : G hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 0 hp : p \u2260 0 hp_top : \u00acp = \u22a4 \u22a2 snorm (Set.indicator s fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** exact snorm_indicator_const hs hp hp_top ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.eq_rnDeriv ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bd s : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hs : s \u27c2\u2098 \u03bd hadd : \u03bc = s + withDensity \u03bd f \u22a2 f =\u1da0[ae \u03bd] rnDeriv \u03bc \u03bd ** refine' ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite hf (measurable_rnDeriv \u03bc \u03bd) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bd s : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hs : s \u27c2\u2098 \u03bd hadd : \u03bc = s + withDensity \u03bd f \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bd s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bd = \u222b\u207b (x : \u03b1) in s, rnDeriv \u03bc \u03bd x \u2202\u03bd ** intro a ha _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bd s : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hs : s \u27c2\u2098 \u03bd hadd : \u03bc = s + withDensity \u03bd f a : Set \u03b1 ha : MeasurableSet a a\u271d : \u2191\u2191\u03bd a < \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in a, f x \u2202\u03bd = \u222b\u207b (x : \u03b1) in a, rnDeriv \u03bc \u03bd x \u2202\u03bd ** calc\n  \u222b\u207b x : \u03b1 in a, f x \u2202\u03bd = \u03bd.withDensity f a := (withDensity_apply f ha).symm\n  _ = \u03bd.withDensity (\u03bc.rnDeriv \u03bd) a := by rw [eq_withDensity_rnDeriv hf hs hadd]\n  _ = \u222b\u207b x : \u03b1 in a, \u03bc.rnDeriv \u03bd x \u2202\u03bd := withDensity_apply _ ha ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bd s : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hs : s \u27c2\u2098 \u03bd hadd : \u03bc = s + withDensity \u03bd f a : Set \u03b1 ha : MeasurableSet a a\u271d : \u2191\u2191\u03bd a < \u22a4 \u22a2 \u2191\u2191(withDensity \u03bd f) a = \u2191\u2191(withDensity \u03bd (rnDeriv \u03bc \u03bd)) a ** rw [eq_withDensity_rnDeriv hf hs hadd] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.support_X_mul ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R s : \u03c3 p : MvPolynomial \u03c3 R \u22a2 \u2200 (y : R), 1 * y = 0 \u2194 y = 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.le_degrees_add ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : Multiset.Disjoint (degrees p) (degrees q) \u22a2 degrees p \u2264 degrees (p + q) ** apply Finset.sup_le ** case a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : Multiset.Disjoint (degrees p) (degrees q) \u22a2 \u2200 (b : \u03c3 \u2192\u2080 \u2115), b \u2208 support p \u2192 \u2191toMultiset b \u2264 degrees (p + q) ** intro d hd ** case a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : Multiset.Disjoint (degrees p) (degrees q) d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p \u22a2 \u2191toMultiset d \u2264 degrees (p + q) ** rw [Multiset.disjoint_iff_ne] at h ** case a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p \u22a2 \u2191toMultiset d \u2264 degrees (p + q) ** rw [Multiset.le_iff_count] ** case a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p \u22a2 \u2200 (a : \u03c3), Multiset.count a (\u2191toMultiset d) \u2264 Multiset.count a (degrees (p + q)) ** intro i ** case a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 \u22a2 Multiset.count i (\u2191toMultiset d) \u2264 Multiset.count i (degrees (p + q)) ** rw [degrees, Multiset.count_finset_sup] ** case a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 \u22a2 Multiset.count i (\u2191toMultiset d) \u2264 Finset.sup (support (p + q)) fun a => Multiset.count i (\u2191toMultiset a) ** simp only [Finsupp.count_toMultiset] ** case a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 \u22a2 \u2191d i \u2264 Finset.sup (support (p + q)) fun a => \u2191a i ** by_cases h0 : d = 0 ** case pos R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 h0 : d = 0 \u22a2 \u2191d i \u2264 Finset.sup (support (p + q)) fun a => \u2191a i ** simp only [h0, zero_le, Finsupp.zero_apply] ** case neg R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 h0 : \u00acd = 0 \u22a2 \u2191d i \u2264 Finset.sup (support (p + q)) fun a => \u2191a i ** refine' @Finset.le_sup _ _ _ _ (p + q).support (fun a => a i) d _ ** case neg R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 h0 : \u00acd = 0 \u22a2 d \u2208 support (p + q) ** rw [mem_support_iff, coeff_add] ** case neg R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 h0 : \u00acd = 0 \u22a2 coeff d p + coeff d q \u2260 0 ** suffices q.coeff d = 0 by rwa [this, add_zero, coeff, \u2190 Finsupp.mem_support_iff] ** case neg R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 h0 : \u00acd = 0 \u22a2 coeff d q = 0 ** rw [\u2190 Finsupp.support_eq_empty, \u2190 Ne.def, \u2190 Finset.nonempty_iff_ne_empty] at h0 ** case neg R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 h0 : Finset.Nonempty d.support \u22a2 coeff d q = 0 ** obtain \u27e8j, hj\u27e9 := h0 ** case neg.intro R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i j : \u03c3 hj : j \u2208 d.support \u22a2 coeff d q = 0 ** contrapose! h ** case neg.intro R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i j : \u03c3 hj : j \u2208 d.support h : coeff d q \u2260 0 \u22a2 \u2203 a, a \u2208 degrees p \u2227 \u2203 b, b \u2208 degrees q \u2227 a = b ** rw [mem_support_iff] at hd ** case neg.intro R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 hd : coeff d p \u2260 0 i j : \u03c3 hj : j \u2208 d.support h : coeff d q \u2260 0 \u22a2 \u2203 a, a \u2208 degrees p \u2227 \u2203 b, b \u2208 degrees q \u2227 a = b ** refine' \u27e8j, _, j, _, rfl\u27e9 ** case neg.intro.refine'_1 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 hd : coeff d p \u2260 0 i j : \u03c3 hj : j \u2208 d.support h : coeff d q \u2260 0 \u22a2 j \u2208 degrees p  case neg.intro.refine'_2 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 hd : coeff d p \u2260 0 i j : \u03c3 hj : j \u2208 d.support h : coeff d q \u2260 0 \u22a2 j \u2208 degrees q ** all_goals rw [mem_degrees]; refine' \u27e8d, _, hj\u27e9; assumption ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R h : \u2200 (a : \u03c3), a \u2208 degrees p \u2192 \u2200 (b : \u03c3), b \u2208 degrees q \u2192 a \u2260 b d : \u03c3 \u2192\u2080 \u2115 hd : d \u2208 support p i : \u03c3 h0 : \u00acd = 0 this : coeff d q = 0 \u22a2 coeff d p + coeff d q \u2260 0 ** rwa [this, add_zero, coeff, \u2190 Finsupp.mem_support_iff] ** case neg.intro.refine'_2 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 hd : coeff d p \u2260 0 i j : \u03c3 hj : j \u2208 d.support h : coeff d q \u2260 0 \u22a2 j \u2208 degrees q ** rw [mem_degrees] ** case neg.intro.refine'_2 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 hd : coeff d p \u2260 0 i j : \u03c3 hj : j \u2208 d.support h : coeff d q \u2260 0 \u22a2 \u2203 d, coeff d q \u2260 0 \u2227 j \u2208 d.support ** refine' \u27e8d, _, hj\u27e9 ** case neg.intro.refine'_2 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p\u271d q\u271d p q : MvPolynomial \u03c3 R d : \u03c3 \u2192\u2080 \u2115 hd : coeff d p \u2260 0 i j : \u03c3 hj : j \u2208 d.support h : coeff d q \u2260 0 \u22a2 coeff d q \u2260 0 ** assumption ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.ext_of_generateFrom_of_cover ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t \u22a2 \u03bc = \u03bd ** refine' ext_of_sUnion_eq_univ hc hU fun t ht => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T \u22a2 restrict \u03bc t = restrict \u03bd t ** ext1 u hu ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u \u22a2 \u2191\u2191(restrict \u03bc t) u = \u2191\u2191(restrict \u03bd t) u ** simp only [restrict_apply hu] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u \u22a2 \u2191\u2191\u03bc (u \u2229 t) = \u2191\u2191\u03bd (u \u2229 t) ** refine' induction_on_inter h_gen h_inter _ (ST_eq t ht) _ _ hu ** case h.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u \u22a2 \u2191\u2191\u03bc (\u2205 \u2229 t) = \u2191\u2191\u03bd (\u2205 \u2229 t) ** simp only [Set.empty_inter, measure_empty] ** case h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u \u22a2 \u2200 (t_1 : Set \u03b1), MeasurableSet t_1 \u2192 \u2191\u2191\u03bc (t_1 \u2229 t) = \u2191\u2191\u03bd (t_1 \u2229 t) \u2192 \u2191\u2191\u03bc (t_1\u1d9c \u2229 t) = \u2191\u2191\u03bd (t_1\u1d9c \u2229 t) ** intro v hv hvt ** case h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u v : Set \u03b1 hv : MeasurableSet v hvt : \u2191\u2191\u03bc (v \u2229 t) = \u2191\u2191\u03bd (v \u2229 t) \u22a2 \u2191\u2191\u03bc (v\u1d9c \u2229 t) = \u2191\u2191\u03bd (v\u1d9c \u2229 t) ** have := T_eq t ht ** case h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u v : Set \u03b1 hv : MeasurableSet v hvt : \u2191\u2191\u03bc (v \u2229 t) = \u2191\u2191\u03bd (v \u2229 t) this : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t \u22a2 \u2191\u2191\u03bc (v\u1d9c \u2229 t) = \u2191\u2191\u03bd (v\u1d9c \u2229 t) ** rw [Set.inter_comm] at hvt \u22a2 ** case h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u v : Set \u03b1 hv : MeasurableSet v hvt : \u2191\u2191\u03bc (t \u2229 v) = \u2191\u2191\u03bd (t \u2229 v) this : \u2191\u2191\u03bc t = \u2191\u2191\u03bd t \u22a2 \u2191\u2191\u03bc (t \u2229 v\u1d9c) = \u2191\u2191\u03bd (t \u2229 v\u1d9c) ** rwa [\u2190 measure_inter_add_diff t hv, \u2190 measure_inter_add_diff t hv, \u2190 hvt,\n  ENNReal.add_right_inj] at this ** case h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u v : Set \u03b1 hv : MeasurableSet v hvt : \u2191\u2191\u03bc (t \u2229 v) = \u2191\u2191\u03bd (t \u2229 v) this : \u2191\u2191\u03bc (t \u2229 v) + \u2191\u2191\u03bc (t \\ v) = \u2191\u2191\u03bc (t \u2229 v) + \u2191\u2191\u03bd (t \\ v) \u22a2 \u2191\u2191\u03bc (t \u2229 v) \u2260 \u22a4 ** exact ne_top_of_le_ne_top (htop t ht) (measure_mono <| Set.inter_subset_left _ _) ** case h.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u \u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),     Pairwise (Disjoint on f) \u2192       (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192         (\u2200 (i : \u2115), \u2191\u2191\u03bc (f i \u2229 t) = \u2191\u2191\u03bd (f i \u2229 t)) \u2192 \u2191\u2191\u03bc ((\u22c3 i, f i) \u2229 t) = \u2191\u2191\u03bd ((\u22c3 i, f i) \u2229 t) ** intro f hfd hfm h_eq ** case h.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u f : \u2115 \u2192 Set \u03b1 hfd : Pairwise (Disjoint on f) hfm : \u2200 (i : \u2115), MeasurableSet (f i) h_eq : \u2200 (i : \u2115), \u2191\u2191\u03bc (f i \u2229 t) = \u2191\u2191\u03bd (f i \u2229 t) \u22a2 \u2191\u2191\u03bc ((\u22c3 i, f i) \u2229 t) = \u2191\u2191\u03bd ((\u22c3 i, f i) \u2229 t) ** simp only [\u2190 restrict_apply (hfm _), \u2190 restrict_apply (MeasurableSet.iUnion hfm)] at h_eq \u22a2 ** case h.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t\u271d : Set \u03b1 S T : Set (Set \u03b1) h_gen : m0 = generateFrom S hc : Set.Countable T h_inter : IsPiSystem S hU : \u22c3\u2080 T = univ htop : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t \u2260 \u22a4 ST_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2200 (s : Set \u03b1), s \u2208 S \u2192 \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bd (s \u2229 t) T_eq : \u2200 (t : Set \u03b1), t \u2208 T \u2192 \u2191\u2191\u03bc t = \u2191\u2191\u03bd t t : Set \u03b1 ht : t \u2208 T u : Set \u03b1 hu : MeasurableSet u f : \u2115 \u2192 Set \u03b1 hfd : Pairwise (Disjoint on f) hfm : \u2200 (i : \u2115), MeasurableSet (f i) h_eq : \u2200 (i : \u2115), \u2191\u2191(restrict \u03bc t) (f i) = \u2191\u2191(restrict \u03bd t) (f i) \u22a2 \u2191\u2191(restrict \u03bc t) (\u22c3 i, f i) = \u2191\u2191(restrict \u03bd t) (\u22c3 i, f i) ** simp only [measure_iUnion hfd hfm, h_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rel.preimage_univ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : Rel \u03b1 \u03b2 \u22a2 preimage r Set.univ = dom r ** rw [preimage, image_univ, codom_inv] ** Qed",
        "informal": ""
    },
    {
        "formal": "Subtype.forall_set_subtype ** \u03b1 : Type u_1 t : Set \u03b1 p : Set \u03b1 \u2192 Prop \u22a2 (\u2200 (s : Set \u2191t), p (val '' s)) \u2194 \u2200 (s : Set \u03b1), s \u2286 t \u2192 p s ** rw [\u2190 forall_subset_range_iff, range_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.isFiniteMeasure_withDensity_ofReal ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192 \u211d hfi : HasFiniteIntegral f \u22a2 IsFiniteMeasure (Measure.withDensity \u03bc fun x => ENNReal.ofReal (f x)) ** refine' isFiniteMeasure_withDensity ((lintegral_mono fun x => _).trans_lt hfi).ne ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192 \u211d hfi : HasFiniteIntegral f x : \u03b1 \u22a2 ENNReal.ofReal (f x) \u2264 \u2191\u2016f x\u2016\u208a ** exact Real.ofReal_le_ennnorm (f x) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.cgf_const ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc c : \u211d \u22a2 cgf (fun x => c) \u03bc t = t * c ** simp only [cgf, mgf_const, log_exp] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.get_cons_addChar ** c : Char cs : List Char i : Pos \u22a2 get { data := c :: cs } (i + c) = get { data := cs } i ** simp [get, utf8GetAux, Pos.zero_ne_addChar, utf8GetAux_addChar_right_cancel] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.measurable_range_sup' ** M : Type u_1 inst\u271d\u00b3 : MeasurableSpace M \u03b1 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 M \u03b4 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : MeasurableSup\u2082 \u03b1 f : \u2115 \u2192 \u03b4 \u2192 \u03b1 n : \u2115 hf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k) \u22a2 Measurable (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) f) ** simp_rw [\u2190 Nat.lt_succ_iff] at hf ** M : Type u_1 inst\u271d\u00b3 : MeasurableSpace M \u03b1 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 M \u03b4 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : MeasurableSup\u2082 \u03b1 f : \u2115 \u2192 \u03b4 \u2192 \u03b1 n : \u2115 hf : \u2200 (k : \u2115), k < Nat.succ n \u2192 Measurable (f k) \u22a2 Measurable (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) f) ** refine' Finset.measurable_sup' _ _ ** M : Type u_1 inst\u271d\u00b3 : MeasurableSpace M \u03b1 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 M \u03b4 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : MeasurableSup\u2082 \u03b1 f : \u2115 \u2192 \u03b4 \u2192 \u03b1 n : \u2115 hf : \u2200 (k : \u2115), k < Nat.succ n \u2192 Measurable (f k) \u22a2 \u2200 (n_1 : \u2115), n_1 \u2208 range (n + 1) \u2192 Measurable (f n_1) ** simpa [Finset.mem_range] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.edist_indicatorConstLp_eq_nnnorm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 edist (indicatorConstLp p hs h\u03bcs c) (indicatorConstLp p ht h\u03bct c) =     \u2191\u2016indicatorConstLp p (_ : MeasurableSet (s \u2206 t)) (_ : \u2191\u2191\u03bc (s \u2206 t) \u2260 \u22a4) c\u2016\u208a ** unfold indicatorConstLp ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 edist (Mem\u2112p.toLp (indicator s fun x => c) (_ : Mem\u2112p (indicator s fun x => c) p))       (Mem\u2112p.toLp (indicator t fun x => c) (_ : Mem\u2112p (indicator t fun x => c) p)) =     \u2191\u2016Mem\u2112p.toLp (indicator (s \u2206 t) fun x => c) (_ : Mem\u2112p (indicator (s \u2206 t) fun x => c) p)\u2016\u208a ** rw [Lp.edist_toLp_toLp, snorm_indicator_sub_indicator, Lp.coe_nnnorm_toLp] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.JordanDecomposition.toSignedMeasure_injective ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 \u22a2 Injective toSignedMeasure ** intro j\u2081 j\u2082 hj ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 \u22a2 j\u2081 = j\u2082 ** obtain \u27e8S, hS\u2081, hS\u2082, hS\u2083, hS\u2084, hS\u2085\u27e9 := j\u2081.exists_compl_positive_negative ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 \u22a2 j\u2081 = j\u2082 ** obtain \u27e8T, hT\u2081, hT\u2082, hT\u2083, hT\u2084, hT\u2085\u27e9 := j\u2082.exists_compl_positive_negative ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2082) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2082) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 \u22a2 j\u2081 = j\u2082 ** rw [\u2190 hj] at hT\u2082 hT\u2083 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 \u22a2 j\u2081 = j\u2082 ** obtain \u27e8hST\u2081, -\u27e9 :=\n  of_symmDiff_compl_positive_negative hS\u2081.compl hT\u2081.compl \u27e8hS\u2083, (compl_compl S).symm \u25b8 hS\u2082\u27e9\n    \u27e8hT\u2083, (compl_compl T).symm \u25b8 hT\u2082\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 \u22a2 j\u2081 = j\u2082 ** refine' eq_of_posPart_eq_posPart _ hj ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 \u22a2 j\u2081.posPart = j\u2082.posPart ** ext1 i hi ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i h\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) h\u03bc\u2082 : ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c) \u22a2 \u2191\u2191j\u2081.posPart i = \u2191\u2191j\u2082.posPart i ** rw [\u2190 ENNReal.toReal_eq_toReal (measure_ne_top _ _) (measure_ne_top _ _), h\u03bc\u2081, h\u03bc\u2082, \u2190 hj] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i h\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) h\u03bc\u2082 : ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c) \u22a2 \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) = \u2191(toSignedMeasure j\u2081) (i \u2229 T\u1d9c) ** exact of_inter_eq_of_symmDiff_eq_zero_positive hS\u2081.compl hT\u2081.compl hi hS\u2083 hT\u2083 hST\u2081 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i \u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) ** rw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hS\u2081.compl),\n  show j\u2081.negPart (i \u2229 S\u1d9c) = 0 from\n    nonpos_iff_eq_zero.1 (hS\u2085 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n  ENNReal.zero_toReal, sub_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i \u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S\u1d9c)) ** conv_lhs => rw [\u2190 Set.inter_union_compl i S] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i \u22a2 ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S \u222a i \u2229 S\u1d9c)) = ENNReal.toReal (\u2191\u2191j\u2081.posPart (i \u2229 S\u1d9c)) ** rw [measure_union,\n  show j\u2081.posPart (i \u2229 S) = 0 from\n    nonpos_iff_eq_zero.1 (hS\u2084 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n  zero_add] ** case hd \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i \u22a2 Disjoint (i \u2229 S) (i \u2229 S\u1d9c) ** refine'\n  Set.disjoint_of_subset_left (Set.inter_subset_right _ _)\n    (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) disjoint_compl_right) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i \u22a2 MeasurableSet (i \u2229 S\u1d9c) ** exact hi.inter hS\u2081.compl ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i h\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) \u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = \u2191(toSignedMeasure j\u2082) (i \u2229 T\u1d9c) ** rw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hT\u2081.compl),\n  show j\u2082.negPart (i \u2229 T\u1d9c) = 0 from\n    nonpos_iff_eq_zero.1 (hT\u2085 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n  ENNReal.zero_toReal, sub_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i h\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) \u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart i) = ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T\u1d9c)) ** conv_lhs => rw [\u2190 Set.inter_union_compl i T] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i h\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) \u22a2 ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T \u222a i \u2229 T\u1d9c)) = ENNReal.toReal (\u2191\u2191j\u2082.posPart (i \u2229 T\u1d9c)) ** rw [measure_union,\n  show j\u2082.posPart (i \u2229 T) = 0 from\n    nonpos_iff_eq_zero.1 (hT\u2084 \u25b8 measure_mono (Set.inter_subset_right _ _)),\n  zero_add] ** case hd \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i h\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) \u22a2 Disjoint (i \u2229 T) (i \u2229 T\u1d9c) ** exact\n  Set.disjoint_of_subset_left (Set.inter_subset_right _ _)\n    (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) disjoint_compl_right) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 S : Set \u03b1 hS\u2081 : MeasurableSet S hS\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) S \u2264 VectorMeasure.restrict 0 S hS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) S\u1d9c hS\u2084 : \u2191\u2191j\u2081.posPart S = 0 hS\u2085 : \u2191\u2191j\u2081.negPart S\u1d9c = 0 T : Set \u03b1 hT\u2081 : MeasurableSet T hT\u2082 : VectorMeasure.restrict (toSignedMeasure j\u2081) T \u2264 VectorMeasure.restrict 0 T hT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict (toSignedMeasure j\u2081) T\u1d9c hT\u2084 : \u2191\u2191j\u2082.posPart T = 0 hT\u2085 : \u2191\u2191j\u2082.negPart T\u1d9c = 0 hST\u2081 : \u2191(toSignedMeasure j\u2081) (S\u1d9c \u2206 T\u1d9c) = 0 i : Set \u03b1 hi : MeasurableSet i h\u03bc\u2081 : ENNReal.toReal (\u2191\u2191j\u2081.posPart i) = \u2191(toSignedMeasure j\u2081) (i \u2229 S\u1d9c) \u22a2 MeasurableSet (i \u2229 T\u1d9c) ** exact hi.inter hT\u2081.compl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 h : t =\u1d50[\u03bc] univ \u22a2 s \u222a t =\u1d50[\u03bc] univ ** convert ae_eq_set_union (ae_eq_refl s) h ** case h.e'_5 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 h : t =\u1d50[\u03bc] univ \u22a2 univ = s \u222a univ ** rw [union_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mem_\u21121_toReal_of_lintegral_ne_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192 \u211d\u22650\u221e hfm : AEMeasurable f hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u22a2 Mem\u2112p (fun x => ENNReal.toReal (f x)) 1 ** rw [Mem\u2112p, snorm_one_eq_lintegral_nnnorm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192 \u211d\u22650\u221e hfm : AEMeasurable f hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (f x)) \u03bc \u2227 \u222b\u207b (x : \u03b1), \u2191\u2016ENNReal.toReal (f x)\u2016\u208a \u2202\u03bc < \u22a4 ** exact\n  \u27e8(AEMeasurable.ennreal_toReal hfm).aestronglyMeasurable,\n    hasFiniteIntegral_toReal_of_lintegral_ne_top hfi\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.integral_eq_integral ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192\u209b E hfi : Integrable \u2191f \u22a2 integral \u03bc f = \u222b (x : \u03b1), \u2191f x \u2202\u03bc ** rw [MeasureTheory.integral_eq f hfi, \u2190 L1.SimpleFunc.toLp_one_eq_toL1,\n  L1.SimpleFunc.integral_L1_eq_integral, L1.SimpleFunc.integral_eq_integral] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192\u209b E hfi : Integrable \u2191f \u22a2 integral \u03bc f = integral \u03bc (Lp.simpleFunc.toSimpleFunc (Lp.simpleFunc.toLp f (_ : Mem\u2112p (\u2191f) 1))) ** exact SimpleFunc.integral_congr hfi (Lp.simpleFunc.toSimpleFunc_toLp _ _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "PEquiv.ofSet_eq_refl ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x s\u271d : Set \u03b1 inst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d s : Set \u03b1 inst\u271d : DecidablePred fun x => x \u2208 s h : ofSet s = PEquiv.refl \u03b1 \u22a2 s = Set.univ ** rw [Set.eq_univ_iff_forall] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x s\u271d : Set \u03b1 inst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d s : Set \u03b1 inst\u271d : DecidablePred fun x => x \u2208 s h : ofSet s = PEquiv.refl \u03b1 \u22a2 \u2200 (x : \u03b1), x \u2208 s ** intro ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x s\u271d : Set \u03b1 inst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d s : Set \u03b1 inst\u271d : DecidablePred fun x => x \u2208 s h : ofSet s = PEquiv.refl \u03b1 x\u271d : \u03b1 \u22a2 x\u271d \u2208 s ** rw [\u2190 mem_ofSet_self_iff, h] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x s\u271d : Set \u03b1 inst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d s : Set \u03b1 inst\u271d : DecidablePred fun x => x \u2208 s h : ofSet s = PEquiv.refl \u03b1 x\u271d : \u03b1 \u22a2 x\u271d \u2208 \u2191(PEquiv.refl \u03b1) x\u271d ** exact rfl ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type x s\u271d : Set \u03b1 inst\u271d\u00b9 : DecidablePred fun x => x \u2208 s\u271d s : Set \u03b1 inst\u271d : DecidablePred fun x => x \u2208 s h : s = Set.univ \u22a2 ofSet s = PEquiv.refl \u03b1 ** simp only [\u2190 ofSet_univ, h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.mul_div_cancel ** a b : Int H\u271d : b \u2260 0 a\u271d b\u271d : Nat H : \u2191b\u271d \u2260 0 \u22a2 div (\u2191a\u271d * \u2191b\u271d) \u2191b\u271d = \u2191a\u271d ** rw [\u2190 ofNat_mul, \u2190 ofNat_div,\n  Nat.mul_div_cancel _ <| Nat.pos_of_ne_zero <| Int.ofNat_ne_zero.1 H] ** a\u271d b\u271d : Int this : \u2200 {a b : Nat}, \u2191b \u2260 0 \u2192 div (\u2191a * \u2191b) \u2191b = \u2191a a b : Nat H : -\u2191b \u2260 0 \u22a2 div (\u2191a * -\u2191b) (-\u2191b) = \u2191a ** rw [Int.mul_neg, Int.neg_div, Int.div_neg, Int.neg_neg,\n  this (Int.neg_ne_zero.1 H)] ** a\u271d b\u271d : Int this : \u2200 {a b : Nat}, \u2191b \u2260 0 \u2192 div (\u2191a * \u2191b) \u2191b = \u2191a a b : Nat H : \u2191b \u2260 0 \u22a2 div (-\u2191a * \u2191b) \u2191b = -\u2191a ** rw [Int.neg_mul, Int.neg_div, this H] ** a\u271d b\u271d : Int this : \u2200 {a b : Nat}, \u2191b \u2260 0 \u2192 div (\u2191a * \u2191b) \u2191b = \u2191a a b : Nat H : -\u2191b \u2260 0 \u22a2 div (-\u2191a * -\u2191b) (-\u2191b) = -\u2191a ** rw [Int.neg_mul_neg, Int.div_neg, this (Int.neg_ne_zero.1 H)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.expand_one_apply ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S f : MvPolynomial \u03c3 R \u22a2 \u2191(expand 1) f = f ** simp only [expand, pow_one, eval\u2082Hom_eq_bind\u2082, bind\u2082_C_left, RingHom.toMonoidHom_eq_coe,\n  RingHom.coe_monoidHom_id, AlgHom.coe_mk, RingHom.coe_mk, MonoidHom.id_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "WithTop.image_coe_Ici ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ici a = Ico \u2191a \u22a4 ** rw [\u2190 preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toOuterMeasure_caratheodory ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 \u22a2 OuterMeasure.caratheodory (toOuterMeasure p) = \u22a4 ** refine' eq_top_iff.2 <| le_trans (le_sInf fun x hx => _) (le_sum_caratheodory _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 x : MeasurableSpace \u03b1 hx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i) \u22a2 \u22a4 \u2264 x ** have \u27e8y, hy\u27e9 := hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s t : Set \u03b1 x : MeasurableSpace \u03b1 hx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i) y : \u03b1 hy : (fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)) y = x \u22a2 \u22a4 \u2264 x ** exact\n  ((le_of_eq (dirac_caratheodory y).symm).trans (le_smul_caratheodory _ _)).trans (le_of_eq hy) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_boolIndicator ** \u03b1 : Type u_1 s : Set \u03b1 t : Set Bool \u22a2 boolIndicator s \u207b\u00b9' t = univ \u2228 boolIndicator s \u207b\u00b9' t = s \u2228 boolIndicator s \u207b\u00b9' t = s\u1d9c \u2228 boolIndicator s \u207b\u00b9' t = \u2205 ** simp only [preimage_boolIndicator_eq_union] ** \u03b1 : Type u_1 s : Set \u03b1 t : Set Bool \u22a2 ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = univ \u2228     ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = s \u2228       ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = s\u1d9c \u2228         ((if true \u2208 t then s else \u2205) \u222a if false \u2208 t then s\u1d9c else \u2205) = \u2205 ** split_ifs <;> simp [s.union_compl_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurableSet_range_of_continuous_injective ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f \u22a2 MeasurableSet (range f) ** letI := upgradePolishSpace \u03b3 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 \u22a2 MeasurableSet (range f) ** obtain \u27e8b, b_count, b_nonempty, hb\u27e9 :\n  \u2203 b : Set (Set \u03b3), b.Countable \u2227 \u2205 \u2209 b \u2227 IsTopologicalBasis b := exists_countable_basis \u03b3 ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b \u22a2 MeasurableSet (range f) ** haveI : Encodable b := b_count.toEncodable ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b \u22a2 MeasurableSet (range f) ** let A := { p : b \u00d7 b // Disjoint (p.1 : Set \u03b3) p.2 } ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } \u22a2 MeasurableSet (range f) ** have : \u2200 p : A, \u2203 q : Set \u03b2,\n    f '' (p.1.1 : Set \u03b3) \u2286 q \u2227 Disjoint (f '' (p.1.2 : Set \u03b3)) q \u2227 MeasurableSet q := by\n  intro p\n  apply\n    AnalyticSet.measurablySeparable ((hb.isOpen p.1.1.2).analyticSet_image f_cont)\n      ((hb.isOpen p.1.2.2).analyticSet_image f_cont)\n  exact Disjoint.image p.2 (f_inj.injOn univ) (subset_univ _) (subset_univ _) ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } this : \u2200 (p : A), \u2203 q, f '' \u2191(\u2191p).1 \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).2) q \u2227 MeasurableSet q \u22a2 MeasurableSet (range f) ** choose q hq1 hq2 q_meas using this ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) \u22a2 MeasurableSet (range f) ** let E : b \u2192 Set \u03b2 := fun s =>\n  closure (f '' s) \u2229 \u22c2 (t : b) (ht : Disjoint s.1 t.1), q \u27e8(s, t), ht\u27e9 \\ q \u27e8(t, s), ht.symm\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } \u22a2 MeasurableSet (range f) ** obtain \u27e8u, u_anti, u_pos, u_lim\u27e9 :\n  \u2203 u : \u2115 \u2192 \u211d, StrictAnti u \u2227 (\u2200 n : \u2115, 0 < u n) \u2227 Tendsto u atTop (\ud835\udcdd 0) :=\n  exists_seq_strictAnti_tendsto (0 : \u211d) ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) \u22a2 MeasurableSet (range f) ** let F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : b) (_ : IsBounded s.1 \u2227 diam s.1 \u2264 u n), E s ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s \u22a2 MeasurableSet (range f) ** suffices range f = \u22c2 n, F n by\n  have E_meas : \u2200 s : b, MeasurableSet (E s) := by\n    intro b\n    refine' isClosed_closure.measurableSet.inter _\n    refine' MeasurableSet.iInter fun s => _\n    exact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _)\n  have F_meas : \u2200 n, MeasurableSet (F n) := by\n    intro n\n    refine' MeasurableSet.iUnion fun s => _\n    exact MeasurableSet.iUnion fun _ => E_meas _\n  rw [this]\n  exact MeasurableSet.iInter fun n => F_meas n ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s \u22a2 range f = \u22c2 n, F n ** apply Subset.antisymm ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } \u22a2 \u2200 (p : A), \u2203 q, f '' \u2191(\u2191p).1 \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).2) q \u2227 MeasurableSet q ** intro p ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } p : A \u22a2 \u2203 q, f '' \u2191(\u2191p).1 \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).2) q \u2227 MeasurableSet q ** apply\n  AnalyticSet.measurablySeparable ((hb.isOpen p.1.1.2).analyticSet_image f_cont)\n    ((hb.isOpen p.1.2.2).analyticSet_image f_cont) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } p : A \u22a2 Disjoint (f '' \u2191(\u2191p).1) (f '' \u2191(\u2191p).2) ** exact Disjoint.image p.2 (f_inj.injOn univ) (subset_univ _) (subset_univ _) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n \u22a2 MeasurableSet (range f) ** have E_meas : \u2200 s : b, MeasurableSet (E s) := by\n  intro b\n  refine' isClosed_closure.measurableSet.inter _\n  refine' MeasurableSet.iInter fun s => _\n  exact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n E_meas : \u2200 (s : \u2191b), MeasurableSet (E s) \u22a2 MeasurableSet (range f) ** have F_meas : \u2200 n, MeasurableSet (F n) := by\n  intro n\n  refine' MeasurableSet.iUnion fun s => _\n  exact MeasurableSet.iUnion fun _ => E_meas _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n E_meas : \u2200 (s : \u2191b), MeasurableSet (E s) F_meas : \u2200 (n : \u2115), MeasurableSet (F n) \u22a2 MeasurableSet (range f) ** rw [this] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n E_meas : \u2200 (s : \u2191b), MeasurableSet (E s) F_meas : \u2200 (n : \u2115), MeasurableSet (F n) \u22a2 MeasurableSet (\u22c2 n, F n) ** exact MeasurableSet.iInter fun n => F_meas n ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n \u22a2 \u2200 (s : \u2191b), MeasurableSet (E s) ** intro b ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b\u271d : Set (Set \u03b3) b_count : Set.Countable b\u271d b_nonempty : \u00ac\u2205 \u2208 b\u271d hb : IsTopologicalBasis b\u271d this\u271d : Encodable \u2191b\u271d A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b\u271d \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n b : \u2191b\u271d \u22a2 MeasurableSet (E b) ** refine' isClosed_closure.measurableSet.inter _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b\u271d : Set (Set \u03b3) b_count : Set.Countable b\u271d b_nonempty : \u00ac\u2205 \u2208 b\u271d hb : IsTopologicalBasis b\u271d this\u271d : Encodable \u2191b\u271d A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b\u271d \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n b : \u2191b\u271d \u22a2 MeasurableSet     (\u22c2 t,       \u22c2 (ht : Disjoint \u2191b \u2191t),         q { val := (b, t), property := ht } \\ q { val := (t, b), property := (_ : Disjoint \u2191t \u2191b) }) ** refine' MeasurableSet.iInter fun s => _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b\u271d : Set (Set \u03b3) b_count : Set.Countable b\u271d b_nonempty : \u00ac\u2205 \u2208 b\u271d hb : IsTopologicalBasis b\u271d this\u271d : Encodable \u2191b\u271d A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b\u271d \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n b s : \u2191b\u271d \u22a2 MeasurableSet     (\u22c2 (ht : Disjoint \u2191b \u2191s),       q { val := (b, s), property := ht } \\ q { val := (s, b), property := (_ : Disjoint \u2191s \u2191b) }) ** exact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n E_meas : \u2200 (s : \u2191b), MeasurableSet (E s) \u22a2 \u2200 (n : \u2115), MeasurableSet (F n) ** intro n ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n E_meas : \u2200 (s : \u2191b), MeasurableSet (E s) n : \u2115 \u22a2 MeasurableSet (F n) ** refine' MeasurableSet.iUnion fun s => _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s this : range f = \u22c2 n, F n E_meas : \u2200 (s : \u2191b), MeasurableSet (E s) n : \u2115 s : \u2191b \u22a2 MeasurableSet (\u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s) ** exact MeasurableSet.iUnion fun _ => E_meas _ ** case intro.intro.intro.intro.intro.intro.h\u2081 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s \u22a2 range f \u2286 \u22c2 n, F n ** rintro x \u27e8y, rfl\u27e9 ** case intro.intro.intro.intro.intro.intro.h\u2081.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 \u22a2 f y \u2208 \u22c2 n, F n ** refine mem_iInter.2 fun n => ?_ ** case intro.intro.intro.intro.intro.intro.h\u2081.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 \u22a2 f y \u2208 F n ** obtain \u27e8s, sb, ys, hs\u27e9 : \u2203 (s : Set \u03b3), s \u2208 b \u2227 y \u2208 s \u2227 s \u2286 ball y (u n / 2) := by\n  apply hb.mem_nhds_iff.1\n  exact ball_mem_nhds _ (half_pos (u_pos n)) ** case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) \u22a2 f y \u2208 F n ** have diam_s : diam s \u2264 u n := by\n  apply (diam_mono hs isBounded_ball).trans\n  convert diam_ball (x := y) (half_pos (u_pos n)).le\n  ring ** case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) diam_s : diam s \u2264 u n \u22a2 f y \u2208 F n ** refine' mem_iUnion.2 \u27e8\u27e8s, sb\u27e9, _\u27e9 ** case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) diam_s : diam s \u2264 u n \u22a2 f y \u2208     \u22c3 (_ : Bornology.IsBounded \u2191{ val := s, property := sb } \u2227 diam \u2191{ val := s, property := sb } \u2264 u n),       E { val := s, property := sb } ** refine' mem_iUnion.2 \u27e8\u27e8isBounded_ball.subset hs, diam_s\u27e9, _\u27e9 ** case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) diam_s : diam s \u2264 u n \u22a2 f y \u2208 E { val := s, property := sb } ** apply mem_inter (subset_closure (mem_image_of_mem _ ys)) ** case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) diam_s : diam s \u2264 u n \u22a2 f y \u2208     \u22c2 t,       \u22c2 (ht : Disjoint \u2191{ val := s, property := sb } \u2191t),         q { val := ({ val := s, property := sb }, t), property := ht } \\           q { val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) } ** refine' mem_iInter.2 fun t => mem_iInter.2 fun ht => \u27e8_, _\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 \u22a2 \u2203 s, s \u2208 b \u2227 y \u2208 s \u2227 s \u2286 ball y (u n / 2) ** apply hb.mem_nhds_iff.1 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 \u22a2 ball y (u n / 2) \u2208 \ud835\udcdd y ** exact ball_mem_nhds _ (half_pos (u_pos n)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) \u22a2 diam s \u2264 u n ** apply (diam_mono hs isBounded_ball).trans ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) \u22a2 diam (ball y (u n / 2)) \u2264 u n ** convert diam_ball (x := y) (half_pos (u_pos n)).le ** case h.e'_4 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) \u22a2 u n = 2 * (u n / 2) ** ring ** case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) diam_s : diam s \u2264 u n t : \u2191b ht : Disjoint \u2191{ val := s, property := sb } \u2191t \u22a2 f y \u2208 q { val := ({ val := s, property := sb }, t), property := ht } ** apply hq1 ** case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_1.a \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) diam_s : diam s \u2264 u n t : \u2191b ht : Disjoint \u2191{ val := s, property := sb } \u2191t \u22a2 f y \u2208 f '' \u2191(\u2191{ val := ({ val := s, property := sb }, t), property := ht }).1 ** exact mem_image_of_mem _ ys ** case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) diam_s : diam s \u2264 u n t : \u2191b ht : Disjoint \u2191{ val := s, property := sb } \u2191t \u22a2 \u00acf y \u2208 q { val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) } ** apply disjoint_left.1 (hq2 \u27e8(t, \u27e8s, sb\u27e9), ht.symm\u27e9) ** case intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_2.a \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s y : \u03b3 n : \u2115 s : Set \u03b3 sb : s \u2208 b ys : y \u2208 s hs : s \u2286 ball y (u n / 2) diam_s : diam s \u2264 u n t : \u2191b ht : Disjoint \u2191{ val := s, property := sb } \u2191t \u22a2 f y \u2208     f '' \u2191(\u2191{ val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }).2 ** exact mem_image_of_mem _ ys ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s \u22a2 \u22c2 n, F n \u2286 range f ** intro x hx ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n \u22a2 x \u2208 range f ** have C1 : \u2200 n, \u2203 (s : b) (_ : IsBounded s.1 \u2227 diam s.1 \u2264 u n), x \u2208 E s := fun n => by\n  simpa only [mem_iUnion] using mem_iInter.1 hx n ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n C1 : \u2200 (n : \u2115), \u2203 s x_1, x \u2208 E s \u22a2 x \u2208 range f ** choose s hs hxs using C1 ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) \u22a2 x \u2208 range f ** have C2 : \u2200 n, (s n).1.Nonempty := by\n  intro n\n  rw [nonempty_iff_ne_empty]\n  intro hn\n  have := (s n).2\n  rw [hn] at this\n  exact b_nonempty this ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) C2 : \u2200 (n : \u2115), Set.Nonempty \u2191(s n) \u22a2 x \u2208 range f ** choose y hy using C2 ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) \u22a2 x \u2208 range f ** have I : \u2200 m n, ((s m).1 \u2229 (s n).1).Nonempty := by\n  intro m n\n  rw [\u2190 not_disjoint_iff_nonempty_inter]\n  by_contra' h\n  have A : x \u2208 q \u27e8(s m, s n), h\u27e9 \\ q \u27e8(s n, s m), h.symm\u27e9 :=\n    haveI := mem_iInter.1 (hxs m).2 (s n)\n    (mem_iInter.1 this h : _)\n  have B : x \u2208 q \u27e8(s n, s m), h.symm\u27e9 \\ q \u27e8(s m, s n), h\u27e9 :=\n    haveI := mem_iInter.1 (hxs n).2 (s m)\n    (mem_iInter.1 this h.symm : _)\n  exact A.2 B.1 ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) \u22a2 x \u2208 range f ** have cauchy_y : CauchySeq y := by\n  have : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) := by\n    simpa only [mul_zero] using u_lim.const_mul 2\n  refine cauchySeq_of_le_tendsto_0' (fun n => 2 * u n) (fun m n hmn => ?_) this\n  rcases I m n with \u27e8z, zsm, zsn\u27e9\n  calc\n    dist (y m) (y n) \u2264 dist (y m) z + dist z (y n) := dist_triangle _ _ _\n    _ \u2264 u m + u n :=\n      (add_le_add ((dist_le_diam_of_mem (hs m).1 (hy m) zsm).trans (hs m).2)\n        ((dist_le_diam_of_mem (hs n).1 zsn (hy n)).trans (hs n).2))\n    _ \u2264 2 * u m := by linarith [u_anti.antitone hmn] ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y \u22a2 x \u2208 range f ** haveI : Nonempty \u03b3 := \u27e8y 0\u27e9 ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 \u22a2 x \u2208 range f ** let z := limUnder atTop y ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y \u22a2 x \u2208 range f ** have y_lim : Tendsto y atTop (\ud835\udcdd z) := cauchy_y.tendsto_limUnder ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) \u22a2 x \u2208 range f ** suffices f z = x by\n  rw [\u2190 this]\n  exact mem_range_self _ ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) \u22a2 f z = x ** by_contra' hne ** case intro.intro.intro.intro.intro.intro.h\u2082 \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x \u22a2 False ** obtain \u27e8v, w, v_open, w_open, fzv, xw, hvw\u27e9 := t2_separation hne ** case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u22a2 False ** obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4 > (0 : \u211d), ball z \u03b4 \u2286 f \u207b\u00b9' v := by\n  apply Metric.mem_nhds_iff.1\n  exact f_cont.continuousAt.preimage_mem_nhds (v_open.mem_nhds fzv) ** case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v \u22a2 False ** obtain \u27e8n, hn\u27e9 : \u2203 n, u n + dist (y n) z < \u03b4 :=\n  haveI : Tendsto (fun n => u n + dist (y n) z) atTop (\ud835\udcdd 0) := by\n    simpa only [add_zero] using u_lim.add (tendsto_iff_dist_tendsto_zero.1 y_lim)\n  ((tendsto_order.1 this).2 _ \u03b4pos).exists ** case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v n : \u2115 hn : u n + dist (y n) z < \u03b4 \u22a2 False ** have fsnv : f '' s n \u2286 v := by\n  rw [image_subset_iff]\n  apply Subset.trans _ h\u03b4\n  intro a ha\n  calc\n    dist a z \u2264 dist a (y n) + dist (y n) z := dist_triangle _ _ _\n    _ \u2264 u n + dist (y n) z :=\n      (add_le_add_right ((dist_le_diam_of_mem (hs n).1 ha (hy n)).trans (hs n).2) _)\n    _ < \u03b4 := hn ** case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v n : \u2115 hn : u n + dist (y n) z < \u03b4 fsnv : f '' \u2191(s n) \u2286 v \u22a2 False ** have : x \u2208 closure v := closure_mono fsnv (hxs n).1 ** case intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d\u00b9 : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this\u271d : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v n : \u2115 hn : u n + dist (y n) z < \u03b4 fsnv : f '' \u2191(s n) \u2286 v this : x \u2208 closure v \u22a2 False ** exact disjoint_left.1 (hvw.closure_left w_open) this xw ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n n : \u2115 \u22a2 \u2203 s x_1, x \u2208 E s ** simpa only [mem_iUnion] using mem_iInter.1 hx n ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) \u22a2 \u2200 (n : \u2115), Set.Nonempty \u2191(s n) ** intro n ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) n : \u2115 \u22a2 Set.Nonempty \u2191(s n) ** rw [nonempty_iff_ne_empty] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) n : \u2115 \u22a2 \u2191(s n) \u2260 \u2205 ** intro hn ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) n : \u2115 hn : \u2191(s n) = \u2205 \u22a2 False ** have := (s n).2 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) n : \u2115 hn : \u2191(s n) = \u2205 this : \u2191(s n) \u2208 b \u22a2 False ** rw [hn] at this ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) n : \u2115 hn : \u2191(s n) = \u2205 this : \u2205 \u2208 b \u22a2 False ** exact b_nonempty this ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) \u22a2 \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) ** intro m n ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) m n : \u2115 \u22a2 Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) ** rw [\u2190 not_disjoint_iff_nonempty_inter] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) m n : \u2115 \u22a2 \u00acDisjoint \u2191(s m) \u2191(s n) ** by_contra' h ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) m n : \u2115 h : Disjoint \u2191(s m) \u2191(s n) \u22a2 False ** have A : x \u2208 q \u27e8(s m, s n), h\u27e9 \\ q \u27e8(s n, s m), h.symm\u27e9 :=\n  haveI := mem_iInter.1 (hxs m).2 (s n)\n  (mem_iInter.1 this h : _) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A\u271d : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A\u271d \u2192 Set \u03b2 hq1 : \u2200 (p : A\u271d), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A\u271d), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A\u271d), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) m n : \u2115 h : Disjoint \u2191(s m) \u2191(s n) A : x \u2208 q { val := (s m, s n), property := h } \\ q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) } \u22a2 False ** have B : x \u2208 q \u27e8(s n, s m), h.symm\u27e9 \\ q \u27e8(s m, s n), h\u27e9 :=\n  haveI := mem_iInter.1 (hxs n).2 (s m)\n  (mem_iInter.1 this h.symm : _) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A\u271d : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A\u271d \u2192 Set \u03b2 hq1 : \u2200 (p : A\u271d), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A\u271d), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A\u271d), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) m n : \u2115 h : Disjoint \u2191(s m) \u2191(s n) A : x \u2208 q { val := (s m, s n), property := h } \\ q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) } B : x \u2208 q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) } \\ q { val := (s m, s n), property := h } \u22a2 False ** exact A.2 B.1 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) \u22a2 CauchySeq y ** have : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) := by\n  simpa only [mul_zero] using u_lim.const_mul 2 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) this : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) \u22a2 CauchySeq y ** refine cauchySeq_of_le_tendsto_0' (fun n => 2 * u n) (fun m n hmn => ?_) this ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) this : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) m n : \u2115 hmn : m \u2264 n \u22a2 dist (y m) (y n) \u2264 (fun n => 2 * u n) m ** rcases I m n with \u27e8z, zsm, zsn\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) this : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) m n : \u2115 hmn : m \u2264 n z : \u03b3 zsm : z \u2208 \u2191(s m) zsn : z \u2208 \u2191(s n) \u22a2 dist (y m) (y n) \u2264 (fun n => 2 * u n) m ** calc\n  dist (y m) (y n) \u2264 dist (y m) z + dist z (y n) := dist_triangle _ _ _\n  _ \u2264 u m + u n :=\n    (add_le_add ((dist_le_diam_of_mem (hs m).1 (hy m) zsm).trans (hs m).2)\n      ((dist_le_diam_of_mem (hs n).1 zsn (hy n)).trans (hs n).2))\n  _ \u2264 2 * u m := by linarith [u_anti.antitone hmn] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) \u22a2 Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) ** simpa only [mul_zero] using u_lim.const_mul 2 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) this : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) m n : \u2115 hmn : m \u2264 n z : \u03b3 zsm : z \u2208 \u2191(s m) zsn : z \u2208 \u2191(s n) \u22a2 u m + u n \u2264 2 * u m ** linarith [u_anti.antitone hmn] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d\u00b9 : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this\u271d : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) this : f z = x \u22a2 x \u2208 range f ** rw [\u2190 this] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d\u00b9 : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this\u271d : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) this : f z = x \u22a2 f z \u2208 range f ** exact mem_range_self _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 ball z \u03b4 \u2286 f \u207b\u00b9' v ** apply Metric.mem_nhds_iff.1 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd z ** exact f_cont.continuousAt.preimage_mem_nhds (v_open.mem_nhds fzv) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v \u22a2 Tendsto (fun n => u n + dist (y n) z) atTop (\ud835\udcdd 0) ** simpa only [add_zero] using u_lim.add (tendsto_iff_dist_tendsto_zero.1 y_lim) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v n : \u2115 hn : u n + dist (y n) z < \u03b4 \u22a2 f '' \u2191(s n) \u2286 v ** rw [image_subset_iff] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v n : \u2115 hn : u n + dist (y n) z < \u03b4 \u22a2 \u2191(s n) \u2286 f \u207b\u00b9' v ** apply Subset.trans _ h\u03b4 ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v n : \u2115 hn : u n + dist (y n) z < \u03b4 \u22a2 \u2191(s n) \u2286 ball z \u03b4 ** intro a ha ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 \u03b2 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 f : \u03b3 \u2192 \u03b2 f_cont : Continuous f f_inj : Injective f this\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3 b : Set (Set \u03b3) b_count : Set.Countable b b_nonempty : \u00ac\u2205 \u2208 b hb : IsTopologicalBasis b this\u271d : Encodable \u2191b A : Type u_3 := { p // Disjoint \u2191p.1 \u2191p.2 } q : A \u2192 Set \u03b2 hq1 : \u2200 (p : A), f '' \u2191(\u2191p).1 \u2286 q p hq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).2) (q p) q_meas : \u2200 (p : A), MeasurableSet (q p) E : \u2191b \u2192 Set \u03b2 :=   fun s =>     closure (f '' \u2191s) \u2229       \u22c2 t,         \u22c2 (ht : Disjoint \u2191s \u2191t),           q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) } u : \u2115 \u2192 \u211d u_anti : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 s, \u22c3 (_ : Bornology.IsBounded \u2191s \u2227 diam \u2191s \u2264 u n), E s x : \u03b2 hx : x \u2208 \u22c2 n, F n s : \u2115 \u2192 \u2191b hs : \u2200 (n : \u2115), Bornology.IsBounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n hxs : \u2200 (n : \u2115), x \u2208 E (s n) y : \u2115 \u2192 \u03b3 hy : \u2200 (n : \u2115), y n \u2208 \u2191(s n) I : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n)) cauchy_y : CauchySeq y this : Nonempty \u03b3 z : \u03b3 := limUnder atTop y y_lim : Tendsto y atTop (\ud835\udcdd z) hne : f z \u2260 x v w : Set \u03b2 v_open : IsOpen v w_open : IsOpen w fzv : f z \u2208 v xw : x \u2208 w hvw : Disjoint v w \u03b4 : \u211d \u03b4pos : \u03b4 > 0 h\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v n : \u2115 hn : u n + dist (y n) z < \u03b4 a : \u03b3 ha : a \u2208 \u2191(s n) \u22a2 a \u2208 ball z \u03b4 ** calc\n  dist a z \u2264 dist a (y n) + dist (y n) z := dist_triangle _ _ _\n  _ \u2264 u n + dist (y n) z :=\n    (add_le_add_right ((dist_le_diam_of_mem (hs n).1 ha (hy n)).trans (hs n).2) _)\n  _ < \u03b4 := hn ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.LocallyIntegrable.integrable_smul_right_of_hasCompactSupport ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g \u22a2 Integrable fun x => f x \u2022 g x ** let K := tsupport g ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g \u22a2 Integrable fun x => f x \u2022 g x ** have hK : IsCompact K := h'g ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K \u22a2 Integrable fun x => f x \u2022 g x ** have : K.indicator (fun x \u21a6 f x \u2022 g x) = (fun x \u21a6 f x \u2022 g x) := by\n  apply indicator_eq_self.2\n  apply support_subset_iff'.2\n  intros x hx\n  simp [image_eq_zero_of_nmem_tsupport hx] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x \u22a2 Integrable fun x => f x \u2022 g x ** rw [\u2190 this, indicator_smul_left] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x \u22a2 Integrable fun x => Set.indicator K (fun x => f x) x \u2022 g x ** apply Integrable.smul_of_top_left ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K \u22a2 (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x ** apply indicator_eq_self.2 ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K \u22a2 (support fun x => f x \u2022 g x) \u2286 K ** apply support_subset_iff'.2 ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K \u22a2 \u2200 (x : X), \u00acx \u2208 K \u2192 f x \u2022 g x = 0 ** intros x hx ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K x : X hx : \u00acx \u2208 K \u22a2 f x \u2022 g x = 0 ** simp [image_eq_zero_of_nmem_tsupport hx] ** case h\u03c6 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x \u22a2 Integrable fun x => Set.indicator K (fun x => f x) x ** rw [integrable_indicator_iff hK.measurableSet] ** case h\u03c6 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x \u22a2 IntegrableOn (fun x => f x) K ** exact hf.integrableOn_isCompact hK ** case hf X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : TopologicalSpace X inst\u271d\u2075 : MeasurableSpace Y inst\u271d\u2074 : TopologicalSpace Y inst\u271d\u00b3 : NormedAddCommGroup E f\u271d g\u271d : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : OpensMeasurableSpace X inst\u271d : T2Space X f : X \u2192 \u211d hf : LocallyIntegrable f g : X \u2192 E hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => f x \u2022 g x) = fun x => f x \u2022 g x \u22a2 Mem\u2112p (fun x => g x) \u22a4 ** exact hg.mem\u2112p_top_of_hasCompactSupport h'g \u03bc ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.castNum_ldiff ** \u03b1 : Type u_1 \u22a2 \u2200 (m n : Num), \u2191(ldiff m n) = Nat.ldiff \u2191m \u2191n ** apply castNum_eq_bitwise PosNum.ldiff <;> intros <;> (try cases_type* Bool) <;> rfl ** case pbb \u03b1 : Type u_1 a\u271d b\u271d : Bool m\u271d n\u271d : PosNum \u22a2 PosNum.ldiff (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && !b\u271d) (PosNum.ldiff m\u271d n\u271d) ** try cases_type* Bool ** case pbb \u03b1 : Type u_1 a\u271d b\u271d : Bool m\u271d n\u271d : PosNum \u22a2 PosNum.ldiff (PosNum.bit a\u271d m\u271d) (PosNum.bit b\u271d n\u271d) = bit (a\u271d && !b\u271d) (PosNum.ldiff m\u271d n\u271d) ** cases_type* Bool ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.IndepFun.mgf_add ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d X Y : \u03a9 \u2192 \u211d h_indep : IndepFun X Y hX : AEStronglyMeasurable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc hY : AEStronglyMeasurable (fun \u03c9 => rexp (t * Y \u03c9)) \u03bc \u22a2 mgf (X + Y) \u03bc t = mgf X \u03bc t * mgf Y \u03bc t ** simp_rw [mgf, Pi.add_apply, mul_add, exp_add] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d X Y : \u03a9 \u2192 \u211d h_indep : IndepFun X Y hX : AEStronglyMeasurable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc hY : AEStronglyMeasurable (fun \u03c9 => rexp (t * Y \u03c9)) \u03bc \u22a2 \u222b (x : \u03a9), rexp (t * X x) * rexp (t * Y x) \u2202\u03bc = (\u222b (x : \u03a9), rexp (t * X x) \u2202\u03bc) * \u222b (x : \u03a9), rexp (t * Y x) \u2202\u03bc ** exact (h_indep.exp_mul t t).integral_mul hX hY ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.any ** l m r : List Char f : Char \u2192 Bool \u22a2 Substring.any       { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },         stopPos := { byteIdx := utf8Len l + utf8Len m } }       f =     List.any m f ** simp [-List.append_assoc, Substring.any, anyAux_of_valid] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.DList.ofList_toList ** \u03b1 : Type u l : DList \u03b1 \u22a2 ofList (toList l) = l ** cases' l with app inv ** case mk \u03b1 : Type u app : List \u03b1 \u2192 List \u03b1 inv : \u2200 (l : List \u03b1), app l = app [] ++ l \u22a2 ofList (toList { apply := app, invariant := inv }) = { apply := app, invariant := inv } ** simp only [ofList, toList, mk.injEq] ** case mk \u03b1 : Type u app : List \u03b1 \u2192 List \u03b1 inv : \u2200 (l : List \u03b1), app l = app [] ++ l \u22a2 (fun x => app [] ++ x) = app ** funext x ** case mk.h \u03b1 : Type u app : List \u03b1 \u2192 List \u03b1 inv : \u2200 (l : List \u03b1), app l = app [] ++ l x : List \u03b1 \u22a2 app [] ++ x = app x ** rw [(inv x)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.iSup_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m : OuterMeasure \u03b1 \u03b9 : Sort u_5 f : \u03b9 \u2192 OuterMeasure \u03b1 s : Set \u03b1 \u22a2 \u2191(\u2a06 i, f i) s = \u2a06 i, \u2191(f i) s ** rw [iSup, sSup_apply, iSup_range, iSup] ** Qed",
        "informal": ""
    },
    {
        "formal": "Partrec.merge ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 h : x \u2208 f a \u2228 x \u2208 g a \u22a2 x \u2208 k a ** have : (k a).Dom := (K _).2.2 (h.imp Exists.fst Exists.fst) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 h : x \u2208 f a \u2228 x \u2208 g a this : (k a).Dom \u22a2 x \u2208 k a ** refine' \u27e8this, _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 h : x \u2208 f a \u2228 x \u2208 g a this : (k a).Dom \u22a2 Part.get (k a) this = x ** cases' h with h h <;> cases' (K _).1 _ \u27e8this, rfl\u27e9 with h' h' ** case inl.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 this : (k a).Dom h : x \u2208 f a h' : Part.get (k a) this \u2208 f a \u22a2 Part.get (k a) this = x ** exact mem_unique h' h ** case inl.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 this : (k a).Dom h : x \u2208 f a h' : Part.get (k a) this \u2208 g a \u22a2 Part.get (k a) this = x ** exact (H _ _ h _ h').symm ** case inr.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 this : (k a).Dom h : x \u2208 g a h' : Part.get (k a) this \u2208 f a \u22a2 Part.get (k a) this = x ** exact H _ _ h' _ h ** case inr.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 this : (k a).Dom h : x \u2208 g a h' : Part.get (k a) this \u2208 g a \u22a2 Part.get (k a) this = x ** exact mem_unique h' h ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) \u22a2 \u2203 t,     Set.Countable t \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227         (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2 ** rcases HasBesicovitchCovering.no_satelliteConfig (\u03b1 := \u03b1) with \u27e8N, \u03c4, h\u03c4, hN\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) \u22a2 \u2203 t,     Set.Countable t \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227         (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2 ** let P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop := fun t =>\n  ((t : Set (\u03b1 \u00d7 \u211d)).PairwiseDisjoint fun p => closedBall p.1 p.2) \u2227\n    (\u2200 p : \u03b1 \u00d7 \u211d, p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 p : \u03b1 \u00d7 \u211d, p \u2208 t \u2192 p.2 \u2208 f p.1 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 this :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) \u22a2 \u2203 t,     Set.Countable t \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227         (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2 ** choose! F hF using this ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) \u22a2 \u2203 t,     Set.Countable t \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227         (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2 ** let u n := F^[n] \u2205 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 \u22a2 \u2203 t,     Set.Countable t \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227         (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2 ** have u_succ : \u2200 n : \u2115, u n.succ = F (u n) := fun n => by\n  simp only [Function.comp_apply, Function.iterate_succ'] ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) \u22a2 \u2203 t,     Set.Countable t \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227         (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1) \u2227           \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) = 0 \u2227 PairwiseDisjoint t fun p => closedBall p.1 p.2 ** refine' \u27e8\u22c3 n, u n, countable_iUnion fun n => (u n).countable_toSet, _, _, _, _\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 \u22a2 \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) ** intro t ht ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t \u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) ** set B := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.1 p.2 with hB ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 \u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ B) ** have B_closed : IsClosed B := isClosed_biUnion_finset fun i _ => isClosed_ball ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B \u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ B) ** set s' := s \\ B ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B this : \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) \u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' ** choose! r hr using this ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) \u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' ** obtain \u27e8v, vs', h\u03bcv, hv\u27e9 :\n  \u2203 v : Finset \u03b1,\n    \u2191v \u2286 s' \u2227\n      \u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 N / (N + 1) * \u03bc s' \u2227\n        (v : Set \u03b1).PairwiseDisjoint fun x : \u03b1 => closedBall x (r x) :=\n  haveI rI : \u2200 x \u2208 s', r x \u2208 Ioo (0 : \u211d) 1 := fun x hx => (hr x hx).1.2\n  exist_finset_disjoint_balls_large_measure \u03bc h\u03c4 hN s' r (fun x hx => (rI x hx).1) fun x hx =>\n    (rI x hx).2.le ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) \u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' ** refine' \u27e8t \u222a Finset.image (fun x => (x, r x)) v, Finset.subset_union_left _ _, \u27e8_, _, _\u27e9, _\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B \u22a2 \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) ** intro x hx ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' \u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) ** have xs : x \u2208 s := ((mem_diff x).1 hx).1 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s \u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) ** rcases eq_empty_or_nonempty B with (hB | hB) ** case inl \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s hB : B = \u2205 \u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) ** rcases hf x xs 1 zero_lt_one with \u27e8r, hr, h'r\u27e9 ** case inl.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s hB : B = \u2205 r : \u211d hr : r \u2208 f x h'r : r \u2208 Ioo 0 1 \u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) ** exact \u27e8r, \u27e8hr, h'r\u27e9, by simp only [hB, empty_disjoint]\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s hB : B = \u2205 r : \u211d hr : r \u2208 f x h'r : r \u2208 Ioo 0 1 \u22a2 Disjoint B (closedBall x r) ** simp only [hB, empty_disjoint] ** case inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s hB : Set.Nonempty B \u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) ** let r := infDist x B ** case inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s hB : Set.Nonempty B r : \u211d := infDist x B \u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) ** have : 0 < min r 1 :=\n  lt_min ((B_closed.not_mem_iff_infDist_pos hB).1 ((mem_diff x).1 hx).2) zero_lt_one ** case inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s hB : Set.Nonempty B r : \u211d := infDist x B this : 0 < min r 1 \u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) ** rcases hf x xs _ this with \u27e8r, hr, h'r\u27e9 ** case inr.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s hB : Set.Nonempty B r\u271d : \u211d := infDist x B this : 0 < min r\u271d 1 r : \u211d hr : r \u2208 f x h'r : r \u2208 Ioo 0 (min r\u271d 1) \u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r) ** refine' \u27e8r, \u27e8hr, \u27e8h'r.1, h'r.2.trans_le (min_le_right _ _)\u27e9\u27e9, _\u27e9 ** case inr.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s hB : Set.Nonempty B r\u271d : \u211d := infDist x B this : 0 < min r\u271d 1 r : \u211d hr : r \u2208 f x h'r : r \u2208 Ioo 0 (min r\u271d 1) \u22a2 Disjoint B (closedBall x r) ** rw [disjoint_comm] ** case inr.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB\u271d : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B x : \u03b1 hx : x \u2208 s' xs : x \u2208 s hB : Set.Nonempty B r\u271d : \u211d := infDist x B this : 0 < min r\u271d 1 r : \u211d hr : r \u2208 f x h'r : r \u2208 Ioo 0 (min r\u271d 1) \u22a2 Disjoint (closedBall x r) B ** exact disjoint_closedBall_of_lt_infDist (h'r.2.trans_le (min_le_left _ _)) ** case intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) \u22a2 PairwiseDisjoint \u2191(t \u222a Finset.image (fun x => (x, r x)) v) fun p => closedBall p.1 p.2 ** simp only [Finset.coe_union, pairwiseDisjoint_union, ht.1, true_and_iff, Finset.coe_image] ** case intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) \u22a2 (PairwiseDisjoint ((fun x => (x, r x)) '' \u2191v) fun p => closedBall p.1 p.2) \u2227     \u2200 \u2983i : \u03b1 \u00d7 \u211d\u2984,       i \u2208 \u2191t \u2192 \u2200 \u2983j : \u03b1 \u00d7 \u211d\u2984, j \u2208 (fun x => (x, r x)) '' \u2191v \u2192 i \u2260 j \u2192 Disjoint (closedBall i.1 i.2) (closedBall j.1 j.2) ** constructor ** case intro.intro.intro.refine'_1.left \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) \u22a2 PairwiseDisjoint ((fun x => (x, r x)) '' \u2191v) fun p => closedBall p.1 p.2 ** intro p hp q hq hpq ** case intro.intro.intro.refine'_1.left \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 (fun x => (x, r x)) '' \u2191v q : \u03b1 \u00d7 \u211d hq : q \u2208 (fun x => (x, r x)) '' \u2191v hpq : p \u2260 q \u22a2 (Disjoint on fun p => closedBall p.1 p.2) p q ** rcases (mem_image _ _ _).1 hp with \u27e8p', p'v, rfl\u27e9 ** case intro.intro.intro.refine'_1.left.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) q : \u03b1 \u00d7 \u211d hq : q \u2208 (fun x => (x, r x)) '' \u2191v p' : \u03b1 p'v : p' \u2208 \u2191v hp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v hpq : (p', r p') \u2260 q \u22a2 (Disjoint on fun p => closedBall p.1 p.2) (p', r p') q ** rcases (mem_image _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9 ** case intro.intro.intro.refine'_1.left.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p' : \u03b1 p'v : p' \u2208 \u2191v hp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v q' : \u03b1 q'v : q' \u2208 \u2191v hq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v hpq : (p', r p') \u2260 (q', r q') \u22a2 (Disjoint on fun p => closedBall p.1 p.2) (p', r p') (q', r q') ** refine' hv p'v q'v fun hp'q' => _ ** case intro.intro.intro.refine'_1.left.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p' : \u03b1 p'v : p' \u2208 \u2191v hp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v q' : \u03b1 q'v : q' \u2208 \u2191v hq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v hpq : (p', r p') \u2260 (q', r q') hp'q' : p' = (q', r q').1 \u22a2 False ** rw [hp'q'] at hpq ** case intro.intro.intro.refine'_1.left.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p' : \u03b1 p'v : p' \u2208 \u2191v hp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v q' : \u03b1 q'v : q' \u2208 \u2191v hq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v hpq : ((q', r q').1, r (q', r q').1) \u2260 (q', r q') hp'q' : p' = (q', r q').1 \u22a2 False ** exact hpq rfl ** case intro.intro.intro.refine'_1.right \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) \u22a2 \u2200 \u2983i : \u03b1 \u00d7 \u211d\u2984,     i \u2208 \u2191t \u2192 \u2200 \u2983j : \u03b1 \u00d7 \u211d\u2984, j \u2208 (fun x => (x, r x)) '' \u2191v \u2192 i \u2260 j \u2192 Disjoint (closedBall i.1 i.2) (closedBall j.1 j.2) ** intro p hp q hq hpq ** case intro.intro.intro.refine'_1.right \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 \u2191t q : \u03b1 \u00d7 \u211d hq : q \u2208 (fun x => (x, r x)) '' \u2191v hpq : p \u2260 q \u22a2 Disjoint (closedBall p.1 p.2) (closedBall q.1 q.2) ** rcases (mem_image _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9 ** case intro.intro.intro.refine'_1.right.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 \u2191t q' : \u03b1 q'v : q' \u2208 \u2191v hq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v hpq : p \u2260 (q', r q') \u22a2 Disjoint (closedBall p.1 p.2) (closedBall (q', r q').1 (q', r q').2) ** apply disjoint_of_subset_left _ (hr q' (vs' q'v)).2 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 \u2191t q' : \u03b1 q'v : q' \u2208 \u2191v hq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v hpq : p \u2260 (q', r q') \u22a2 closedBall p.1 p.2 \u2286 B ** rw [hB, \u2190 Finset.set_biUnion_coe] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 \u2191t q' : \u03b1 q'v : q' \u2208 \u2191v hq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v hpq : p \u2260 (q', r q') \u22a2 closedBall p.1 p.2 \u2286 \u22c3 x \u2208 \u2191t, closedBall x.1 x.2 ** exact subset_biUnion_of_mem (u := fun x : \u03b1 \u00d7 \u211d => closedBall x.1 x.2) hp ** case intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) \u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u2192 p.1 \u2208 s ** intro p hp ** case intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u22a2 p.1 \u2208 s ** rcases Finset.mem_union.1 hp with (h'p | h'p) ** case intro.intro.intro.refine'_2.inl \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v h'p : p \u2208 t \u22a2 p.1 \u2208 s ** exact ht.2.1 p h'p ** case intro.intro.intro.refine'_2.inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v h'p : p \u2208 Finset.image (fun x => (x, r x)) v \u22a2 p.1 \u2208 s ** rcases Finset.mem_image.1 h'p with \u27e8p', p'v, rfl\u27e9 ** case intro.intro.intro.refine'_2.inr.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p' : \u03b1 p'v : p' \u2208 v hp : (p', r p') \u2208 t \u222a Finset.image (fun x => (x, r x)) v h'p : (p', r p') \u2208 Finset.image (fun x => (x, r x)) v \u22a2 (p', r p').1 \u2208 s ** exact ((mem_diff _).1 (vs' (Finset.mem_coe.2 p'v))).1 ** case intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) \u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u2192 p.2 \u2208 f p.1 ** intro p hp ** case intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u22a2 p.2 \u2208 f p.1 ** rcases Finset.mem_union.1 hp with (h'p | h'p) ** case intro.intro.intro.refine'_3.inl \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v h'p : p \u2208 t \u22a2 p.2 \u2208 f p.1 ** exact ht.2.2 p h'p ** case intro.intro.intro.refine'_3.inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p : \u03b1 \u00d7 \u211d hp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v h'p : p \u2208 Finset.image (fun x => (x, r x)) v \u22a2 p.2 \u2208 f p.1 ** rcases Finset.mem_image.1 h'p with \u27e8p', p'v, rfl\u27e9 ** case intro.intro.intro.refine'_3.inr.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) p' : \u03b1 p'v : p' \u2208 v hp : (p', r p') \u2208 t \u222a Finset.image (fun x => (x, r x)) v h'p : (p', r p') \u2208 Finset.image (fun x => (x, r x)) v \u22a2 (p', r p').2 \u2208 f (p', r p').1 ** exact (hr p' (vs' p'v)).1.1 ** case intro.intro.intro.refine'_4 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t \u222a Finset.image (fun x => (x, r x)) v, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' ** convert h\u03bcv using 2 ** case h.e'_3.h.e'_3 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 t : Finset (\u03b1 \u00d7 \u211d) ht : P t B : Set \u03b1 := \u22c3 p \u2208 t, closedBall p.1 p.2 hB : B = \u22c3 p \u2208 t, closedBall p.1 p.2 B_closed : IsClosed B s' : Set \u03b1 := s \\ B r : \u03b1 \u2192 \u211d hr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x)) v : Finset \u03b1 vs' : \u2191v \u2286 s' h\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s' hv : PairwiseDisjoint \u2191v fun x => closedBall x (r x) \u22a2 s \\ \u22c3 p \u2208 t \u222a Finset.image (fun x => (x, r x)) v, closedBall p.1 p.2 = s' \\ \u22c3 x \u2208 v, closedBall x (r x) ** rw [Finset.set_biUnion_union, \u2190 diff_diff, Finset.set_biUnion_finset_image] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 n : \u2115 \u22a2 u (Nat.succ n) = F (u n) ** simp only [Function.comp_apply, Function.iterate_succ'] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) \u22a2 \u2200 (n : \u2115), P (u n) ** intro n ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) n : \u2115 \u22a2 P (u n) ** induction' n with n IH ** case zero \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) \u22a2 P (u Nat.zero) ** simp only [Prod.forall, id.def, Function.iterate_zero, Nat.zero_eq] ** case zero \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) \u22a2 (PairwiseDisjoint \u2191\u2205 fun p => closedBall p.1 p.2) \u2227     (\u2200 (a : \u03b1) (b : \u211d), (a, b) \u2208 \u2205 \u2192 a \u2208 s) \u2227 \u2200 (a : \u03b1) (b : \u211d), (a, b) \u2208 \u2205 \u2192 b \u2208 f a ** simp only [Finset.not_mem_empty, IsEmpty.forall_iff, Finset.coe_empty, forall\u2082_true_iff,\n  and_self_iff, pairwiseDisjoint_empty] ** case succ \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) n : \u2115 IH : P (u n) \u22a2 P (u (Nat.succ n)) ** rw [u_succ] ** case succ \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) n : \u2115 IH : P (u n) \u22a2 P (F (u n)) ** exact (hF (u n) IH).2.1 ** case intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) \u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 \u22c3 n, \u2191(u n) \u2192 p.1 \u2208 s ** intro p hp ** case intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) p : \u03b1 \u00d7 \u211d hp : p \u2208 \u22c3 n, \u2191(u n) \u22a2 p.1 \u2208 s ** rcases mem_iUnion.1 hp with \u27e8n, hn\u27e9 ** case intro.intro.intro.refine'_1.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) p : \u03b1 \u00d7 \u211d hp : p \u2208 \u22c3 n, \u2191(u n) n : \u2115 hn : p \u2208 \u2191(u n) \u22a2 p.1 \u2208 s ** exact (Pu n).2.1 p (Finset.mem_coe.1 hn) ** case intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) \u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 \u22c3 n, \u2191(u n) \u2192 p.2 \u2208 f p.1 ** intro p hp ** case intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) p : \u03b1 \u00d7 \u211d hp : p \u2208 \u22c3 n, \u2191(u n) \u22a2 p.2 \u2208 f p.1 ** rcases mem_iUnion.1 hp with \u27e8n, hn\u27e9 ** case intro.intro.intro.refine'_2.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) p : \u03b1 \u00d7 \u211d hp : p \u2208 \u22c3 n, \u2191(u n) n : \u2115 hn : p \u2208 \u2191(u n) \u22a2 p.2 \u2208 f p.1 ** exact (Pu n).2.2 p (Finset.mem_coe.1 hn) ** case intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) = 0 ** have A :\n  \u2200 n,\n    \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 n : \u2115, (u n : Set (\u03b1 \u00d7 \u211d))), closedBall p.fst p.snd) \u2264\n      \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) := by\n  intro n\n  apply measure_mono\n  apply diff_subset_diff (Subset.refl _)\n  exact biUnion_subset_biUnion_left (subset_iUnion (fun i => (u i : Set (\u03b1 \u00d7 \u211d))) n) ** case intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s C : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd (0 * \u2191\u2191\u03bc s)) \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) = 0 ** rw [zero_mul] at C ** case intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s C : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd 0) \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) = 0 ** apply le_bot_iff.1 ** case intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s C : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd 0) \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u22a5 ** exact le_of_tendsto_of_tendsto' tendsto_const_nhds C fun n => (A n).trans (B n) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) \u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) ** intro n ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) n : \u2115 \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) ** apply measure_mono ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) n : \u2115 \u22a2 s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2 \u2286 s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2 ** apply diff_subset_diff (Subset.refl _) ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) n : \u2115 \u22a2 \u22c3 p \u2208 u n, closedBall p.1 p.2 \u2286 \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2 ** exact biUnion_subset_biUnion_left (subset_iUnion (fun i => (u i : Set (\u03b1 \u00d7 \u211d))) n) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s ** intro n ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) n : \u2115 \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s ** induction' n with n IH ** case succ \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) n : \u2115 IH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u (Nat.succ n), closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.succ n * \u2191\u2191\u03bc s ** calc\n  \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n.succ), closedBall p.fst p.snd) \u2264\n      N / (N + 1) * \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) := by\n    rw [u_succ]; exact (hF (u n) (Pu n)).2.2\n  _ \u2264 (N / (N + 1) : \u211d\u22650\u221e) ^ n.succ * \u03bc s := by\n    rw [pow_succ, mul_assoc]; exact mul_le_mul_left' IH _ ** case zero \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u Nat.zero, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.zero * \u2191\u2191\u03bc s ** simp only [le_refl, diff_empty, one_mul, iUnion_false, iUnion_empty, pow_zero, Nat.zero_eq,\n  Function.iterate_zero, id.def, Finset.not_mem_empty] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) n : \u2115 IH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u (Nat.succ n), closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) ** rw [u_succ] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) n : \u2115 IH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F (u n), closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) ** exact (hF (u n) (Pu n)).2.2 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) n : \u2115 IH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.succ n * \u2191\u2191\u03bc s ** rw [pow_succ, mul_assoc] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) n : \u2115 IH : \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * ((\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) ** exact mul_le_mul_left' IH _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd (0 * \u2191\u2191\u03bc s)) ** apply ENNReal.Tendsto.mul_const _ (Or.inr (measure_lt_top \u03bc s).ne) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 Tendsto (fun x => (\u2191N / (\u2191N + 1)) ^ x) atTop (\ud835\udcdd 0) ** apply ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 ** case hr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191N / (\u2191N + 1) < 1 ** rw [ENNReal.div_lt_iff, one_mul] ** case hr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191N < \u2191N + 1 ** conv_lhs => rw [\u2190 add_zero (N : \u211d\u22650\u221e)] ** case hr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191N + 0 < \u2191N + 1 ** exact ENNReal.add_lt_add_left (ENNReal.nat_ne_top N) zero_lt_one ** case hr.h0 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191N + 1 \u2260 0 \u2228 \u2191N \u2260 0 ** simp only [true_or_iff, add_eq_zero_iff, Ne.def, not_false_iff, one_ne_zero, and_false_iff] ** case hr.ht \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) A : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 \u22c3 n, \u2191(u n), closedBall p.1 p.2) \u2264 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) B : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 u n, closedBall p.1 p.2) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s \u22a2 \u2191N + 1 \u2260 \u22a4 \u2228 \u2191N \u2260 \u22a4 ** simp only [ENNReal.nat_ne_top, Ne.def, not_false_iff, or_true_iff] ** case intro.intro.intro.refine'_4 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) \u22a2 PairwiseDisjoint (\u22c3 n, \u2191(u n)) fun p => closedBall p.1 p.2 ** refine' (pairwiseDisjoint_iUnion _).2 fun n => (Pu n).1 ** case intro.intro.intro.refine'_4 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) \u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => \u2191(u n) ** apply (monotone_nat_of_le_succ fun n => ?_).directed_le ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) n : \u2115 \u22a2 \u2191(u n) \u2264 \u2191(u (n + 1)) ** rw [\u2190 Nat.succ_eq_add_one, u_succ] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=   fun t =>     (PairwiseDisjoint \u2191t fun p => closedBall p.1 p.2) \u2227       (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.1 \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.2 \u2208 f p.1 F : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d) hF :   \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),     P t \u2192       t \u2286 F t \u2227         P (F t) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 F t, closedBall p.1 p.2) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 p \u2208 t, closedBall p.1 p.2) u : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205 u_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n) Pu : \u2200 (n : \u2115), P (u n) n : \u2115 \u22a2 \u2191(u n) \u2264 \u2191(F (u n)) ** exact (hF (u n) (Pu n)).1 ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.integrable_of_summable_norm_Icc ** E : Type u_1 inst\u271d : NormedAddCommGroup E f : C(\u211d, E) hf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 \u22a2 Integrable \u2191f ** refine'\n  @integrable_of_summable_norm_restrict \u211d \u2124 E _ volume _ _ _ _ _ _ _ _\n    (summable_of_nonneg_of_le\n      (fun n : \u2124 => mul_nonneg (norm_nonneg\n          (f.restrict (\u27e8Icc (n : \u211d) ((n : \u211d) + 1), isCompact_Icc\u27e9 : Compacts \u211d)))\n          ENNReal.toReal_nonneg)\n      (fun n => _) hf) _ ** case refine'_1 E : Type u_1 inst\u271d : NormedAddCommGroup E f : C(\u211d, E) hf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 \u22a2 \u2124 \u2192 Compacts \u211d ** intro n ** case refine'_1 E : Type u_1 inst\u271d : NormedAddCommGroup E f : C(\u211d, E) hf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 n : \u2124 \u22a2 Compacts \u211d ** exact \u27e8Icc (n : \u211d) ((n : \u211d) + 1), isCompact_Icc\u27e9 ** case refine'_2 E : Type u_1 inst\u271d : NormedAddCommGroup E f : C(\u211d, E) hf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 n : \u2124 \u22a2 \u2016ContinuousMap.restrict (\u2191{ carrier := Icc (\u2191n) (\u2191n + 1), isCompact' := (_ : IsCompact (Icc (\u2191n) (\u2191n + 1))) }) f\u2016 *       ENNReal.toReal (\u2191\u2191volume \u2191{ carrier := Icc (\u2191n) (\u2191n + 1), isCompact' := (_ : IsCompact (Icc (\u2191n) (\u2191n + 1))) }) \u2264     \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 ** simp only [Compacts.coe_mk, Real.volume_Icc, add_sub_cancel', ENNReal.toReal_ofReal zero_le_one,\n  mul_one, norm_le _ (norm_nonneg _)] ** case refine'_2 E : Type u_1 inst\u271d : NormedAddCommGroup E f : C(\u211d, E) hf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 n : \u2124 \u22a2 \u2200 (x : \u2191(Icc (\u2191n) (\u2191n + 1))),     \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264       \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 ** intro x ** case refine'_2 E : Type u_1 inst\u271d : NormedAddCommGroup E f : C(\u211d, E) hf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 n : \u2124 x : \u2191(Icc (\u2191n) (\u2191n + 1)) \u22a2 \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264     \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 ** have := ((f.comp <| ContinuousMap.addRight n).restrict (Icc 0 1)).norm_coe_le_norm\n    \u27e8x - n, \u27e8sub_nonneg.mpr x.2.1, sub_le_iff_le_add'.mpr x.2.2\u27e9\u27e9 ** case refine'_2 E : Type u_1 inst\u271d : NormedAddCommGroup E f : C(\u211d, E) hf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 n : \u2124 x : \u2191(Icc (\u2191n) (\u2191n + 1)) this :   \u2016\u2191(ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n)))         { val := \u2191x - \u2191n, property := (_ : 0 \u2264 \u2191x - \u2191n \u2227 \u2191x - \u2191n \u2264 1) }\u2016 \u2264     \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 \u22a2 \u2016\u2191(ContinuousMap.restrict (Icc (\u2191n) (\u2191n + 1)) f) x\u2016 \u2264     \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 ** simpa only [ContinuousMap.restrict_apply, comp_apply, coe_addRight, Subtype.coe_mk,\n  sub_add_cancel] using this ** case refine'_3 E : Type u_1 inst\u271d : NormedAddCommGroup E f : C(\u211d, E) hf : Summable fun n => \u2016ContinuousMap.restrict (Icc 0 1) (comp f (ContinuousMap.addRight \u2191n))\u2016 \u22a2 \u22c3 i, \u2191{ carrier := Icc (\u2191i) (\u2191i + 1), isCompact' := (_ : IsCompact (Icc (\u2191i) (\u2191i + 1))) } = univ ** exact iUnion_Icc_int_cast \u211d ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.bind\u2082_monomial ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S inst\u271d : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R f : R \u2192+* MvPolynomial \u03c3 S d : \u03c3 \u2192\u2080 \u2115 r : R \u22a2 \u2191(bind\u2082 f) (\u2191(monomial d) r) = \u2191f r * \u2191(monomial d) 1 ** simp only [monomial_eq, RingHom.map_mul, bind\u2082_C_right, Finsupp.prod, map_prod,\n  map_pow, bind\u2082_X_right, C_1, one_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.shiftRight_neg ** m n : \u2124 \u22a2 m >>> (-n) = m <<< n ** rw [\u2190 shiftLeft_neg, neg_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.valMinAbs_neg_of_ne_half ** n : \u2115 a : ZMod n ha : 2 * val a \u2260 n \u22a2 valMinAbs (-a) = -valMinAbs a ** cases' eq_zero_or_neZero n with h h ** case inr n : \u2115 a : ZMod n ha : 2 * val a \u2260 n h : NeZero n \u22a2 valMinAbs (-a) = -valMinAbs a ** refine' (valMinAbs_spec _ _).2 \u27e8_, _, _\u27e9 ** case inl n : \u2115 a : ZMod n ha : 2 * val a \u2260 n h : n = 0 \u22a2 valMinAbs (-a) = -valMinAbs a ** subst h ** case inl a : ZMod 0 ha : 2 * val a \u2260 0 \u22a2 valMinAbs (-a) = -valMinAbs a ** rfl ** case inr.refine'_1 n : \u2115 a : ZMod n ha : 2 * val a \u2260 n h : NeZero n \u22a2 -a = \u2191(-valMinAbs a) ** rw [Int.cast_neg, coe_valMinAbs] ** case inr.refine'_2 n : \u2115 a : ZMod n ha : 2 * val a \u2260 n h : NeZero n \u22a2 -\u2191n < -valMinAbs a * 2 ** rw [neg_mul, neg_lt_neg_iff] ** case inr.refine'_2 n : \u2115 a : ZMod n ha : 2 * val a \u2260 n h : NeZero n \u22a2 valMinAbs a * 2 < \u2191n ** exact a.valMinAbs_mem_Ioc.2.lt_of_ne (mt a.valMinAbs_mul_two_eq_iff.1 ha) ** case inr.refine'_3 n : \u2115 a : ZMod n ha : 2 * val a \u2260 n h : NeZero n \u22a2 -valMinAbs a * 2 \u2264 \u2191n ** linarith only [a.valMinAbs_mem_Ioc.1] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pi_eq_empty_iff' ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 inst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i) \u22a2 pi s t = \u2205 \u2194 \u2203 i, i \u2208 s \u2227 t i = \u2205 ** simp [pi_eq_empty_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_deriv_comp_mul_deriv ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g'\u271d g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' g g' : \u211d \u2192 \u211d hf : \u2200 (x : \u211d), x \u2208 [[a, b]] \u2192 HasDerivAt f (f' x) x hg : \u2200 (x : \u211d), x \u2208 [[a, b]] \u2192 HasDerivAt g (g' (f x)) (f x) hf' : ContinuousOn f' [[a, b]] hg' : Continuous g' \u22a2 \u222b (x : \u211d) in a..b, (g' \u2218 f) x * f' x = (g \u2218 f) b - (g \u2218 f) a ** simpa [mul_comm] using integral_deriv_comp_smul_deriv hf hg hf' hg' ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_mul_Ico_of_neg ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a b c : \u03b1 h : c < 0 \u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Ico a b = Ioc (b / c) (a / c) ** simpa only [mul_comm] using preimage_mul_const_Ico_of_neg a b h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.uniformIntegrable_average ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf : UniformIntegrable f p \u03bc \u22a2 UniformIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) p \u03bc ** obtain \u27e8hf\u2081, hf\u2082, hf\u2083\u27e9 := hf ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u22a2 UniformIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) p \u03bc ** refine' \u27e8fun n => _, fun \u03b5 h\u03b5 => _, _\u27e9 ** case intro.intro.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 \u22a2 AEStronglyMeasurable ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) n) \u03bc ** exact (Finset.aestronglyMeasurable_sum' _ fun i _ => hf\u2081 i).const_smul _ ** case intro.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 \u03b4 x,     \u2200 (i : \u2115) (s : Set \u03b1),       MeasurableSet s \u2192         \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192           snorm (indicator s ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8\u03b4, h\u03b4\u2081, h\u03b4\u2082\u27e9 := hf\u2082 h\u03b5 ** case intro.intro.refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 \u03b4 x,     \u2200 (i : \u2115) (s : Set \u03b1),       MeasurableSet s \u2192         \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192           snorm (indicator s ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e8\u03b4, h\u03b4\u2081, fun n s hs hle => _\u27e9 ** case intro.intro.refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 snorm (indicator s ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) n)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** simp_rw [Finset.smul_sum, Set.indicator_finset_sum] ** case intro.intro.refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 snorm (\u2211 i in Finset.range n, indicator s ((\u2191n)\u207b\u00b9 \u2022 f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' le_trans (snorm_sum_le (fun i _ => ((hf\u2081 i).const_smul _).indicator hs) hp) _ ** case intro.intro.refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 \u2211 i in Finset.range n, snorm (indicator s ((\u2191n)\u207b\u00b9 \u2022 f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** have : \u2200 i, s.indicator ((n : \u211d) \u207b\u00b9 \u2022 f i) = (\u2191n : \u211d)\u207b\u00b9 \u2022 s.indicator (f i) :=\n  fun i \u21a6 indicator_const_smul _ _ _ ** case intro.intro.refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) \u22a2 \u2211 i in Finset.range n, snorm (indicator s ((\u2191n)\u207b\u00b9 \u2022 f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** simp_rw [this, snorm_const_smul, \u2190 Finset.mul_sum, nnnorm_inv, Real.nnnorm_coe_nat] ** case intro.intro.refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** by_cases hn : (\u2191(\u2191n : \u211d\u22650)\u207b\u00b9 : \u211d\u22650\u221e) = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' le_trans _ (_ : \u2191(\u2191n : \u211d\u22650)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5 \u2264 ENNReal.ofReal \u03b5) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** simp only [hn, zero_mul, zero_le] ** case neg.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 \u2191(\u2191n)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5 ** refine' (ENNReal.mul_le_mul_left hn ENNReal.coe_ne_top).2 _ ** case neg.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 n \u2022 ENNReal.ofReal \u03b5 ** conv_rhs => rw [\u2190 Finset.card_range n] ** case neg.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2211 x in Finset.range n, snorm (indicator s (f x)) p \u03bc \u2264 Finset.card (Finset.range n) \u2022 ENNReal.ofReal \u03b5 ** exact Finset.sum_le_card_nsmul _ _ _ fun i _ => h\u03b4\u2082 _ _ hs hle ** case neg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5 \u2264 ENNReal.ofReal \u03b5 ** simp only [ENNReal.coe_eq_zero, inv_eq_zero, Nat.cast_eq_zero] at hn ** case neg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u00acn = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * n \u2022 ENNReal.ofReal \u03b5 \u2264 ENNReal.ofReal \u03b5 ** rw [nsmul_eq_mul, \u2190 mul_assoc, ENNReal.coe_inv, ENNReal.coe_nat,\n  ENNReal.inv_mul_cancel _ (ENNReal.nat_ne_top _), one_mul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u00acn = 0 \u22a2 \u2191n \u2260 0  case neg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u00acn = 0 \u22a2 \u2191n \u2260 0 ** all_goals simpa only [Ne.def, Nat.cast_eq_zero] ** case neg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u03b4 : \u211d h\u03b4\u2081 : 0 < \u03b4 h\u03b4\u2082 :   \u2200 (i : \u2115) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 n : \u2115 s : Set \u03b1 hs : MeasurableSet s hle : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 this : \u2200 (i : \u2115), indicator s ((\u2191n)\u207b\u00b9 \u2022 f i) = (\u2191n)\u207b\u00b9 \u2022 indicator s (f i) hn : \u00acn = 0 \u22a2 \u2191n \u2260 0 ** simpa only [Ne.def, Nat.cast_eq_zero] ** case intro.intro.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc hf\u2083 : \u2203 C, \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u22a2 \u2203 C, \u2200 (i : \u2115), snorm ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i) p \u03bc \u2264 \u2191C ** obtain \u27e8C, hC\u27e9 := hf\u2083 ** case intro.intro.refine'_3.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u22a2 \u2203 C, \u2200 (i : \u2115), snorm ((fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in Finset.range n, f i) i) p \u03bc \u2264 \u2191C ** simp_rw [Finset.smul_sum] ** case intro.intro.refine'_3.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C \u22a2 \u2203 C, \u2200 (i : \u2115), snorm (\u2211 x in Finset.range i, (\u2191i)\u207b\u00b9 \u2022 f x) p \u03bc \u2264 \u2191C ** refine' \u27e8C, fun n => (snorm_sum_le (fun i _ => (hf\u2081 i).const_smul _) hp).trans _\u27e9 ** case intro.intro.refine'_3.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 \u22a2 \u2211 i in Finset.range n, snorm ((\u2191n)\u207b\u00b9 \u2022 f i) p \u03bc \u2264 \u2191C ** simp_rw [snorm_const_smul, \u2190 Finset.mul_sum, nnnorm_inv, Real.nnnorm_coe_nat] ** case intro.intro.refine'_3.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191C ** by_cases hn : (\u2191(\u2191n : \u211d\u22650)\u207b\u00b9 : \u211d\u22650\u221e) = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191C ** refine' le_trans _ (_ : \u2191(\u2191n : \u211d\u22650)\u207b\u00b9 * (n \u2022 C : \u211d\u22650\u221e) \u2264 C) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191C ** simp only [hn, zero_mul, zero_le] ** case neg.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191(\u2191n)\u207b\u00b9 * \u2191(n \u2022 C) ** refine' (ENNReal.mul_le_mul_left hn ENNReal.coe_ne_top).2 _ ** case neg.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191(n \u2022 C) ** conv_rhs => rw [\u2190 Finset.card_range n] ** case neg.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2211 x in Finset.range n, snorm (f x) p \u03bc \u2264 \u2191(Finset.card (Finset.range n) \u2022 C) ** convert Finset.sum_le_card_nsmul _ _ _ fun i _ => hC i ** case h.e'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2191(Finset.card (Finset.range n) \u2022 C) = Finset.card (Finset.range n) \u2022 \u2191C ** rw [ENNReal.coe_smul] ** case neg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00ac\u2191(\u2191n)\u207b\u00b9 = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2191(n \u2022 C) \u2264 \u2191C ** simp only [ENNReal.coe_eq_zero, inv_eq_zero, Nat.cast_eq_zero] at hn ** case neg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00acn = 0 \u22a2 \u2191(\u2191n)\u207b\u00b9 * \u2191(n \u2022 C) \u2264 \u2191C ** rw [ENNReal.coe_smul, nsmul_eq_mul, \u2190 mul_assoc, ENNReal.coe_inv, ENNReal.coe_nat,\n  ENNReal.inv_mul_cancel _ (ENNReal.nat_ne_top _), one_mul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00acn = 0 \u22a2 \u2191n \u2260 0  case neg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00acn = 0 \u22a2 \u2191n \u2260 0 ** all_goals simpa only [Ne.def, Nat.cast_eq_zero] ** case neg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u03b9 \u2192 \u03b1 \u2192 \u03b2 E : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E hp : 1 \u2264 p f : \u2115 \u2192 \u03b1 \u2192 E hf\u2081 : \u2200 (i : \u2115), AEStronglyMeasurable (f i) \u03bc hf\u2082 : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u2115), snorm (f i) p \u03bc \u2264 \u2191C n : \u2115 hn : \u00acn = 0 \u22a2 \u2191n \u2260 0 ** simpa only [Ne.def, Nat.cast_eq_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.coeff_rename_ne_zero ** \u03c3 : Type u_1 \u03c4 : Type u_2 \u03b1 : Type u_3 R : Type u_4 S : Type u_5 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R d : \u03c4 \u2192\u2080 \u2115 h : coeff d (\u2191(rename f) \u03c6) \u2260 0 \u22a2 \u2203 u, Finsupp.mapDomain f u = d \u2227 coeff u \u03c6 \u2260 0 ** contrapose! h ** \u03c3 : Type u_1 \u03c4 : Type u_2 \u03b1 : Type u_3 R : Type u_4 S : Type u_5 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S f : \u03c3 \u2192 \u03c4 \u03c6 : MvPolynomial \u03c3 R d : \u03c4 \u2192\u2080 \u2115 h : \u2200 (u : \u03c3 \u2192\u2080 \u2115), Finsupp.mapDomain f u = d \u2192 coeff u \u03c6 = 0 \u22a2 coeff d (\u2191(rename f) \u03c6) = 0 ** apply coeff_rename_eq_zero _ _ _ h ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.DList.toList_append ** \u03b1 : Type u l\u2081 l\u2082 : DList \u03b1 \u22a2 toList (append l\u2081 l\u2082) = toList l\u2081 ++ toList l\u2082 ** cases' l\u2081 with _ l\u2081_invariant ** case mk \u03b1 : Type u l\u2082 : DList \u03b1 apply\u271d : List \u03b1 \u2192 List \u03b1 l\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l \u22a2 toList (append { apply := apply\u271d, invariant := l\u2081_invariant } l\u2082) =     toList { apply := apply\u271d, invariant := l\u2081_invariant } ++ toList l\u2082 ** cases' l\u2082 ** case mk.mk \u03b1 : Type u apply\u271d\u00b9 : List \u03b1 \u2192 List \u03b1 l\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d\u00b9 l = apply\u271d\u00b9 [] ++ l apply\u271d : List \u03b1 \u2192 List \u03b1 invariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l \u22a2 toList (append { apply := apply\u271d\u00b9, invariant := l\u2081_invariant } { apply := apply\u271d, invariant := invariant\u271d }) =     toList { apply := apply\u271d\u00b9, invariant := l\u2081_invariant } ++ toList { apply := apply\u271d, invariant := invariant\u271d } ** simp ** case mk.mk \u03b1 : Type u apply\u271d\u00b9 : List \u03b1 \u2192 List \u03b1 l\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d\u00b9 l = apply\u271d\u00b9 [] ++ l apply\u271d : List \u03b1 \u2192 List \u03b1 invariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l \u22a2 apply\u271d\u00b9 (apply\u271d []) = apply\u271d\u00b9 [] ++ apply\u271d [] ** rw [l\u2081_invariant] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.neg_add_le_right_of_le_add ** a b c : Int h : a \u2264 b + c \u22a2 -c + a \u2264 b ** rw [Int.add_comm] at h ** a b c : Int h : a \u2264 c + b \u22a2 -c + a \u2264 b ** exact Int.neg_add_le_left_of_le_add h ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.exists_isOpen_lt_of_lt ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : OuterRegular \u03bc A : Set \u03b1 r : \u211d\u22650\u221e hr : \u2191\u2191\u03bc A < r \u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r ** rcases OuterRegular.outerRegular (measurableSet_toMeasurable \u03bc A) r\n    (by rwa [measure_toMeasurable]) with\n  \u27e8U, hAU, hUo, hU\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : OuterRegular \u03bc A : Set \u03b1 r : \u211d\u22650\u221e hr : \u2191\u2191\u03bc A < r U : Set \u03b1 hAU : U \u2287 toMeasurable \u03bc A hUo : IsOpen U hU : \u2191\u2191\u03bc U < r \u22a2 \u2203 U, U \u2287 A \u2227 IsOpen U \u2227 \u2191\u2191\u03bc U < r ** exact \u27e8U, (subset_toMeasurable _ _).trans hAU, hUo, hU\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : OuterRegular \u03bc A : Set \u03b1 r : \u211d\u22650\u221e hr : \u2191\u2191\u03bc A < r \u22a2 r > \u2191\u2191?m.8585 (toMeasurable \u03bc A) ** rwa [measure_toMeasurable] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.set_lintegral_condCdf_rat ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r : \u211a s : Set \u03b1 hs : MeasurableSet s \u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r) ** have : \u2200\u1d50 a \u2202\u03c1.fst, a \u2208 s \u2192 ENNReal.ofReal (condCdf \u03c1 a r) = preCdf \u03c1 r a := by\n  filter_upwards [ofReal_condCdf_ae_eq \u03c1 r] with a ha using fun _ => ha ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r : \u211a s : Set \u03b1 hs : MeasurableSet s this : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a \u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r) ** rw [set_lintegral_congr_fun hs this, set_lintegral_preCdf_fst \u03c1 r hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r : \u211a s : Set \u03b1 hs : MeasurableSet s this : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a \u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r) ** exact \u03c1.IicSnd_apply r hs ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r : \u211a s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 s \u2192 ENNReal.ofReal (\u2191(condCdf \u03c1 a) \u2191r) = preCdf \u03c1 r a ** filter_upwards [ofReal_condCdf_ae_eq \u03c1 r] with a ha using fun _ => ha ** Qed",
        "informal": ""
    },
    {
        "formal": "List.mem_range ** m n : Nat \u22a2 m \u2208 range n \u2194 m < n ** simp only [range_eq_range', mem_range'_1, Nat.zero_le, true_and, Nat.zero_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.countable_pi ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x \u03c0 : \u03b1 \u2192 Type u_1 inst\u271d : Finite \u03b1 s : (a : \u03b1) \u2192 Set (\u03c0 a) hs : \u2200 (a : \u03b1), Set.Countable (s a) \u22a2 Set.Countable {f | \u2200 (a : \u03b1), f a \u2208 s a} ** simpa only [\u2190 mem_univ_pi] using countable_univ_pi hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.count_apply_infinite ** \u03b1 : Type u_1 \u03b2 : Type ?u.8763 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 hs : Set.Infinite s \u22a2 \u2191\u2191count s = \u22a4 ** refine' top_unique (le_of_tendsto' ENNReal.tendsto_nat_nhds_top fun n => _) ** \u03b1 : Type u_1 \u03b2 : Type ?u.8763 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 hs : Set.Infinite s n : \u2115 \u22a2 \u2191n \u2264 \u2191\u2191count s ** rcases hs.exists_subset_card_eq n with \u27e8t, ht, rfl\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type ?u.8763 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 hs : Set.Infinite s t : Finset \u03b1 ht : \u2191t \u2286 s \u22a2 \u2191(Finset.card t) \u2264 \u2191\u2191count s ** calc\n  (t.card : \u211d\u22650\u221e) = \u2211 i in t, 1 := by simp\n  _ = \u2211' i : (t : Set \u03b1), 1 := (t.tsum_subtype 1).symm\n  _ \u2264 count (t : Set \u03b1) := le_count_apply\n  _ \u2264 count s := measure_mono ht ** \u03b1 : Type u_1 \u03b2 : Type ?u.8763 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 hs : Set.Infinite s t : Finset \u03b1 ht : \u2191t \u2286 s \u22a2 \u2191(Finset.card t) = \u2211 i in t, 1 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5 ** rcases exists_between h\u03b5.bot_lt with \u27e8\u03b5\u2082, h\u03b5\u20820 : 0 < \u03b5\u2082, h\u03b5\u2082\u03b5\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5 ** rcases exists_between h\u03b5\u20820 with \u27e8\u03b5\u2081, h\u03b5\u20810, h\u03b5\u2081\u2082\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5 ** rcases exists_simpleFunc_forall_lintegral_sub_lt_of_pos h h\u03b5\u20810.ne' with \u27e8\u03c6, _, h\u03c6\u27e9 ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 \u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5 ** rcases \u03c6.exists_forall_le with \u27e8C, hC\u27e9 ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C \u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < \u03b4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5 ** use (\u03b5\u2082 - \u03b5\u2081) / C, ENNReal.div_pos_iff.2 \u27e8(tsub_pos_iff_lt.2 h\u03b5\u2081\u2082).ne', ENNReal.coe_ne_top\u27e9 ** case right \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C \u22a2 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc < \u03b5 ** refine' fun s hs => lt_of_le_of_lt _ h\u03b5\u2082\u03b5 ** case right \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u03b5\u2082 ** simp only [lintegral_eq_nnreal, iSup_le_iff] ** case right \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u22a2 \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650),     (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) (Measure.restrict \u03bc s) \u2264 \u03b5\u2082 ** intro \u03c8 h\u03c8 ** case right \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264 \u03b5\u2082 ** calc\n  (map (\u2191) \u03c8).lintegral (\u03bc.restrict s) \u2264\n      (map (\u2191) \u03c6).lintegral (\u03bc.restrict s) + (map (\u2191) (\u03c8 - \u03c6)).lintegral (\u03bc.restrict s) := by\n    rw [\u2190 SimpleFunc.add_lintegral, \u2190 SimpleFunc.map_add @ENNReal.coe_add]\n    refine' SimpleFunc.lintegral_mono (fun x => _) le_rfl\n    simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add,\n      SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, WithTop.coe_max (\u03c6 x) (\u03c8 x), ENNReal.some]\n  _ \u2264 (map (\u2191) \u03c6).lintegral (\u03bc.restrict s) + \u03b5\u2081 := by\n    refine' add_le_add le_rfl (le_trans _ (h\u03c6 _ h\u03c8).le)\n    exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self\n  _ \u2264 (SimpleFunc.const \u03b1 (C : \u211d\u22650\u221e)).lintegral (\u03bc.restrict s) + \u03b5\u2081 :=\n    (add_le_add (SimpleFunc.lintegral_mono (fun x => by exact coe_le_coe.2 (hC x)) le_rfl) le_rfl)\n  _ = C * \u03bc s + \u03b5\u2081 := by\n    simp only [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const,\n      Measure.restrict_apply, MeasurableSet.univ, univ_inter, Function.const]\n  _ \u2264 C * ((\u03b5\u2082 - \u03b5\u2081) / C) + \u03b5\u2081 := by gcongr\n  _ \u2264 \u03b5\u2082 - \u03b5\u2081 + \u03b5\u2081 := by gcongr; apply mul_div_le\n  _ = \u03b5\u2082 := tsub_add_cancel_of_le h\u03b5\u2081\u2082.le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264     SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) (Measure.restrict \u03bc s) +       SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s) ** rw [\u2190 SimpleFunc.add_lintegral, \u2190 SimpleFunc.map_add @ENNReal.coe_add] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c8) (Measure.restrict \u03bc s) \u2264     SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) (Measure.restrict \u03bc s) ** refine' SimpleFunc.lintegral_mono (fun x => _) le_rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x x : \u03b1 \u22a2 \u2191(SimpleFunc.map ENNReal.some \u03c8) x \u2264 \u2191(SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) x ** simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add,\n  SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, WithTop.coe_max (\u03c6 x) (\u03c8 x), ENNReal.some] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) (Measure.restrict \u03bc s) +       SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s) \u2264     SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) (Measure.restrict \u03bc s) + \u03b5\u2081 ** refine' add_le_add le_rfl (le_trans _ (h\u03c6 _ h\u03c8).le) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) (Measure.restrict \u03bc s) \u2264     SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc ** exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x x : \u03b1 \u22a2 \u2191(SimpleFunc.map ENNReal.some \u03c6) x \u2264 \u2191(const \u03b1 \u2191C) x ** exact coe_le_coe.2 (hC x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 SimpleFunc.lintegral (const \u03b1 \u2191C) (Measure.restrict \u03bc s) + \u03b5\u2081 = \u2191C * \u2191\u2191\u03bc s + \u03b5\u2081 ** simp only [\u2190 SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const,\n  Measure.restrict_apply, MeasurableSet.univ, univ_inter, Function.const] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 \u2191C * \u2191\u2191\u03bc s + \u03b5\u2081 \u2264 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) + \u03b5\u2081 ** gcongr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) + \u03b5\u2081 \u2264 \u03b5\u2082 - \u03b5\u2081 + \u03b5\u2081 ** gcongr ** case bc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03b5\u2082 : \u211d\u22650\u221e h\u03b5\u20820 : 0 < \u03b5\u2082 h\u03b5\u2082\u03b5 : \u03b5\u2082 < \u03b5 \u03b5\u2081 : \u211d\u22650\u221e h\u03b5\u20810 : 0 < \u03b5\u2081 h\u03b5\u2081\u2082 : \u03b5\u2081 < \u03b5\u2082 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 left\u271d : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x h\u03c6 : \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5\u2081 C : \u211d\u22650 hC : \u2200 (x : \u03b1), \u2191\u03c6 x \u2264 C s : Set \u03b1 hs : \u2191\u2191\u03bc s < (\u03b5\u2082 - \u03b5\u2081) / \u2191C \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 \u2191C * ((\u03b5\u2082 - \u03b5\u2081) / \u2191C) \u2264 \u03b5\u2082 - \u03b5\u2081 ** apply mul_div_le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.le_sum_caratheodory ** \u03b1 : Type u_1 \u03b9 : Type u_2 m : \u03b9 \u2192 OuterMeasure \u03b1 s : Set \u03b1 h : MeasurableSet s t : Set \u03b1 \u22a2 \u2191(sum m) t = \u2191(sum m) (t \u2229 s) + \u2191(sum m) (t \\ s) ** simp [fun i => MeasurableSpace.measurableSet_iInf.1 h i t, ENNReal.tsum_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.recF_eq_of_wEquiv ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 \u22a2 WEquiv x y \u2192 recF u x = recF u y ** apply q.P.w_cases _ x ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 \u22a2 \u2200 (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1)     (f : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1),     WEquiv (MvPFunctor.wMk (P F) a f' f) y \u2192 recF u (MvPFunctor.wMk (P F) a f' f) = recF u y ** intro a\u2080 f'\u2080 f\u2080 ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 \u22a2 WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) y \u2192 recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u y ** apply q.P.w_cases _ y ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 \u22a2 \u2200 (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1)     (f : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1),     WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a f' f) \u2192       recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a f' f) ** intro a\u2081 f'\u2081 f\u2081 ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 \u22a2 WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) \u2192     recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) ** intro h ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) \u22a2 recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) ** refine' @WEquiv.recOn _ _ _ _ _ (\u03bb a a' _ => recF u a = recF u a') _ _ h _ _ _ ** case refine'_1 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) \u22a2 \u2200 (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1)     (f\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1)     (a_1 : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x)),     (\u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),         (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x))) \u2192       (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a f' f\u2080) (MvPFunctor.wMk (P F) a f' f\u2081)         (_ : WEquiv (MvPFunctor.wMk (P F) a f' f\u2080) (MvPFunctor.wMk (P F) a f' f\u2081)) ** intros a f' f\u2080 f\u2081 _h ih ** case refine'_1 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d) a : (P F).A f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1 f\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1 _h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x) ih :   \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),     (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x)) \u22a2 recF u (MvPFunctor.wMk (P F) a f' f\u2080) = recF u (MvPFunctor.wMk (P F) a f' f\u2081) ** simp only [recF_eq, Function.comp] ** case refine'_1 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d) a : (P F).A f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1 f\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1 _h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x) ih :   \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),     (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x)) \u22a2 u (abs { fst := a, snd := splitFun f' fun x => recF u (f\u2080 x) }) =     u (abs { fst := a, snd := splitFun f' fun x => recF u (f\u2081 x) }) ** congr ** case refine'_1.e_a.e_a.e_snd n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d) a : (P F).A f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1 f\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1 _h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x) ih :   \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),     (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x)) \u22a2 (splitFun f' fun x => recF u (f\u2080 x)) = splitFun f' fun x => recF u (f\u2081 x) ** funext ** case refine'_1.e_a.e_a.e_snd.h.h n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d) a : (P F).A f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1 f\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1 _h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x) ih :   \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),     (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x)) x\u271d\u00b9 : Fin2 (n + 1) x\u271d : MvPFunctor.B (P F) a x\u271d\u00b9 \u22a2 splitFun f' (fun x => recF u (f\u2080 x)) x\u271d\u00b9 x\u271d = splitFun f' (fun x => recF u (f\u2081 x)) x\u271d\u00b9 x\u271d ** congr ** case refine'_1.e_a.e_a.e_snd.h.h.e_g n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d) a : (P F).A f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1 f\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1 _h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x) ih :   \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),     (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x)) x\u271d\u00b9 : Fin2 (n + 1) x\u271d : MvPFunctor.B (P F) a x\u271d\u00b9 \u22a2 (fun x => recF u (f\u2080 x)) = fun x => recF u (f\u2081 x) ** funext ** case refine'_1.e_a.e_a.e_snd.h.h.e_g.h n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081\u271d) a : (P F).A f' : MvPFunctor.B (MvPFunctor.drop (P F)) a \u27f9 \u03b1 f\u2080 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a \u2192 MvPFunctor.W (P F) \u03b1 _h : \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f\u2080 x) (f\u2081 x) ih :   \u2200 (x : PFunctor.B (MvPFunctor.last (P F)) a),     (fun a a' x => recF u a = recF u a') (f\u2080 x) (f\u2081 x) (_ : WEquiv (f\u2080 x) (f\u2081 x)) x\u271d\u00b2 : Fin2 (n + 1) x\u271d\u00b9 : MvPFunctor.B (P F) a x\u271d\u00b2 x\u271d : last (MvPFunctor.B (P F) a) \u22a2 recF u (f\u2080 x\u271d) = recF u (f\u2081 x\u271d) ** apply ih ** case refine'_2 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) \u22a2 \u2200 (a\u2080 : (P F).A) (f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1)     (f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1) (a\u2081 : (P F).A)     (f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1)     (f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1)     (a :       abs { fst := a\u2080, snd := MvPFunctor.appendContents (P F) f'\u2080 f\u2080 } =         abs { fst := a\u2081, snd := MvPFunctor.appendContents (P F) f'\u2081 f\u2081 }),     (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)       (_ : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081)) ** intros a\u2080 f'\u2080 f\u2080 a\u2081 f'\u2081 f\u2081 h ** case refine'_2 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080\u271d : (P F).A f'\u2080\u271d : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080\u271d \u27f9 \u03b1 f\u2080\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2080\u271d \u2192 MvPFunctor.W (P F) \u03b1 a\u2081\u271d : (P F).A f'\u2081\u271d : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081\u271d \u27f9 \u03b1 f\u2081\u271d : PFunctor.B (MvPFunctor.last (P F)) a\u2081\u271d \u2192 MvPFunctor.W (P F) \u03b1 h\u271d : WEquiv (MvPFunctor.wMk (P F) a\u2080\u271d f'\u2080\u271d f\u2080\u271d) (MvPFunctor.wMk (P F) a\u2081\u271d f'\u2081\u271d f\u2081\u271d) a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h :   abs { fst := a\u2080, snd := MvPFunctor.appendContents (P F) f'\u2080 f\u2080 } =     abs { fst := a\u2081, snd := MvPFunctor.appendContents (P F) f'\u2081 f\u2081 } \u22a2 recF u (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) = recF u (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) ** simp only [recF_eq', abs_map, MvPFunctor.wDest'_wMk, h] ** case refine'_3 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x y : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) \u22a2 \u2200 (u_1 v w : MvPFunctor.W (P F) \u03b1) (a : WEquiv u_1 v) (a_1 : WEquiv v w),     (fun a a' x => recF u a = recF u a') u_1 v a \u2192       (fun a a' x => recF u a = recF u a') v w a_1 \u2192 (fun a a' x => recF u a = recF u a') u_1 w (_ : WEquiv u_1 w) ** intros x y z _e\u2081 _e\u2082 ih\u2081 ih\u2082 ** case refine'_3 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n \u03b2 : Type u u : F (\u03b1 ::: \u03b2) \u2192 \u03b2 x\u271d y\u271d : MvPFunctor.W (P F) \u03b1 a\u2080 : (P F).A f'\u2080 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2080 \u27f9 \u03b1 f\u2080 : PFunctor.B (MvPFunctor.last (P F)) a\u2080 \u2192 MvPFunctor.W (P F) \u03b1 a\u2081 : (P F).A f'\u2081 : MvPFunctor.B (MvPFunctor.drop (P F)) a\u2081 \u27f9 \u03b1 f\u2081 : PFunctor.B (MvPFunctor.last (P F)) a\u2081 \u2192 MvPFunctor.W (P F) \u03b1 h : WEquiv (MvPFunctor.wMk (P F) a\u2080 f'\u2080 f\u2080) (MvPFunctor.wMk (P F) a\u2081 f'\u2081 f\u2081) x y z : MvPFunctor.W (P F) \u03b1 _e\u2081 : WEquiv x y _e\u2082 : WEquiv y z ih\u2081 : recF u x = recF u y ih\u2082 : recF u y = recF u z \u22a2 recF u x = recF u z ** exact Eq.trans ih\u2081 ih\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condCount_add_compl_eq ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t\u271d u\u271d u t : Set \u03a9 hs : Set.Finite s \u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t ** have : condCount s t = (condCount (s \u2229 u) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u) +\n    condCount (s \u2229 u\u1d9c) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u\u1d9c)) := by\n  rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n    (disjoint_compl_right.mono inf_le_right inf_le_right), Set.inter_union_compl] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t\u271d u\u271d u t : Set \u03a9 hs : Set.Finite s this :   \u2191\u2191(condCount s) t =     \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +       \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c) \u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t ** rw [this] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t\u271d u\u271d u t : Set \u03a9 hs : Set.Finite s this :   \u2191\u2191(condCount s) t =     \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +       \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c) \u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c =     \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +       \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c) ** simp [condCount_inter_self hs] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t\u271d u\u271d u t : Set \u03a9 hs : Set.Finite s \u22a2 \u2191\u2191(condCount s) t =     \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +       \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c) ** rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n  (disjoint_compl_right.mono inf_le_right inf_le_right), Set.inter_union_compl] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsStoppingTime.measurableSet_eq_stopping_time_of_countable ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} ** rw [h\u03c4.measurableSet] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 \u22a2 \u2200 (i : \u03b9), MeasurableSet ({\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) ** intro j ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 this : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j} \u22a2 MeasurableSet ({\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j}) ** rw [this] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 this : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j} \u22a2 MeasurableSet ({\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j}) ** refine'\n  MeasurableSet.inter (MeasurableSet.inter _ (h\u03c4.measurableSet_le j)) (h\u03c0.measurableSet_le j) ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 this : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j} \u22a2 MeasurableSet {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} ** apply measurableSet_eq_fun_of_countable ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 \u22a2 {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j} ** ext1 \u03c9 ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 \u03c9 : \u03a9 \u22a2 \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2194 \u03c9 \u2208 {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j} ** simp only [Set.mem_inter_iff, Set.mem_setOf_eq] ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 \u03c9 : \u03a9 \u22a2 \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j \u2194 (min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j \u2227 \u03c4 \u03c9 \u2264 j) \u2227 \u03c0 \u03c9 \u2264 j ** refine' \u27e8fun h => \u27e8\u27e8_, h.2\u27e9, _\u27e9, fun h => \u27e8_, h.1.2\u27e9\u27e9 ** case h.refine'_1 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 \u03c9 : \u03a9 h : \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j \u22a2 min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j ** rw [h.1] ** case h.refine'_2 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 \u03c9 : \u03a9 h : \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j \u22a2 \u03c0 \u03c9 \u2264 j ** rw [\u2190 h.1] ** case h.refine'_2 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 \u03c9 : \u03a9 h : \u03c4 \u03c9 = \u03c0 \u03c9 \u2227 \u03c4 \u03c9 \u2264 j \u22a2 \u03c4 \u03c9 \u2264 j ** exact h.2 ** case h.refine'_3 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 \u03c9 : \u03a9 h : (min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j \u2227 \u03c4 \u03c9 \u2264 j) \u2227 \u03c0 \u03c9 \u2264 j \u22a2 \u03c4 \u03c9 = \u03c0 \u03c9 ** cases' h with h' h\u03c0_le ** case h.refine'_3.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 \u03c9 : \u03a9 h' : min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j \u2227 \u03c4 \u03c9 \u2264 j h\u03c0_le : \u03c0 \u03c9 \u2264 j \u22a2 \u03c4 \u03c9 = \u03c0 \u03c9 ** cases' h' with h_eq h\u03c4_le ** case h.refine'_3.intro.intro \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 \u03c9 : \u03a9 h\u03c0_le : \u03c0 \u03c9 \u2264 j h_eq : min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j h\u03c4_le : \u03c4 \u03c9 \u2264 j \u22a2 \u03c4 \u03c9 = \u03c0 \u03c9 ** rwa [min_eq_left h\u03c4_le, min_eq_left h\u03c0_le] at h_eq ** case hf \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 this : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j} \u22a2 Measurable fun x => min (\u03c4 x) j ** exact (h\u03c4.min_const j).measurable_of_le fun _ => min_le_right _ _ ** case hg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2076 : Countable \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : MeasurableSpace \u03b9 inst\u271d\u00b3 : BorelSpace \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSingletonClass \u03b9 inst\u271d : SecondCountableTopology \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 j : \u03b9 this : {\u03c9 | \u03c4 \u03c9 = \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} = {\u03c9 | min (\u03c4 \u03c9) j = min (\u03c0 \u03c9) j} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 j} \u2229 {\u03c9 | \u03c0 \u03c9 \u2264 j} \u22a2 Measurable fun x => min (\u03c0 x) j ** exact (h\u03c0.min_const j).measurable_of_le fun _ => min_le_right _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.toJordanDecomposition_neg ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u22a2 toJordanDecomposition (-s) = -toJordanDecomposition s ** apply toSignedMeasure_injective ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u22a2 toSignedMeasure (toJordanDecomposition (-s)) = toSignedMeasure (-toJordanDecomposition s) ** simp [toSignedMeasure_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.iIndepSets.iIndep ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba \u22a2 kernel.iIndep m \u03ba ** intro s f ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 \u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 s, f i) = \u220f i in s, \u2191\u2191(\u2191\u03ba a) (f i) ** refine Finset.induction ?_ ?_ s ** case refine_1 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 \u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 \u2205, f i) = \u220f i in \u2205, \u2191\u2191(\u2191\u03ba a) (f i) ** simp only [Finset.not_mem_empty, Set.mem_setOf_eq, IsEmpty.forall_iff, implies_true,\n  Set.iInter_of_empty, Set.iInter_univ, measure_univ, Finset.prod_empty,\n  Filter.eventually_true, forall_true_left] ** case refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 \u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},     \u00aca \u2208 s \u2192       ((\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192           \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 s, f i) = \u220f i in s, \u2191\u2191(\u2191\u03ba a) (f i)) \u2192         (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192           \u2200\u1d50 (a_3 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_3) (\u22c2 i \u2208 insert a s, f i) = \u220f i in insert a s, \u2191\u2191(\u2191\u03ba a_3) (f i) ** intro a S ha_notin_S h_rec hf_m ** case refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i \u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i) ** have hf_m_S : \u2200 x \u2208 S, MeasurableSet[m x] (f x) := fun x hx => hf_m x (by simp [hx]) ** case refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) \u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i) ** let p := piiUnionInter \u03c0 S ** case refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S \u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i) ** set m_p := generateFrom p with hS_eq_generate ** case refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p \u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i) ** have h_indep : Indep m_p (m a) \u03ba \u03bc := by\n  have hp : IsPiSystem p := isPiSystem_piiUnionInter \u03c0 h_pi S\n  have h_le' : \u2200 i, generateFrom (\u03c0 i) \u2264 _m\u03a9 := fun i \u21a6 (h_generate i).symm.trans_le (h_le i)\n  have hm_p : m_p \u2264 _m\u03a9 := generateFrom_piiUnionInter_le \u03c0 h_le' S\n  exact IndepSets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)\n    (iIndepSets.piiUnionInter_of_not_mem h_ind ha_notin_S) ** case refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p h_indep : Indep m_p (m a) \u03ba \u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i) ** have h := h_indep.symm (f a) (\u22c2 n \u2208 S, f n) (hf_m a (Finset.mem_insert_self a S)) ?_ ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i x : \u03b9 hx : x \u2208 S \u22a2 x \u2208 insert a S ** simp [hx] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p \u22a2 Indep m_p (m a) \u03ba ** have hp : IsPiSystem p := isPiSystem_piiUnionInter \u03c0 h_pi S ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p hp : IsPiSystem p \u22a2 Indep m_p (m a) \u03ba ** have h_le' : \u2200 i, generateFrom (\u03c0 i) \u2264 _m\u03a9 := fun i \u21a6 (h_generate i).symm.trans_le (h_le i) ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p hp : IsPiSystem p h_le' : \u2200 (i : \u03b9), generateFrom (\u03c0 i) \u2264 _m\u03a9 \u22a2 Indep m_p (m a) \u03ba ** have hm_p : m_p \u2264 _m\u03a9 := generateFrom_piiUnionInter_le \u03c0 h_le' S ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p hp : IsPiSystem p h_le' : \u2200 (i : \u03b9), generateFrom (\u03c0 i) \u2264 _m\u03a9 hm_p : m_p \u2264 _m\u03a9 \u22a2 Indep m_p (m a) \u03ba ** exact IndepSets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)\n  (iIndepSets.piiUnionInter_of_not_mem h_ind ha_notin_S) ** case refine_2.refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p h_indep : Indep m_p (m a) \u03ba h : \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (f a \u2229 \u22c2 n \u2208 S, f n) = \u2191\u2191(\u2191\u03ba a_1) (f a) * \u2191\u2191(\u2191\u03ba a_1) (\u22c2 n \u2208 S, f n) \u22a2 \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a_1) (f i) ** filter_upwards [h_rec hf_m_S, h] with a' ha' h' ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p h_indep : Indep m_p (m a) \u03ba h : \u2200\u1d50 (a_1 : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a_1) (f a \u2229 \u22c2 n \u2208 S, f n) = \u2191\u2191(\u2191\u03ba a_1) (f a) * \u2191\u2191(\u2191\u03ba a_1) (\u22c2 n \u2208 S, f n) a' : \u03b1 ha' : \u2191\u2191(\u2191\u03ba a') (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a') (f i) h' : \u2191\u2191(\u2191\u03ba a') (f a \u2229 \u22c2 n \u2208 S, f n) = \u2191\u2191(\u2191\u03ba a') (f a) * \u2191\u2191(\u2191\u03ba a') (\u22c2 n \u2208 S, f n) \u22a2 \u2191\u2191(\u2191\u03ba a') (\u22c2 i \u2208 insert a S, f i) = \u220f i in insert a S, \u2191\u2191(\u2191\u03ba a') (f i) ** rwa [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, \u2190 ha'] ** case refine_2.refine_1 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p h_indep : Indep m_p (m a) \u03ba \u22a2 \u22c2 n \u2208 S, f n \u2208 {s | MeasurableSet s} ** have h_le_p : \u2200 i \u2208 S, m i \u2264 m_p := by\n  intros n hn\n  rw [hS_eq_generate, h_generate n]\n  refine le_generateFrom_piiUnionInter (S : Set \u03b9) hn ** case refine_2.refine_1 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p h_indep : Indep m_p (m a) \u03ba h_le_p : \u2200 (i : \u03b9), i \u2208 S \u2192 m i \u2264 m_p \u22a2 \u22c2 n \u2208 S, f n \u2208 {s | MeasurableSet s} ** have h_S_f : \u2200 i \u2208 S, MeasurableSet[m_p] (f i) :=\n  fun i hi \u21a6 (h_le_p i hi) (f i) (hf_m_S i hi) ** case refine_2.refine_1 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p h_indep : Indep m_p (m a) \u03ba h_le_p : \u2200 (i : \u03b9), i \u2208 S \u2192 m i \u2264 m_p h_S_f : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (f i) \u22a2 \u22c2 n \u2208 S, f n \u2208 {s | MeasurableSet s} ** exact S.measurableSet_biInter h_S_f ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p h_indep : Indep m_p (m a) \u03ba \u22a2 \u2200 (i : \u03b9), i \u2208 S \u2192 m i \u2264 m_p ** intros n hn ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p h_indep : Indep m_p (m a) \u03ba n : \u03b9 hn : n \u2208 S \u22a2 m n \u2264 m_p ** rw [hS_eq_generate, h_generate n] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 _m\u03a9 : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba m : \u03b9 \u2192 MeasurableSpace \u03a9 h_le : \u2200 (i : \u03b9), m i \u2264 _m\u03a9 \u03c0 : \u03b9 \u2192 Set (Set \u03a9) h_pi : \u2200 (n : \u03b9), IsPiSystem (\u03c0 n) h_generate : \u2200 (i : \u03b9), m i = generateFrom (\u03c0 i) h_ind : iIndepSets \u03c0 \u03ba s : Finset \u03b9 f : \u03b9 \u2192 Set \u03a9 a : \u03b9 S : Finset \u03b9 ha_notin_S : \u00aca \u2208 S h_rec :   (\u2200 (i : \u03b9), i \u2208 S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i) \u2192     \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (\u22c2 i \u2208 S, f i) = \u220f i in S, \u2191\u2191(\u2191\u03ba a) (f i) hf_m : \u2200 (i : \u03b9), i \u2208 insert a S \u2192 f i \u2208 (fun x => {s | MeasurableSet s}) i hf_m_S : \u2200 (x : \u03b9), x \u2208 S \u2192 MeasurableSet (f x) p : Set (Set \u03a9) := piiUnionInter \u03c0 \u2191S m_p : MeasurableSpace \u03a9 := generateFrom p hS_eq_generate : m_p = generateFrom p h_indep : Indep m_p (m a) \u03ba n : \u03b9 hn : n \u2208 S \u22a2 generateFrom (\u03c0 n) \u2264 generateFrom p ** refine le_generateFrom_piiUnionInter (S : Set \u03b9) hn ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM0to1.tr_respects ** \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 \u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T }) ** cases' e : M q T.1 with val ** case some \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 val : \u039b \u00d7 TM0.Stmt \u0393 e : M q T.head = some val \u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T }) ** cases' val with q' s ** case some.mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 e : M q T.head = some (q', s) \u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T }) ** simp only [FRespects, TM0.step, trCfg, e, Option.isSome, cond, Option.map_some'] ** case some.mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 e : M q T.head = some (q', s) \u22a2 Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }     {       l :=         match           match             M q'               (match s with                 | TM0.Stmt.move d => Tape.move d T                 | TM0.Stmt.write a => Tape.write a T).head with           | some val => true           | none => false with         | true => some (\u039b'.normal q')         | false => none,       var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T } ** revert e ** case some.mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 \u22a2 M q T.head = some (q', s) \u2192     Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }       {         l :=           match             match               M q'                 (match s with                   | TM0.Stmt.move d => Tape.move d T                   | TM0.Stmt.write a => Tape.write a T).head with             | some val => true             | none => false with           | true => some (\u039b'.normal q')           | false => none,         var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } ** have : TM1.step (tr M) \u27e8some (\u039b'.act s q'), (), T\u27e9 = some \u27e8some (\u039b'.normal q'), (), match s with\n    | TM0.Stmt.move d => T.move d\n    | TM0.Stmt.write a => T.write a\u27e9 := by\n  cases' s with d a <;> rfl ** case some.mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } \u22a2 M q T.head = some (q', s) \u2192     Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }       {         l :=           match             match               M q'                 (match s with                   | TM0.Stmt.move d => Tape.move d T                   | TM0.Stmt.write a => Tape.write a T).head with             | some val => true             | none => false with           | true => some (\u039b'.normal q')           | false => none,         var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } ** intro e ** case some.mk \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) \u22a2 Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }     {       l :=         match           match             M q'               (match s with                 | TM0.Stmt.move d => Tape.move d T                 | TM0.Stmt.write a => Tape.write a T).head with           | some val => true           | none => false with         | true => some (\u039b'.normal q')         | false => none,       var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T } ** refine' TransGen.head _ (TransGen.head' this _) ** case some.mk.refine'_2 \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) \u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)     { l := some (\u039b'.normal q'), var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T }     {       l :=         match           match             M q'               (match s with                 | TM0.Stmt.move d => Tape.move d T                 | TM0.Stmt.write a => Tape.write a T).head with           | some val => true           | none => false with         | true => some (\u039b'.normal q')         | false => none,       var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T } ** cases e' : M q' _ ** case none \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 e : M q T.head = none \u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T }) ** simp only [TM0.step, trCfg, e] ** case none \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 e : M q T.head = none \u22a2 FRespects (TM1.step (tr M))     (fun a =>       { l := bif Option.isSome (M a.q a.Tape.head) then some (\u039b'.normal a.q) else none, var := (), Tape := a.Tape })     { l := bif Option.isSome none then some (\u039b'.normal q) else none, var := (), Tape := T }     (Option.map       (fun x =>         { q := x.1,           Tape :=             match x.2 with             | TM0.Stmt.move d => Tape.move d T             | TM0.Stmt.write a => Tape.write a T })       none) ** exact Eq.refl none ** \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 \u22a2 TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } ** cases' s with d a <;> rfl ** case some.mk.refine'_1 \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) \u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208 TM1.step (tr M) { l := some (\u039b'.normal q), var := (), Tape := T } ** simp only [TM1.step, TM1.stepAux] ** case some.mk.refine'_1 \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) \u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208     some       (bif Option.isNone (M q T.head) then { l := none, var := (), Tape := T }       else         {           l :=             some               (match M q T.head with               | none => default               | some (q', s) => \u039b'.act s q'),           var := (), Tape := T }) ** rw [e] ** case some.mk.refine'_1 \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) \u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208     some       (bif Option.isNone (some (q', s)) then { l := none, var := (), Tape := T }       else         {           l :=             some               (match some (q', s) with               | none => default               | some (q', s) => \u039b'.act s q'),           var := (), Tape := T }) ** rfl ** case some.mk.refine'_2.none \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) e' :   M q'       (match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T).head =     none \u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)     { l := some (\u039b'.normal q'), var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T }     {       l :=         match           match none with           | some val => true           | none => false with         | true => some (\u039b'.normal q')         | false => none,       var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T } ** apply ReflTransGen.single ** case some.mk.refine'_2.none.hab \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) e' :   M q'       (match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T).head =     none \u22a2 {       l :=         match           match none with           | some val => true           | none => false with         | true => some (\u039b'.normal q')         | false => none,       var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T } \u2208     TM1.step (tr M)       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } ** simp only [TM1.step, TM1.stepAux] ** case some.mk.refine'_2.none.hab \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) e' :   M q'       (match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T).head =     none \u22a2 { l := none, var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T } \u2208     some       (bif           Option.isNone             (M q'               (match s with                 | TM0.Stmt.move d => Tape.move d T                 | TM0.Stmt.write a => Tape.write a T).head) then         { l := none, var := (),           Tape :=             match s with             | TM0.Stmt.move d => Tape.move d T             | TM0.Stmt.write a => Tape.write a T }       else         {           l :=             some               (match                 M q'                   (match s with                     | TM0.Stmt.move d => Tape.move d T                     | TM0.Stmt.write a => Tape.write a T).head with               | none => default               | some (q', s) => \u039b'.act s q'),           var := (),           Tape :=             match s with             | TM0.Stmt.move d => Tape.move d T             | TM0.Stmt.write a => Tape.write a T }) ** rw [e'] ** case some.mk.refine'_2.none.hab \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) e' :   M q'       (match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T).head =     none \u22a2 { l := none, var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T } \u2208     some       (bif Option.isNone none then         { l := none, var := (),           Tape :=             match s with             | TM0.Stmt.move d => Tape.move d T             | TM0.Stmt.write a => Tape.write a T }       else         {           l :=             some               (match none with               | none => default               | some (q', s) => \u039b'.act s q'),           var := (),           Tape :=             match s with             | TM0.Stmt.move d => Tape.move d T             | TM0.Stmt.write a => Tape.write a T }) ** rfl ** case some.mk.refine'_2.some \u0393 : Type u_1 inst\u271d\u00b9 : Inhabited \u0393 \u039b : Type u_2 inst\u271d : Inhabited \u039b M : TM0.Machine \u0393 \u039b x\u271d : Cfg\u2080 q : \u039b T : Tape \u0393 q' : \u039b s : TM0.Stmt \u0393 this :   TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =     some       { l := some (\u039b'.normal q'), var := (),         Tape :=           match s with           | TM0.Stmt.move d => Tape.move d T           | TM0.Stmt.write a => Tape.write a T } e : M q T.head = some (q', s) val\u271d : \u039b \u00d7 TM0.Stmt \u0393 e' :   M q'       (match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T).head =     some val\u271d \u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)     { l := some (\u039b'.normal q'), var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T }     {       l :=         match           match some val\u271d with           | some val => true           | none => false with         | true => some (\u039b'.normal q')         | false => none,       var := (),       Tape :=         match s with         | TM0.Stmt.move d => Tape.move d T         | TM0.Stmt.write a => Tape.write a T } ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "List.scanl_eq_scanlTR ** \u22a2 @scanl = @scanlTR ** funext \u03b1 f n l ** case h.h.h.h \u03b1 : Type u_2 f : Type u_1 n : \u03b1 \u2192 f \u2192 \u03b1 l : \u03b1 \u22a2 scanl n l = scanlTR n l ** simp [scanlTR, scanlTR_go_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.coeff_X ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R i : \u03c3 \u22a2 coeff (fun\u2080 | i => 1) (X i) = 1 ** classical rw [coeff_X', if_pos rfl] ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R i : \u03c3 \u22a2 coeff (fun\u2080 | i => 1) (X i) = 1 ** rw [coeff_X', if_pos rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Finite.subset ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 hs : Set.Finite s t : Set \u03b1 ht : t \u2286 s \u22a2 Set.Finite t ** cases hs ** case intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s t : Set \u03b1 ht : t \u2286 s a\u271d : Fintype \u2191s \u22a2 Set.Finite t ** haveI := Finite.Set.subset _ ht ** case intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s t : Set \u03b1 ht : t \u2286 s a\u271d : Fintype \u2191s this : Finite \u2191t \u22a2 Set.Finite t ** apply toFinite ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.of_int_cast ** \u03b1 : Type u_1 inst\u271d : AddGroupWithOne \u03b1 n : \u2124 \u22a2 \u2191\u2191n = \u2191n ** rw [\u2190 cast_to_int, to_of_int] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.is_left_invariant_chaar ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G \u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K ** let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (K.map _ <| continuous_mul_left g) - f K ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K \u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K ** have : Continuous eval := (continuous_apply (K.map _ _)).sub (continuous_apply K) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval \u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K\u2080 K ** rw [\u2190 sub_eq_zero] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval \u22a2 chaar K\u2080 (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - chaar K\u2080 K = 0 ** show chaar K\u2080 \u2208 eval \u207b\u00b9' {(0 : \u211d)} ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval \u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {0} ** apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval \u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' {0} ** unfold clPrehaar ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval \u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' {0} ** rw [IsClosed.closure_subset_iff] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval \u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' {0} ** rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9 ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {0} ** simp only [mem_singleton_iff, mem_preimage, sub_eq_zero] ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (\u2191K\u2080) U K ** apply is_left_invariant_prehaar ** case intro.intro.intro.intro.hU G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Set.Nonempty (interior U) ** rw [h2U.interior_eq] ** case intro.intro.intro.intro.hU G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Set.Nonempty U ** exact \u27e81, h3U\u27e9 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval \u22a2 IsClosed (eval \u207b\u00b9' {0}) ** apply continuous_iff_isClosed.mp this ** case a G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G g : G K : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K this : Continuous eval \u22a2 IsClosed {0} ** exact isClosed_singleton ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsStoppingTime.measurableSet_inter_le_iff ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 \u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u2194 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) ** constructor <;> intro h ** case mp \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 h : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) ** have : s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} := by\n  rw [Set.inter_assoc, Set.inter_self] ** case mp \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 h : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) this : s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) ** rw [this] ** case mp \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 h : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) this : s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) ** exact measurableSet_inter_le _ h\u03c0 _ h ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 h : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u22a2 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} = s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} ** rw [Set.inter_assoc, Set.inter_self] ** case mpr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 h : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) ** rw [measurableSet_min_iff h\u03c4 h\u03c0] at h ** case mpr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 h : MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u2227 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) ** exact h.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_eq_nnreal ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc = \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** rw [lintegral] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u22a2 \u2a06 g, \u2a06 (_ : \u2191g \u2264 fun a => f a), SimpleFunc.lintegral g \u03bc =     \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** refine'\n  le_antisymm (iSup\u2082_le fun \u03c6 h\u03c6 => _) (iSup_mono' fun \u03c6 => \u27e8\u03c6.map ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e), le_rfl\u27e9) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a \u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264     \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** by_cases h : \u2200\u1d50 a \u2202\u03bc, \u03c6 a \u2260 \u221e ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 \u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264     \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** let \u03c8 := \u03c6.map ENNReal.toNNReal ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6 \u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264     \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** replace h : \u03c8.map ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e) =\u1d50[\u03bc] \u03c6 := h.mono fun a => ENNReal.coe_toNNReal ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6 h : \u2191(SimpleFunc.map ENNReal.some \u03c8) =\u1d50[\u03bc] \u2191\u03c6 \u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264     \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** have : \u2200 x, \u2191(\u03c8 x) \u2264 f x := fun x => le_trans ENNReal.coe_toNNReal_le_self (h\u03c6 x) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map ENNReal.toNNReal \u03c6 h : \u2191(SimpleFunc.map ENNReal.some \u03c8) =\u1d50[\u03bc] \u2191\u03c6 this : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264     \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** exact\n  le_iSup_of_le (\u03c6.map ENNReal.toNNReal) (le_iSup_of_le this (ge_of_eq <| lintegral_congr h)) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 \u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264     \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** have h_meas : \u03bc (\u03c6 \u207b\u00b9' {\u221e}) \u2260 0 := mt measure_zero_iff_ae_nmem.1 h ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 h_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0 \u22a2 SimpleFunc.lintegral \u03c6 \u03bc \u2264     \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** refine' le_trans le_top (ge_of_eq <| (iSup_eq_top _).2 fun b hb => _) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 h_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0 b : \u211d\u22650\u221e hb : b < \u22a4 \u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc ** obtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, b < n * \u03bc (\u03c6 \u207b\u00b9' {\u221e}) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 h_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0 b : \u211d\u22650\u221e hb : b < \u22a4 \u22a2 \u2203 n, b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4})  case neg.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 h_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0 b : \u211d\u22650\u221e hb : b < \u22a4 n : \u2115 hn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc ** exact exists_nat_mul_gt h_meas (ne_of_lt hb) ** case neg.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 h_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0 b : \u211d\u22650\u221e hb : b < \u22a4 n : \u2115 hn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u22a2 \u2203 i, b < \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc ** use (const \u03b1 (n : \u211d\u22650)).restrict (\u03c6 \u207b\u00b9' {\u221e}) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 h_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0 b : \u211d\u22650\u221e hb : b < \u22a4 n : \u2115 hn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u22a2 b <     \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191(restrict (const \u03b1 \u2191n) (\u2191\u03c6 \u207b\u00b9' {\u22a4})) x) \u2264 f x),       SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (restrict (const \u03b1 \u2191n) (\u2191\u03c6 \u207b\u00b9' {\u22a4}))) \u03bc ** simp only [lt_iSup_iff, exists_prop, coe_restrict, \u03c6.measurableSet_preimage, coe_const,\n  ENNReal.coe_indicator, map_coe_ennreal_restrict, SimpleFunc.map_const, ENNReal.coe_nat,\n  restrict_const_lintegral] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 h_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0 b : \u211d\u22650\u221e hb : b < \u22a4 n : \u2115 hn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u22a2 (\u2200 (x : \u03b1), indicator (\u2191\u03c6 \u207b\u00b9' {\u22a4}) (fun x => \u2191(Function.const \u03b1 (\u2191n) x)) x \u2264 f x) \u2227 b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) ** refine' \u27e8indicator_le fun x hx => le_trans _ (h\u03c6 _), hn\u27e9 ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 h_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0 b : \u211d\u22650\u221e hb : b < \u22a4 n : \u2115 hn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) x : \u03b1 hx : x \u2208 \u2191\u03c6 \u207b\u00b9' {\u22a4} \u22a2 \u2191(Function.const \u03b1 (\u2191n) x) \u2264 \u2191\u03c6 x ** simp only [mem_preimage, mem_singleton_iff] at hx ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b1 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650\u221e h\u03c6 : \u2191\u03c6 \u2264 fun a => f a h : \u00ac\u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u03c6 a \u2260 \u22a4 h_meas : \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) \u2260 0 b : \u211d\u22650\u221e hb : b < \u22a4 n : \u2115 hn : b < \u2191n * \u2191\u2191\u03bc (\u2191\u03c6 \u207b\u00b9' {\u22a4}) x : \u03b1 hx : \u2191\u03c6 x = \u22a4 \u22a2 \u2191(Function.const \u03b1 (\u2191n) x) \u2264 \u2191\u03c6 x ** simp only [hx, le_top] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpInd_nonneg ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2074 : NormedAddCommGroup F inst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : CompleteSpace F' inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u2074 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b3 : SigmaFinite (Measure.trim \u03bc hm) E : Type u_8 inst\u271d\u00b2 : NormedLatticeAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E hx : 0 \u2264 x \u22a2 0 \u2264 \u2191(condexpInd E hm \u03bc s) x ** rw [\u2190 coeFn_le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2074 : NormedAddCommGroup F inst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : CompleteSpace F' inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u2074 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b3 : SigmaFinite (Measure.trim \u03bc hm) E : Type u_8 inst\u271d\u00b2 : NormedLatticeAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E hx : 0 \u2264 x \u22a2 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(\u2191(condexpInd E hm \u03bc s) x) ** refine' EventuallyLE.trans_eq _ (condexpInd_ae_eq_condexpIndSMul hm hs h\u03bcs x).symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2075 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2074 : NormedAddCommGroup F inst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F inst\u271d\u00b9\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9\u2070 : NormedSpace \u211d F' inst\u271d\u2079 : CompleteSpace F' inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u2074 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d\u00b3 : SigmaFinite (Measure.trim \u03bc hm) E : Type u_8 inst\u271d\u00b2 : NormedLatticeAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E hx : 0 \u2264 x \u22a2 \u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) ** exact (coeFn_zero E 1 \u03bc).trans_le (condexpIndSMul_nonneg hs h\u03bcs x hx) ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toMeasure_apply_eq_of_inter_support_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b1 p : PMF \u03b1 s\u271d t\u271d s t : Set \u03b1 hs : MeasurableSet s ht : MeasurableSet t h : s \u2229 support p = t \u2229 support p \u22a2 \u2191\u2191(toMeasure p) s = \u2191\u2191(toMeasure p) t ** simpa only [p.toMeasure_apply_eq_toOuterMeasure_apply, hs, ht] using\n  toOuterMeasure_apply_eq_of_inter_support_eq p h ** Qed",
        "informal": ""
    },
    {
        "formal": "WithBot.image_coe_Iio ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Iio a = Ioo \u22a5 \u2191a ** rw [\u2190 preimage_coe_Iio, image_preimage_eq_inter_range, range_coe, inter_comm, Ioi_inter_Iio] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.insert_WF ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets \u22a2 Buckets.WF (insert m k v).buckets ** dsimp [insert, cond] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets \u22a2 Buckets.WF     (match         AssocList.contains k           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with       | true =>         { size := m.size,           buckets :=             Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.replace k v                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val) }       | false =>         if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then           { size := m.size + 1,             buckets :=               Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val                 (AssocList.cons k v                   m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])                 (_ :                   USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                     Array.size m.buckets.val) }         else           expand (m.size + 1)             (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.cons k v                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val))).buckets ** split ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true \u22a2 Buckets.WF     { size := m.size,         buckets :=           Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.replace k v               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val) }.buckets ** simp at h\u2081 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true \u22a2 Buckets.WF     { size := m.size,         buckets :=           Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.replace k v               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val) }.buckets ** have \u27e8x, hx\u2081, hx\u2082\u27e9 := h\u2081 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true x : \u03b1 \u00d7 \u03b2 hx\u2081 :   x \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] hx\u2082 : (x.fst == k) = true \u22a2 Buckets.WF     { size := m.size,         buckets :=           Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.replace k v               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val) }.buckets ** refine h.update (fun H => ?_) (fun H a h => ?_) ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true x : \u03b1 \u00d7 \u03b2 hx\u2081 :   x \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] hx\u2082 : (x.fst == k) = true inst\u271d\u00b9 : PartialEquivBEq \u03b1 inst\u271d : LawfulHashable \u03b1 H :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (AssocList.toList       (AssocList.replace k v         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) ** simp ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true x : \u03b1 \u00d7 \u03b2 hx\u2081 :   x \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] hx\u2082 : (x.fst == k) = true inst\u271d\u00b9 : PartialEquivBEq \u03b1 inst\u271d : LawfulHashable \u03b1 H :   List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) \u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true)     (List.replaceF (fun x => bif x.fst == k then some (k, v) else none)       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) ** exact pairwise_replaceF H ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h\u271d : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true x : \u03b1 \u00d7 \u03b2 hx\u2081 :   x \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] hx\u2082 : (x.fst == k) = true H :   AssocList.All     (fun k_1 x =>       USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =         USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)     m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] a : \u03b1 \u00d7 \u03b2 h :   a \u2208     AssocList.toList       (AssocList.replace k v         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) \u22a2 (fun k_1 x =>       USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =         USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)     a.fst a.snd ** simp [AssocList.All] at H h \u22a2 ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h\u271d : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true x : \u03b1 \u00d7 \u03b2 hx\u2081 :   x \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] hx\u2082 : (x.fst == k) = true H :   \u2200 (a : \u03b1 \u00d7 \u03b2),     a \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =         USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val a : \u03b1 \u00d7 \u03b2 h :   a \u2208     List.replaceF (fun x => bif x.fst == k then some (k, v) else none)       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) \u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =     USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val ** match mem_replaceF h with\n| .inl rfl => rfl\n| .inr h => exact H _ h ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 v : \u03b2 h\u271d : Buckets.WF m.buckets c\u271d : Bool x a : \u03b1 \u00d7 \u03b2 h\u2081 :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val] \u2227       (x.fst == a.fst) = true hx\u2081 :   x \u2208     AssocList.toList       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val] hx\u2082 : (x.fst == a.fst) = true H :   \u2200 (a_1 : \u03b1 \u00d7 \u03b2),     a_1 \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val] \u2192       USize.toNat (UInt64.toUSize (hash a_1.fst) % Array.size m.buckets.val) =         USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val h :   a \u2208     List.replaceF (fun x => bif x.fst == a.fst then some (a.fst, v) else none)       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val]) \u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =     USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash a.fst))).val ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 v : \u03b2 h\u271d\u00b9 : Buckets.WF m.buckets c\u271d : Bool x a : \u03b1 \u00d7 \u03b2 k : \u03b1 h :   a \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] h\u2081 :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true hx\u2081 :   x \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] hx\u2082 : (x.fst == k) = true H :   \u2200 (a : \u03b1 \u00d7 \u03b2),     a \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =         USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val h\u271d :   a \u2208     List.replaceF (fun x => bif x.fst == k then some (k, v) else none)       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) \u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =     USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val ** exact H _ h ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     false \u22a2 Buckets.WF     (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then         { size := m.size + 1,           buckets :=             Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.cons k v                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val) }       else         expand (m.size + 1)           (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.cons k v               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val))).buckets ** rw [Bool.eq_false_iff] at h\u2081 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2260     true \u22a2 Buckets.WF     (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then         { size := m.size + 1,           buckets :=             Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.cons k v                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val) }       else         expand (m.size + 1)           (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.cons k v               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val))).buckets ** simp at h\u2081 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2200 (x : \u03b1 \u00d7 \u03b2),     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       \u00ac(x.fst == k) = true \u22a2 Buckets.WF     (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then         { size := m.size + 1,           buckets :=             Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.cons k v                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val) }       else         expand (m.size + 1)           (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.cons k v               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val))).buckets ** suffices _ by split <;> [exact this; refine expand_WF this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2200 (x : \u03b1 \u00d7 \u03b2),     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       \u00ac(x.fst == k) = true \u22a2 Buckets.WF     { size := m.size + 1,         buckets :=           Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.cons k v               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val) }.buckets ** refine h.update (.cons ?_) (fun H a h => ?_) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2200 (x : \u03b1 \u00d7 \u03b2),     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       \u00ac(x.fst == k) = true this : ?m.86533 \u22a2 Buckets.WF     (if numBucketsForCapacity (m.size + 1) \u2264 Array.size m.buckets.val then         { size := m.size + 1,           buckets :=             Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.cons k v                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val) }       else         expand (m.size + 1)           (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.cons k v               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val))).buckets ** split <;> [exact this; refine expand_WF this] ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : BEq \u03b1 inst\u271d\u00b2 : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2200 (x : \u03b1 \u00d7 \u03b2),     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       \u00ac(x.fst == k) = true inst\u271d\u00b9 : PartialEquivBEq \u03b1 inst\u271d : LawfulHashable \u03b1 \u22a2 \u2200 (a' : \u03b1 \u00d7 \u03b2),     a' \u2208 AssocList.toList m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       \u00ac((k, v).fst == a'.fst) = true ** exact fun a h h' => h\u2081 a h (PartialEquivBEq.symm h') ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h\u271d : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2200 (x : \u03b1 \u00d7 \u03b2),     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       \u00ac(x.fst == k) = true H :   AssocList.All     (fun k_1 x =>       USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =         USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)     m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] a : \u03b1 \u00d7 \u03b2 h :   a \u2208     AssocList.toList       (AssocList.cons k v         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) \u22a2 (fun k_1 x =>       USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =         USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)     a.fst a.snd ** cases h with\n| head => rfl\n| tail _ h => exact H _ h ** case refine_2.head \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2200 (x : \u03b1 \u00d7 \u03b2),     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       \u00ac(x.fst == k) = true H :   AssocList.All     (fun k_1 x =>       USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =         USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)     m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u22a2 USize.toNat (UInt64.toUSize (hash (k, v).fst) % Array.size m.buckets.val) =     USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val ** rfl ** case refine_2.tail \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 v : \u03b2 h\u271d : Buckets.WF m.buckets c\u271d : Bool h\u2081 :   \u2200 (x : \u03b1 \u00d7 \u03b2),     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2192       \u00ac(x.fst == k) = true H :   AssocList.All     (fun k_1 x =>       USize.toNat (UInt64.toUSize (hash k_1) % Array.size m.buckets.val) =         USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val)     m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] a : \u03b1 \u00d7 \u03b2 h :   List.Mem a     (AssocList.toList       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) \u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) =     USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val ** exact H _ h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.div_mem_nhds_one_of_haar_pos ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E \u22a2 E / E \u2208 \ud835\udcdd 1 ** obtain \u27e8L, hL, hLE, hLpos, hLtop\u27e9 : \u2203 L : Set G, MeasurableSet L \u2227 L \u2286 E \u2227 0 < \u03bc L \u2227 \u03bc L < \u221e :=\n  exists_subset_measure_lt_top hE hEpos ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 \u22a2 E / E \u2208 \ud835\udcdd 1 ** obtain \u27e8K, hKL, hK, hKpos\u27e9 : \u2203 (K : Set G), K \u2286 L \u2227 IsCompact K \u2227 0 < \u03bc K :=\n  MeasurableSet.exists_lt_isCompact_of_ne_top hL (ne_of_lt hLtop) hLpos ** case intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K \u22a2 E / E \u2208 \ud835\udcdd 1 ** have hKtop : \u03bc K \u2260 \u221e := by\n  apply ne_top_of_le_ne_top (ne_of_lt hLtop)\n  apply measure_mono hKL ** case intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 \u22a2 E / E \u2208 \ud835\udcdd 1 ** obtain \u27e8U, hUK, hU, h\u03bcUK\u27e9 : \u2203 (U : Set G), U \u2287 K \u2227 IsOpen U \u2227 \u03bc U < \u03bc K + \u03bc K :=\n  Set.exists_isOpen_lt_add K hKtop hKpos.ne' ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K \u22a2 E / E \u2208 \ud835\udcdd 1 ** obtain \u27e8V, hV1, hVKU\u27e9 : \u2203 V \u2208 \ud835\udcdd (1 : G), V * K \u2286 U :=\n  compact_open_separated_mul_left hK hU hUK ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U \u22a2 E / E \u2208 \ud835\udcdd 1 ** have hv : \u2200 v : G, v \u2208 V \u2192 \u00acDisjoint ({v} * K) K := by\n  intro v hv hKv\n  have hKvsub : {v} * K \u222a K \u2286 U := by\n    apply Set.union_subset _ hUK\n    apply _root_.subset_trans _ hVKU\n    apply Set.mul_subset_mul _ (Set.Subset.refl K)\n    simp only [Set.singleton_subset_iff, hv]\n  replace hKvsub := @measure_mono _ _ \u03bc _ _ hKvsub\n  have hcontr := lt_of_le_of_lt hKvsub h\u03bcUK\n  rw [measure_union hKv (IsCompact.measurableSet hK)] at hcontr\n  have hKtranslate : \u03bc ({v} * K) = \u03bc K := by\n    simp only [singleton_mul, image_mul_left, measure_preimage_mul]\n  rw [hKtranslate, lt_self_iff_false] at hcontr\n  assumption ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U hv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K \u22a2 E / E \u2208 \ud835\udcdd 1 ** suffices V \u2286 E / E from Filter.mem_of_superset hV1 this ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U hv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K \u22a2 V \u2286 E / E ** intro v hvV ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U hv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K v : G hvV : v \u2208 V \u22a2 v \u2208 E / E ** obtain \u27e8x, hxK, hxvK\u27e9 : \u2203 x : G, x \u2208 {v} * K \u2227 x \u2208 K := Set.not_disjoint_iff.1 (hv v hvV) ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U hv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K v : G hvV : v \u2208 V x : G hxK : x \u2208 {v} * K hxvK : x \u2208 K \u22a2 v \u2208 E / E ** refine' \u27e8x, v\u207b\u00b9 * x, hLE (hKL hxvK), _, _\u27e9 ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K \u22a2 \u2191\u2191\u03bc K \u2260 \u22a4 ** apply ne_top_of_le_ne_top (ne_of_lt hLtop) ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K \u22a2 \u2191\u2191\u03bc K \u2264 \u2191\u2191\u03bc L ** apply measure_mono hKL ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U \u22a2 \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K ** intro v hv hKv ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K \u22a2 False ** have hKvsub : {v} * K \u222a K \u2286 U := by\n  apply Set.union_subset _ hUK\n  apply _root_.subset_trans _ hVKU\n  apply Set.mul_subset_mul _ (Set.Subset.refl K)\n  simp only [Set.singleton_subset_iff, hv] ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K hKvsub : {v} * K \u222a K \u2286 U \u22a2 False ** replace hKvsub := @measure_mono _ _ \u03bc _ _ hKvsub ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K hKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U \u22a2 False ** have hcontr := lt_of_le_of_lt hKvsub h\u03bcUK ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K hKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U hcontr : \u2191\u2191\u03bc ({v} * K \u222a K) < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K \u22a2 False ** rw [measure_union hKv (IsCompact.measurableSet hK)] at hcontr ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K hKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U hcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K \u22a2 False ** have hKtranslate : \u03bc ({v} * K) = \u03bc K := by\n  simp only [singleton_mul, image_mul_left, measure_preimage_mul] ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K hKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U hcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K hKtranslate : \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K \u22a2 False ** rw [hKtranslate, lt_self_iff_false] at hcontr ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K hKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U hcontr : False hKtranslate : \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K \u22a2 False ** assumption ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K \u22a2 {v} * K \u222a K \u2286 U ** apply Set.union_subset _ hUK ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K \u22a2 {v} * K \u2286 U ** apply _root_.subset_trans _ hVKU ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K \u22a2 {v} * K \u2286 V * K ** apply Set.mul_subset_mul _ (Set.Subset.refl K) ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K \u22a2 {v} \u2286 V ** simp only [Set.singleton_subset_iff, hv] ** G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U v : G hv : v \u2208 V hKv : Disjoint ({v} * K) K hKvsub : \u2191\u2191\u03bc ({v} * K \u222a K) \u2264 \u2191\u2191\u03bc U hcontr : \u2191\u2191\u03bc ({v} * K) + \u2191\u2191\u03bc K < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K \u22a2 \u2191\u2191\u03bc ({v} * K) = \u2191\u2191\u03bc K ** simp only [singleton_mul, image_mul_left, measure_preimage_mul] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U hv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K v : G hvV : v \u2208 V x : G hxK : x \u2208 {v} * K hxvK : x \u2208 K \u22a2 v\u207b\u00b9 * x \u2208 E ** apply hKL.trans hLE ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.a G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U hv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K v : G hvV : v \u2208 V x : G hxK : x \u2208 {v} * K hxvK : x \u2208 K \u22a2 v\u207b\u00b9 * x \u2208 K ** simpa only [singleton_mul, image_mul_left, mem_preimage] using hxK ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 G : Type u_1 inst\u271d\u2078 : Group G inst\u271d\u2077 : TopologicalSpace G inst\u271d\u2076 : T2Space G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : BorelSpace G inst\u271d\u00b2 : SecondCountableTopology G \u03bc : Measure G inst\u271d\u00b9 : IsHaarMeasure \u03bc inst\u271d : LocallyCompactSpace G E : Set G hE : MeasurableSet E hEpos : 0 < \u2191\u2191\u03bc E L : Set G hL : MeasurableSet L hLE : L \u2286 E hLpos : 0 < \u2191\u2191\u03bc L hLtop : \u2191\u2191\u03bc L < \u22a4 K : Set G hKL : K \u2286 L hK : IsCompact K hKpos : 0 < \u2191\u2191\u03bc K hKtop : \u2191\u2191\u03bc K \u2260 \u22a4 U : Set G hUK : U \u2287 K hU : IsOpen U h\u03bcUK : \u2191\u2191\u03bc U < \u2191\u2191\u03bc K + \u2191\u2191\u03bc K V : Set G hV1 : V \u2208 \ud835\udcdd 1 hVKU : V * K \u2286 U hv : \u2200 (v : G), v \u2208 V \u2192 \u00acDisjoint ({v} * K) K v : G hvV : v \u2208 V x : G hxK : x \u2208 {v} * K hxvK : x \u2208 K \u22a2 (fun x x_1 => x / x_1) x (v\u207b\u00b9 * x) = v ** simp only [div_eq_iff_eq_mul, \u2190 mul_assoc, mul_right_inv, one_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.toDual_min' ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 hs : Finset.Nonempty s \u22a2 \u2191toDual (min' s hs) = max' (image (\u2191toDual) s) (_ : Finset.Nonempty (image (\u2191toDual) s)) ** rw [\u2190 WithBot.coe_eq_coe] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 hs : Finset.Nonempty s \u22a2 \u2191(\u2191toDual (min' s hs)) = \u2191(max' (image (\u2191toDual) s) (_ : Finset.Nonempty (image (\u2191toDual) s))) ** simp only [min'_eq_inf', id_eq, toDual_inf', Function.comp_apply, coe_sup', max'_eq_sup',\n  sup_image] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1 hs : Finset.Nonempty s \u22a2 sup s (WithBot.some \u2218 fun x => \u2191toDual x) = sup s ((WithBot.some \u2218 fun x => x) \u2218 \u2191toDual) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.dirac_bind ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure \u03b1 \u22a2 bind m dirac = m ** ext1 s hs ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 m : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(bind m dirac) s = \u2191\u2191m s ** simp only [bind_apply hs measurable_dirac, dirac_apply' _ hs, lintegral_indicator 1 hs,\n  Pi.one_apply, lintegral_one, restrict_apply, MeasurableSet.univ, univ_inter] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_eq_lintegral_pos_part_sub_lintegral_neg_part ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =     ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) ** let f\u2081 := hf.toL1 f ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =     ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) ** have eq\u2081 : ENNReal.toReal (\u222b\u207b a, ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 := by\n  rw [L1.norm_def]\n  congr 1\n  apply lintegral_congr_ae\n  filter_upwards [Lp.coeFn_posPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n  rw [h\u2081, h\u2082, ENNReal.ofReal]\n  congr 1\n  apply NNReal.eq\n  rw [Real.nnnorm_of_nonneg (le_max_right _ _)]\n  rw [Real.coe_toNNReal', NNReal.coe_mk] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =     ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) ** have eq\u2082 : ENNReal.toReal (\u222b\u207b a, ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 := by\n  rw [L1.norm_def]\n  congr 1\n  apply lintegral_congr_ae\n  filter_upwards [Lp.coeFn_negPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n  rw [h\u2081, h\u2082, ENNReal.ofReal]\n  congr 1\n  apply NNReal.eq\n  simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg]\n  rw [Real.norm_of_nonpos (min_le_right _ _), \u2190 max_neg_neg, neg_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc =     ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) ** rw [eq\u2081, eq\u2082, integral, dif_pos, dif_pos] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 L1.integral (Integrable.toL1 (fun a => f a) ?hc) = \u2016Lp.posPart f\u2081\u2016 - \u2016Lp.negPart f\u2081\u2016  case hc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 Integrable fun a => f a  case hc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 CompleteSpace \u211d  case hc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 CompleteSpace \u211d ** exact L1.integral_eq_norm_posPart_sub _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 ** rw [L1.norm_def] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc) ** congr 1 ** case e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc ** apply lintegral_congr_ae ** case e_a.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 (fun a => ENNReal.ofReal (f a)) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a ** filter_upwards [Lp.coeFn_posPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 ENNReal.ofReal (f a\u271d) = \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u271d\u2016\u208a ** rw [h\u2081, h\u2082, ENNReal.ofReal] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191\u2016max (f a\u271d) 0\u2016\u208a ** congr 1 ** case h.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 Real.toNNReal (f a\u271d) = \u2016max (f a\u271d) 0\u2016\u208a ** apply NNReal.eq ** case h.e_a.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191\u2016max (f a\u271d) 0\u2016\u208a ** rw [Real.nnnorm_of_nonneg (le_max_right _ _)] ** case h.e_a.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191{ val := max (f a\u271d) 0, property := (_ : 0 \u2264 max (f a\u271d) 0) } ** rw [Real.coe_toNNReal', NNReal.coe_mk] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 ** rw [L1.norm_def] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc) ** congr 1 ** case e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc ** apply lintegral_congr_ae ** case e_a.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 (fun a => ENNReal.ofReal (-f a)) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a ** filter_upwards [Lp.coeFn_negPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 ENNReal.ofReal (-f a\u271d) = \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u271d\u2016\u208a ** rw [h\u2081, h\u2082, ENNReal.ofReal] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (-f a\u271d)) = \u2191\u2016-min (f a\u271d) 0\u2016\u208a ** congr 1 ** case h.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 Real.toNNReal (-f a\u271d) = \u2016-min (f a\u271d) 0\u2016\u208a ** apply NNReal.eq ** case h.e_a.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (-f a\u271d)) = \u2191\u2016-min (f a\u271d) 0\u2016\u208a ** simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg] ** case h.e_a.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 max (-f a\u271d) 0 = \u2016min (f a\u271d) 0\u2016 ** rw [Real.norm_of_nonpos (min_le_right _ _), \u2190 max_neg_neg, neg_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.HasFiniteIntegral.tendsto_set_integral_nhds_zero ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 f : \u03b1 \u2192 G hf : HasFiniteIntegral f l : Filter \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0) \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s i, f x \u2202\u03bc) l (\ud835\udcdd 0) ** rw [tendsto_zero_iff_norm_tendsto_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 f : \u03b1 \u2192 G hf : HasFiniteIntegral f l : Filter \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0) \u22a2 Tendsto (fun x => \u2016\u222b (x : \u03b1) in s x, f x \u2202\u03bc\u2016) l (\ud835\udcdd 0) ** simp_rw [\u2190 coe_nnnorm, \u2190 NNReal.coe_zero, NNReal.tendsto_coe, \u2190 ENNReal.tendsto_coe,\n  ENNReal.coe_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 f : \u03b1 \u2192 G hf : HasFiniteIntegral f l : Filter \u03b9 s : \u03b9 \u2192 Set \u03b1 hs : Tendsto (\u2191\u2191\u03bc \u2218 s) l (\ud835\udcdd 0) \u22a2 Tendsto (fun a => \u2191\u2016\u222b (x : \u03b1) in s a, f x \u2202\u03bc\u2016\u208a) l (\ud835\udcdd 0) ** exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds\n  (tendsto_set_lintegral_zero (ne_of_lt hf) hs) (fun i => zero_le _)\n  fun i => ennnorm_integral_le_lintegral_ennnorm _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.to_of_nat ** \u22a2 \u2191\u21910 = 0 ** rw [Nat.cast_zero, cast_zero] ** n : \u2115 \u22a2 \u2191\u2191(n + 1) = n + 1 ** rw [Nat.cast_succ, add_one, succ_to_nat, to_of_nat n] ** Qed",
        "informal": ""
    },
    {
        "formal": "Condensed.StoneanProfinite.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily ** X : Stonean S : Sieve X \u22a2 (\u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a)) \u2194     S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X ** refine \u27e8fun \u27e8\u03b1, _, Y, \u03c0, \u27e8H\u2081, H\u2082\u27e9\u27e9 \u21a6 ?_, fun hS \u21a6 ?_\u27e9 ** case refine_1 X : Stonean S : Sieve X x\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Stonean \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a) \u22a2 S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X ** apply (coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily (Sieve.functorPushforward _ S)).mpr ** case refine_1 X : Stonean S : Sieve X x\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Stonean \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a) \u22a2 \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) ** refine \u27e8\u03b1, inferInstance, fun i => Stonean.toProfinite.obj (Y i),\n  fun i => Stonean.toProfinite.map (\u03c0 i), \u27e8?_,\n  fun a => Sieve.image_mem_functorPushforward Stonean.toCompHaus S (H\u2082 a)\u27e9\u27e9 ** case refine_1 X : Stonean S : Sieve X x\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Stonean \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a) \u22a2 EffectiveEpiFamily (fun i => Stonean.toProfinite.obj (Y i)) fun i => Stonean.toProfinite.map (\u03c0 i) ** simp only [(Stonean.effectiveEpiFamily_tfae _ _).out 0 2] at H\u2081 ** case refine_1 X : Stonean S : Sieve X x\u271d : \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Stonean \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 X H\u2082 : \u2200 (a : \u03b1), S.arrows (\u03c0 a) H\u2081 : \u2200 (b : CoeSort.coe X), \u2203 a x, \u2191(\u03c0 a) x = b \u22a2 EffectiveEpiFamily (fun i => Stonean.toProfinite.obj (Y i)) fun i => Stonean.toProfinite.map (\u03c0 i) ** exact Profinite.effectiveEpiFamily_of_jointly_surjective\n    (fun i => Stonean.toProfinite.obj (Y i)) (fun i => Stonean.toProfinite.map (\u03c0 i)) H\u2081 ** case refine_2 X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u22a2 \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) ** obtain \u27e8\u03b1, _, Y, \u03c0, \u27e8H\u2081, H\u2082\u27e9\u27e9 := (coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily _).mp hS ** case refine_2.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) \u22a2 \u2203 \u03b1 x Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) ** refine \u27e8\u03b1, inferInstance, ?_\u27e9 ** case refine_2.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) \u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) ** obtain \u27e8Y\u2080, H\u2082\u27e9 := Classical.skolem.mp H\u2082 ** case refine_2.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082\u271d : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) ** obtain \u27e8\u03c0\u2080, H\u2082\u27e9 := Classical.skolem.mp H\u2082 ** case refine_2.intro.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082\u271d\u00b9 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082\u271d : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X H\u2082 : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x) \u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) ** obtain \u27e8f\u2080, H\u2082\u27e9 := Classical.skolem.mp H\u2082 ** case refine_2.intro.intro.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X H\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x) f\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x) H\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x) \u22a2 \u2203 Y \u03c0, EffectiveEpiFamily Y \u03c0 \u2227 \u2200 (a : \u03b1), S.arrows (\u03c0 a) ** refine \u27e8Y\u2080 , \u03c0\u2080, \u27e8?_, fun i \u21a6 (H\u2082 i).1\u27e9\u27e9 ** case refine_2.intro.intro.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X H\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x) f\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x) H\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x) \u22a2 EffectiveEpiFamily Y\u2080 \u03c0\u2080 ** simp only [(Stonean.effectiveEpiFamily_tfae _ _).out 0 2] ** case refine_2.intro.intro.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2081 : EffectiveEpiFamily Y \u03c0 H\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X H\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x) f\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x) H\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x) \u22a2 \u2200 (b : CoeSort.coe X), \u2203 a x, \u2191(\u03c0\u2080 a) x = b ** simp only [(Profinite.effectiveEpiFamily_tfae _ _).out 0 2] at H\u2081 ** case refine_2.intro.intro.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X H\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x) f\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x) H\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x) H\u2081 : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b \u22a2 \u2200 (b : CoeSort.coe X), \u2203 a x, \u2191(\u03c0\u2080 a) x = b ** intro b ** case refine_2.intro.intro.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X H\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x) f\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x) H\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x) H\u2081 : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b b : CoeSort.coe X \u22a2 \u2203 a x, \u2191(\u03c0\u2080 a) x = b ** obtain \u27e8i, x, H\u2081\u27e9 := H\u2081 b ** case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X H\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x) f\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x) H\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x) H\u2081\u271d : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b b : CoeSort.coe X i : \u03b1 x : \u2191(Y i).toCompHaus.toTop H\u2081 : \u2191(\u03c0 i) x = b \u22a2 \u2203 a x, \u2191(\u03c0\u2080 a) x = b ** refine \u27e8i, f\u2080 i x, ?_\u27e9 ** case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X H\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x) f\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x) H\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x) H\u2081\u271d : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b b : CoeSort.coe X i : \u03b1 x : \u2191(Y i).toCompHaus.toTop H\u2081 : \u2191(\u03c0 i) x = b \u22a2 \u2191(\u03c0\u2080 i) (\u2191(f\u2080 i) x) = b ** rw [\u2190 H\u2081, (H\u2082 i).2] ** case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro X : Stonean S : Sieve X hS : S \u2208 GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X \u03b1 : Type w\u271d : Fintype \u03b1 Y : \u03b1 \u2192 Profinite \u03c0 : (a : \u03b1) \u2192 Y a \u27f6 Stonean.toProfinite.obj X H\u2082\u271d\u00b2 : \u2200 (a : \u03b1), (Sieve.functorPushforward Stonean.toProfinite S).arrows (\u03c0 a) Y\u2080 : \u03b1 \u2192 Stonean H\u2082\u271d\u00b9 : \u2200 (x : \u03b1), \u2203 g h, S.arrows g \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map g \u03c0\u2080 : (x : \u03b1) \u2192 Y\u2080 x \u27f6 X H\u2082\u271d : \u2200 (x : \u03b1), \u2203 h, S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = h \u226b Stonean.toProfinite.map (\u03c0\u2080 x) f\u2080 : (x : \u03b1) \u2192 Y x \u27f6 Stonean.toProfinite.obj (Y\u2080 x) H\u2082 : \u2200 (x : \u03b1), S.arrows (\u03c0\u2080 x) \u2227 \u03c0 x = f\u2080 x \u226b Stonean.toProfinite.map (\u03c0\u2080 x) H\u2081\u271d : \u2200 (b : \u2191(Stonean.toProfinite.obj X).toCompHaus.toTop), \u2203 a x, \u2191(\u03c0 a) x = b b : CoeSort.coe X i : \u03b1 x : \u2191(Y i).toCompHaus.toTop H\u2081 : \u2191(\u03c0 i) x = b \u22a2 \u2191(\u03c0\u2080 i) (\u2191(f\u2080 i) x) = \u2191(f\u2080 i \u226b Stonean.toProfinite.map (\u03c0\u2080 i)) x ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.foldr ** \u03b1 : Type u_1 l m r : List Char f : Char \u2192 \u03b1 \u2192 \u03b1 init : \u03b1 \u22a2 Substring.foldr f init       { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },         stopPos := { byteIdx := utf8Len l + utf8Len m } } =     List.foldr f init m ** simp [-List.append_assoc, Substring.foldr, foldrAux_of_valid] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.strong_law_aux3 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 \u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) =o[atTop]     Nat.cast ** have A : Tendsto (fun i => \ud835\udd3c[truncation (X i) i]) atTop (\ud835\udcdd \ud835\udd3c[X 0]) := by\n  convert (tendsto_integral_truncation hint).comp tendsto_nat_cast_atTop_atTop using 1\n  ext i\n  exact (hident i).truncation.integral_eq ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 A : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) =o[atTop]     Nat.cast ** convert Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 A) using 1 ** case h.e'_7 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 A : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u22a2 (fun n => (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a) = fun n =>     \u2211 i in range n, ((\u222b (a : \u03a9), truncation (X i) (\u2191i) a) - \u222b (a : \u03a9), X 0 a) ** ext1 n ** case h.e'_7.h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 A : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) n : \u2115 \u22a2 (\u222b (a : \u03a9), Finset.sum (range n) (fun i => truncation (X i) \u2191i) a) - \u2191n * \u222b (a : \u03a9), X 0 a =     \u2211 i in range n, ((\u222b (a : \u03a9), truncation (X i) (\u2191i) a) - \u222b (a : \u03a9), X 0 a) ** simp only [sum_sub_distrib, sum_const, card_range, nsmul_eq_mul, sum_apply, sub_left_inj] ** case h.e'_7.h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 A : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) n : \u2115 \u22a2 \u222b (a : \u03a9), \u2211 c in range n, truncation (X c) (\u2191c) a = \u2211 x in range n, \u222b (a : \u03a9), truncation (X x) (\u2191x) a ** rw [integral_finset_sum _ fun i _ => ?_] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 A : Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) n i : \u2115 x\u271d : i \u2208 range n \u22a2 Integrable fun a => truncation (X i) (\u2191i) a ** exact ((hident i).symm.integrable_snd hint).1.integrable_truncation ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 \u22a2 Tendsto (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** convert (tendsto_integral_truncation hint).comp tendsto_nat_cast_atTop_atTop using 1 ** case h.e'_3 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 \u22a2 (fun i => \u222b (a : \u03a9), truncation (X i) (\u2191i) a) = (fun A => \u222b (x : \u03a9), truncation (X 0) A x) \u2218 Nat.cast ** ext i ** case h.e'_3.h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 i : \u2115 \u22a2 \u222b (a : \u03a9), truncation (X i) (\u2191i) a = ((fun A => \u222b (x : \u03a9), truncation (X 0) A x) \u2218 Nat.cast) i ** exact (hident i).truncation.integral_eq ** Qed",
        "informal": ""
    },
    {
        "formal": "ContinuousMap.toLp_norm_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E inst\u271d\u2074 : CompactSpace \u03b1 inst\u271d\u00b3 : IsFiniteMeasure \u03bc \ud835\udd5c : Type u_5 inst\u271d\u00b2 : Fact (1 \u2264 p) inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E \u22a2 \u2016toLp p \u03bc \ud835\udd5c\u2016 \u2264 \u2191(measureUnivNNReal \u03bc ^ (ENNReal.toReal p)\u207b\u00b9) ** rw [toLp_norm_eq_toLp_norm_coe] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E inst\u271d\u2074 : CompactSpace \u03b1 inst\u271d\u00b3 : IsFiniteMeasure \u03bc \ud835\udd5c : Type u_5 inst\u271d\u00b2 : Fact (1 \u2264 p) inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E \u22a2 \u2016BoundedContinuousFunction.toLp p \u03bc \ud835\udd5c\u2016 \u2264 \u2191(measureUnivNNReal \u03bc ^ (ENNReal.toReal p)\u207b\u00b9) ** exact BoundedContinuousFunction.toLp_norm_le \u03bc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_prod_symm ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2' inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : AEMeasurable f \u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd ** simp_rw [\u2190 lintegral_prod_swap f] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MeasurableSpace \u03b1' inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace \u03b2' inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e hf : AEMeasurable f \u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b1), f (Prod.swap z) \u2202Measure.prod \u03bd \u03bc = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd ** exact lintegral_prod _ hf.prod_swap ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_const_add_Ioi ** M : Type u_1 inst\u271d\u00b9 : OrderedCancelAddCommMonoid M inst\u271d : ExistsAddOfLE M a b c d : M \u22a2 (fun x => a + x) '' Ioi b = Ioi (a + b) ** simp only [add_comm a, image_add_const_Ioi] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.disjoint_or_nonempty_inter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1 a b : \u03b1 s t : Finset \u03b1 \u22a2 _root_.Disjoint s t \u2228 Finset.Nonempty (s \u2229 t) ** rw [\u2190 not_disjoint_iff_nonempty_inter] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1 a b : \u03b1 s t : Finset \u03b1 \u22a2 _root_.Disjoint s t \u2228 \u00ac_root_.Disjoint s t ** exact em _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.map_matrix_volume_pi_eq_smul_volume_pi ** \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 \u22a2 Measure.map (\u2191(\u2191toLin' M)) volume = ofReal |(det M)\u207b\u00b9| \u2022 volume ** apply diagonal_transvection_induction_of_det_ne_zero _ M hM ** case hdiag \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 \u22a2 \u2200 (D : \u03b9 \u2192 \u211d),     det (Matrix.diagonal D) \u2260 0 \u2192       Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume ** intro D hD ** case hdiag \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 D : \u03b9 \u2192 \u211d hD : det (Matrix.diagonal D) \u2260 0 \u22a2 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume ** conv_rhs => rw [\u2190 smul_map_diagonal_volume_pi hD] ** case hdiag \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 D : \u03b9 \u2192 \u211d hD : det (Matrix.diagonal D) \u2260 0 \u22a2 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume =     ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022       ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume ** rw [smul_smul, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, inv_mul_cancel hD, abs_one,\n  ENNReal.ofReal_one, one_smul] ** case htransvec \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 \u22a2 \u2200 (t : TransvectionStruct \u03b9 \u211d),     Measure.map (\u2191(\u2191toLin' (TransvectionStruct.toMatrix t))) volume =       ofReal |(det (TransvectionStruct.toMatrix t))\u207b\u00b9| \u2022 volume ** intro t ** case htransvec \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 t : TransvectionStruct \u03b9 \u211d \u22a2 Measure.map (\u2191(\u2191toLin' (TransvectionStruct.toMatrix t))) volume =     ofReal |(det (TransvectionStruct.toMatrix t))\u207b\u00b9| \u2022 volume ** simp only [Matrix.TransvectionStruct.det, ENNReal.ofReal_one,\n  (volume_preserving_transvectionStruct _).map_eq, one_smul, _root_.inv_one, abs_one] ** case hmul \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 \u22a2 \u2200 (A B : Matrix \u03b9 \u03b9 \u211d),     det A \u2260 0 \u2192       det B \u2260 0 \u2192         Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume \u2192           Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume \u2192             Measure.map (\u2191(\u2191toLin' (A * B))) volume = ofReal |(det (A * B))\u207b\u00b9| \u2022 volume ** intro A B _ _ IHA IHB ** case hmul \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 A B : Matrix \u03b9 \u03b9 \u211d a\u271d\u00b9 : det A \u2260 0 a\u271d : det B \u2260 0 IHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume IHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume \u22a2 Measure.map (\u2191(\u2191toLin' (A * B))) volume = ofReal |(det (A * B))\u207b\u00b9| \u2022 volume ** rw [toLin'_mul, det_mul, LinearMap.coe_comp, \u2190 Measure.map_map, IHB, Measure.map_smul, IHA,\n  smul_smul, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, mul_comm, mul_inv] ** case hmul.hg \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 A B : Matrix \u03b9 \u03b9 \u211d a\u271d\u00b9 : det A \u2260 0 a\u271d : det B \u2260 0 IHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume IHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume \u22a2 Measurable \u2191(\u2191toLin' A) ** apply Continuous.measurable ** case hmul.hg.hf \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 A B : Matrix \u03b9 \u03b9 \u211d a\u271d\u00b9 : det A \u2260 0 a\u271d : det B \u2260 0 IHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume IHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume \u22a2 Continuous \u2191(\u2191toLin' A) ** apply LinearMap.continuous_on_pi ** case hmul.hf \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 A B : Matrix \u03b9 \u03b9 \u211d a\u271d\u00b9 : det A \u2260 0 a\u271d : det B \u2260 0 IHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume IHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume \u22a2 Measurable \u2191(\u2191toLin' B) ** apply Continuous.measurable ** case hmul.hf.hf \u03b9 : Type u_1 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 M : Matrix \u03b9 \u03b9 \u211d hM : det M \u2260 0 A B : Matrix \u03b9 \u03b9 \u211d a\u271d\u00b9 : det A \u2260 0 a\u271d : det B \u2260 0 IHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume IHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume \u22a2 Continuous \u2191(\u2191toLin' B) ** apply LinearMap.continuous_on_pi ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Path.insertNew_toList ** \u03b1 : Type u_1 v : \u03b1 p : Path \u03b1 \u22a2 toList (insertNew p v) = withList p [v] ** simp [insertNew] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.countable_meas_le_ne_meas_lt ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : LinearOrder R g : \u03b1 \u2192 R \u22a2 Set.Countable {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}} ** let F : R \u2192 \u211d\u22650\u221e := fun t \u21a6 \u03bc {a : \u03b1 | t \u2264 g a} ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : LinearOrder R g : \u03b1 \u2192 R F : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a} \u22a2 Set.Countable {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}} ** apply (countable_image_gt_image_Ioi F).mono ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : LinearOrder R g : \u03b1 \u2192 R F : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a} \u22a2 {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}} \u2286 {x | \u2203 z, z < F x \u2227 \u2200 (y : R), x < y \u2192 F y \u2264 z} ** intro t ht ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : LinearOrder R g : \u03b1 \u2192 R F : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a} t : R ht : t \u2208 {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}} \u22a2 t \u2208 {x | \u2203 z, z < F x \u2227 \u2200 (y : R), x < y \u2192 F y \u2264 z} ** have : \u03bc {a | t < g a} < \u03bc {a | t \u2264 g a} :=\n  lt_of_le_of_ne (measure_mono (fun a ha \u21a6 le_of_lt ha)) (Ne.symm ht) ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : LinearOrder R g : \u03b1 \u2192 R F : R \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191\u03bc {a | t \u2264 g a} t : R ht : t \u2208 {t | \u2191\u2191\u03bc {a | t \u2264 g a} \u2260 \u2191\u2191\u03bc {a | t < g a}} this : \u2191\u2191\u03bc {a | t < g a} < \u2191\u2191\u03bc {a | t \u2264 g a} \u22a2 t \u2208 {x | \u2203 z, z < F x \u2227 \u2200 (y : R), x < y \u2192 F y \u2264 z} ** refine \u27e8\u03bc {a | t < g a}, this, fun s hs \u21a6 measure_mono (fun a ha \u21a6 hs.trans_le ha)\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.martingalePart_add_ae_eq ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 \u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n ** set h := f - martingalePart (f + g) \u2131 \u03bc with hhdef ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc \u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n ** have hh : h = predictablePart (f + g) \u2131 \u03bc - g := by\n  rw [hhdef, sub_eq_sub_iff_add_eq_add, add_comm (predictablePart (f + g) \u2131 \u03bc),\n    martingalePart_add_predictablePart] ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g \u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n ** have hhpred : Adapted \u2131 fun n => h (n + 1) := by\n  rw [hh]\n  exact adapted_predictablePart.sub hg ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g hhpred : Adapted \u2131 fun n => h (n + 1) \u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n ** have hhmgle : Martingale h \u2131 \u03bc := hf.sub (martingale_martingalePart\n  (hf.adapted.add <| Predictable.adapted hg <| hg0.symm \u25b8 stronglyMeasurable_zero) fun n =>\n  (hf.integrable n).add <| hgint n) ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g hhpred : Adapted \u2131 fun n => h (n + 1) hhmgle : Martingale h \u2131 \u03bc \u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n ** refine' (eventuallyEq_iff_sub.2 _).symm ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g hhpred : Adapted \u2131 fun n => h (n + 1) hhmgle : Martingale h \u2131 \u03bc \u22a2 f n - martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] 0 ** filter_upwards [hhmgle.eq_zero_of_predictable hhpred n] with \u03c9 h\u03c9 ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g hhpred : Adapted \u2131 fun n => h (n + 1) hhmgle : Martingale h \u2131 \u03bc \u03c9 : \u03a9 h\u03c9 : h n \u03c9 = h 0 \u03c9 \u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9 ** unfold_let h at h\u03c9 ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g hhpred : Adapted \u2131 fun n => h (n + 1) hhmgle : Martingale h \u2131 \u03bc \u03c9 : \u03a9 h\u03c9 : (f - martingalePart (f + g) \u2131 \u03bc) n \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9 \u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9 ** rw [Pi.sub_apply] at h\u03c9 ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g hhpred : Adapted \u2131 fun n => h (n + 1) hhmgle : Martingale h \u2131 \u03bc \u03c9 : \u03a9 h\u03c9 : (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9 \u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9 ** rw [h\u03c9, Pi.sub_apply, martingalePart] ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g hhpred : Adapted \u2131 fun n => h (n + 1) hhmgle : Martingale h \u2131 \u03bc \u03c9 : \u03a9 h\u03c9 : (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9 \u22a2 (f 0 - ((f + g) 0 - predictablePart (f + g) \u2131 \u03bc 0)) \u03c9 = OfNat.ofNat 0 \u03c9 ** simp [hg0] ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc \u22a2 h = predictablePart (f + g) \u2131 \u03bc - g ** rw [hhdef, sub_eq_sub_iff_add_eq_add, add_comm (predictablePart (f + g) \u2131 \u03bc),\n  martingalePart_add_predictablePart] ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g \u22a2 Adapted \u2131 fun n => h (n + 1) ** rw [hh] ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 h : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc hhdef : h = f - martingalePart (f + g) \u2131 \u03bc hh : h = predictablePart (f + g) \u2131 \u03bc - g \u22a2 Adapted \u2131 fun n => (predictablePart (f + g) \u2131 \u03bc - g) (n + 1) ** exact adapted_predictablePart.sub hg ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d g\u271d : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 \u211d a b : \u211d hab : a < b hfc : ContinuousOn f (Icc a b) hgc : ContinuousOn g (Icc a b) hle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x hlt : \u2203 c, c \u2208 Icc a b \u2227 f c < g c \u22a2 \u222b (x : \u211d) in a..b, f x < \u222b (x : \u211d) in a..b, g x ** apply integral_lt_integral_of_ae_le_of_measure_setOf_lt_ne_zero hab.le\n  (hfc.intervalIntegrable_of_Icc hab.le) (hgc.intervalIntegrable_of_Icc hab.le) ** case hlt \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d g\u271d : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 \u211d a b : \u211d hab : a < b hfc : ContinuousOn f (Icc a b) hgc : ContinuousOn g (Icc a b) hle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x hlt : \u2203 c, c \u2208 Icc a b \u2227 f c < g c \u22a2 \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} \u2260 0 ** contrapose! hlt ** case hlt \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d g\u271d : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 \u211d a b : \u211d hab : a < b hfc : ContinuousOn f (Icc a b) hgc : ContinuousOn g (Icc a b) hle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x hlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0 \u22a2 \u2200 (c : \u211d), c \u2208 Icc a b \u2192 g c \u2264 f c ** have h_eq : f =\u1d50[volume.restrict (Ioc a b)] g := by\n  simp only [\u2190 not_le, \u2190 ae_iff] at hlt\n  exact EventuallyLE.antisymm ((ae_restrict_iff' measurableSet_Ioc).2 <|\n    eventually_of_forall hle) hlt ** case hlt \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d g\u271d : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 \u211d a b : \u211d hab : a < b hfc : ContinuousOn f (Icc a b) hgc : ContinuousOn g (Icc a b) hle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x hlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0 h_eq : f =\u1d50[Measure.restrict volume (Ioc a b)] g \u22a2 \u2200 (c : \u211d), c \u2208 Icc a b \u2192 g c \u2264 f c ** rw [Measure.restrict_congr_set Ioc_ae_eq_Icc] at h_eq ** case hlt \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d g\u271d : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 \u211d a b : \u211d hab : a < b hfc : ContinuousOn f (Icc a b) hgc : ContinuousOn g (Icc a b) hle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x hlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0 h_eq : f =\u1d50[Measure.restrict volume (Icc a b)] g \u22a2 \u2200 (c : \u211d), c \u2208 Icc a b \u2192 g c \u2264 f c ** exact fun c hc \u21a6 (Measure.eqOn_Icc_of_ae_eq volume hab.ne h_eq hfc hgc hc).ge ** case hle \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d g\u271d : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 \u211d a b : \u211d hab : a < b hfc : ContinuousOn f (Icc a b) hgc : ContinuousOn g (Icc a b) hle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x hlt : \u2203 c, c \u2208 Icc a b \u2227 f c < g c \u22a2 f \u2264\u1d50[Measure.restrict volume (Ioc a b)] g ** simpa only [gt_iff_lt, not_lt, ge_iff_le, measurableSet_Ioc, ae_restrict_eq, le_principal_iff]\n  using (ae_restrict_mem measurableSet_Ioc).mono hle ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d g\u271d : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 \u211d a b : \u211d hab : a < b hfc : ContinuousOn f (Icc a b) hgc : ContinuousOn g (Icc a b) hle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x hlt : \u2191\u2191(Measure.restrict volume (Ioc a b)) {x | f x < g x} = 0 \u22a2 f =\u1d50[Measure.restrict volume (Ioc a b)] g ** simp only [\u2190 not_le, \u2190 ae_iff] at hlt ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d g\u271d : \u211d \u2192 \u211d a\u271d b\u271d : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 \u211d a b : \u211d hab : a < b hfc : ContinuousOn f (Icc a b) hgc : ContinuousOn g (Icc a b) hle : \u2200 (x : \u211d), x \u2208 Ioc a b \u2192 f x \u2264 g x hlt : \u2200\u1d50 (a : \u211d) \u2202Measure.restrict volume (Ioc a b), g a \u2264 f a \u22a2 f =\u1d50[Measure.restrict volume (Ioc a b)] g ** exact EventuallyLE.antisymm ((ae_restrict_iff' measurableSet_Ioc).2 <|\n  eventually_of_forall hle) hlt ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.toSignedMeasure_toMeasureOfZeroLE ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc \u22a2 SignedMeasure.toMeasureOfZeroLE (toSignedMeasure \u03bc) univ (_ : MeasurableSet univ)       (_ : VectorMeasure.restrict 0 univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) univ) =     \u03bc ** refine' Measure.ext fun i hi => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191\u2191(SignedMeasure.toMeasureOfZeroLE (toSignedMeasure \u03bc) univ (_ : MeasurableSet univ)             (_ : VectorMeasure.restrict 0 univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) univ))       i =     \u2191\u2191\u03bc i ** lift \u03bc i to \u211d\u22650 using (measure_lt_top _ _).ne with m hm ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc i : Set \u03b1 hi : MeasurableSet i m : \u211d\u22650 hm : \u2191m = \u2191\u2191\u03bc i \u22a2 \u2191\u2191(SignedMeasure.toMeasureOfZeroLE (toSignedMeasure \u03bc) univ (_ : MeasurableSet univ)             (_ : VectorMeasure.restrict 0 univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) univ))       i =     \u2191m ** rw [SignedMeasure.toMeasureOfZeroLE_apply _ _ _ hi, coe_eq_coe] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc i : Set \u03b1 hi : MeasurableSet i m : \u211d\u22650 hm : \u2191m = \u2191\u2191\u03bc i \u22a2 { val := \u2191(toSignedMeasure \u03bc) (univ \u2229 i), property := (_ : 0 \u2264 \u2191(toSignedMeasure \u03bc) (univ \u2229 i)) } = m ** congr ** case intro.e_val \u03b1 : Type u_1 \u03b2 : Type u_2 m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc i : Set \u03b1 hi : MeasurableSet i m : \u211d\u22650 hm : \u2191m = \u2191\u2191\u03bc i \u22a2 \u2191(toSignedMeasure \u03bc) (univ \u2229 i) = \u2191m ** simp [hi, \u2190 hm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Semiquot.get_mem ** \u03b1 : Type u_1 \u03b2 : Type u_2 q : Semiquot \u03b1 p : IsPure q \u22a2 get q p \u2208 q ** let \u27e8a, h\u27e9 := exists_mem q ** \u03b1 : Type u_1 \u03b2 : Type u_2 q : Semiquot \u03b1 p : IsPure q a : \u03b1 h : a \u2208 q \u22a2 get q p \u2208 q ** unfold get ** \u03b1 : Type u_1 \u03b2 : Type u_2 q : Semiquot \u03b1 p : IsPure q a : \u03b1 h : a \u2208 q \u22a2 liftOn q id p \u2208 q ** rw [liftOn_ofMem q _ _ a h] ** \u03b1 : Type u_1 \u03b2 : Type u_2 q : Semiquot \u03b1 p : IsPure q a : \u03b1 h : a \u2208 q \u22a2 id a \u2208 q ** exact h ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toPMF_dirac ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a a' : \u03b1 s : Set \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : Countable \u03b1 h : MeasurableSingletonClass \u03b1 \u22a2 Measure.toPMF (Measure.dirac a) = pure a ** rw [toPMF_eq_iff_toMeasure_eq, toMeasure_pure] ** Qed",
        "informal": ""
    },
    {
        "formal": "Measurable.iInf_Prop ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : p \u22a2 Measurable fun b => \u2a05 (_ : p), f b ** convert hf ** case h.e'_5.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : p x\u271d : \u03b4 \u22a2 \u2a05 (_ : p), f x\u271d = f x\u271d ** exact ciInf_pos h ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp \u22a2 Measurable fun b => \u2a05 (_ : p), f b ** convert measurable_const using 1 ** case h.e'_5 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp \u22a2 (fun b => \u2a05 (_ : p), f b) = fun x => ?convert_5  case convert_5 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp \u22a2 \u03b1 ** funext ** case h.e'_5.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp x\u271d : \u03b4 \u22a2 \u2a05 (_ : p), f x\u271d = ?convert_5  case convert_5 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp \u22a2 \u03b1 ** exact ciInf_neg h ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.infinite_prod ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 t : Set \u03b2 \u22a2 Set.Infinite (s \u00d7\u02e2 t) \u2194 Set.Infinite s \u2227 Set.Nonempty t \u2228 Set.Infinite t \u2227 Set.Nonempty s ** refine' \u27e8fun h => _, _\u27e9 ** case refine'_1 \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 t : Set \u03b2 h : Set.Infinite (s \u00d7\u02e2 t) \u22a2 Set.Infinite s \u2227 Set.Nonempty t \u2228 Set.Infinite t \u2227 Set.Nonempty s ** simp_rw [Set.Infinite, @and_comm \u00ac_, \u2190 not_imp] ** case refine'_1 \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 t : Set \u03b2 h : Set.Infinite (s \u00d7\u02e2 t) \u22a2 \u00ac(Set.Nonempty t \u2192 Set.Finite s) \u2228 \u00ac(Set.Nonempty s \u2192 Set.Finite t) ** by_contra' ** case refine'_1 \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 t : Set \u03b2 h : Set.Infinite (s \u00d7\u02e2 t) this : (Set.Nonempty t \u2192 Set.Finite s) \u2227 (Set.Nonempty s \u2192 Set.Finite t) \u22a2 False ** exact h ((this.1 h.nonempty.snd).prod $ this.2 h.nonempty.fst) ** case refine'_2 \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 t : Set \u03b2 \u22a2 Set.Infinite s \u2227 Set.Nonempty t \u2228 Set.Infinite t \u2227 Set.Nonempty s \u2192 Set.Infinite (s \u00d7\u02e2 t) ** rintro (h | h) ** case refine'_2.inl \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 t : Set \u03b2 h : Set.Infinite s \u2227 Set.Nonempty t \u22a2 Set.Infinite (s \u00d7\u02e2 t) ** exact h.1.prod_left h.2 ** case refine'_2.inr \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 t : Set \u03b2 h : Set.Infinite t \u2227 Set.Nonempty s \u22a2 Set.Infinite (s \u00d7\u02e2 t) ** exact h.1.prod_right h.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "BinaryHeap.size_pos_of_max ** \u03b1 : Type u_1 x : \u03b1 lt : \u03b1 \u2192 \u03b1 \u2192 Bool self : BinaryHeap \u03b1 lt e : max self = some x h : \u00ac0 < Array.size self.arr \u22a2 False ** simp [BinaryHeap.max, Array.get?, h] at e ** Qed",
        "informal": ""
    },
    {
        "formal": "Complex.continuousOn_abs_circleTransformBoundingFunction ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 \u22a2 ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) ** have : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 (\u22a4 : Set \u211d)) := by\n  apply_rules [ContinuousOn.smul, continuousOn_const]\n  simp only [deriv_circleMap]\n  have c := (continuous_circleMap 0 R).continuousOn (s := \u22a4)\n  apply_rules [ContinuousOn.mul, c.comp continuousOn_snd fun _ => And.right, continuousOn_const]\n  simp_rw [\u2190 inv_pow]\n  apply continuousOn_prod_circle_transform_function hr ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 this : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4) \u22a2 ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) ** refine' continuous_abs.continuousOn (s := \u22a4).comp this _ ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 this : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4) \u22a2 MapsTo (fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) \u22a4 ** simp [MapsTo] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 \u22a2 ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4) ** apply_rules [ContinuousOn.smul, continuousOn_const] ** case hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 \u22a2 ContinuousOn (fun x => deriv (circleMap z R) x.2) (closedBall z r \u00d7\u02e2 \u22a4)  case hg.hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 \u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4) ** simp only [deriv_circleMap] ** case hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 \u22a2 ContinuousOn (fun x => circleMap 0 R x.2 * I) (closedBall z r \u00d7\u02e2 \u22a4)  case hg.hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 \u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4) ** have c := (continuous_circleMap 0 R).continuousOn (s := \u22a4) ** case hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 c : ContinuousOn (circleMap 0 R) \u22a4 \u22a2 ContinuousOn (fun x => circleMap 0 R x.2 * I) (closedBall z r \u00d7\u02e2 \u22a4)  case hg.hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 \u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4) ** apply_rules [ContinuousOn.mul, c.comp continuousOn_snd fun _ => And.right, continuousOn_const] ** case hg.hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 \u22a2 ContinuousOn (fun x => ((circleMap z R x.2 - x.1) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4) ** simp_rw [\u2190 inv_pow] ** case hg.hg.hf E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R z : \u2102 \u22a2 ContinuousOn (fun x => (circleMap z R x.2 - x.1)\u207b\u00b9 ^ 2) (closedBall z r \u00d7\u02e2 \u22a4) ** apply continuousOn_prod_circle_transform_function hr ** Qed",
        "informal": ""
    },
    {
        "formal": "AEMeasurable.nnreal_tsum ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b2 : MeasurableSpace \u03b1\u271d \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650 \u03bc : Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable (f i) \u22a2 AEMeasurable fun x => \u2211' (i : \u03b9), f i x ** simp_rw [NNReal.tsum_eq_toNNReal_tsum] ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b2 : MeasurableSpace \u03b1\u271d \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03b9 : Type u_7 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650 \u03bc : Measure \u03b1 h : \u2200 (i : \u03b9), AEMeasurable (f i) \u22a2 AEMeasurable fun x => ENNReal.toNNReal (\u2211' (b : \u03b9), \u2191(f b x)) ** exact (AEMeasurable.ennreal_tsum fun i => (h i).coe_nnreal_ennreal).ennreal_toNNReal ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.self_subset_generateMeasurableRec ** \u03b1 : Type u s : Set (Set \u03b1) i : (Quotient.out (ord (aleph 1))).\u03b1 \u22a2 s \u2286 generateMeasurableRec s i ** unfold generateMeasurableRec ** \u03b1 : Type u s : Set (Set \u03b1) i : (Quotient.out (ord (aleph 1))).\u03b1 \u22a2 s \u2286     let i := i;     let S := \u22c3 j, generateMeasurableRec s \u2191j;     s \u222a {\u2205} \u222a compl '' S \u222a range fun f => \u22c3 n, \u2191(f n) ** apply_rules [subset_union_of_subset_left] ** case h.h.h \u03b1 : Type u s : Set (Set \u03b1) i : (Quotient.out (ord (aleph 1))).\u03b1 \u22a2 s \u2286 s ** exact subset_rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.uniformIntegrable_condexp ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 \u22a2 UniformIntegrable (fun i => \u03bc[g|\u2131 i]) 1 \u03bc ** have hmeas : \u2200 n, \u2200 C, MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} := fun n C =>\n  measurableSet_le measurable_const (stronglyMeasurable_condexp.mono (h\u2131 n)).measurable.nnnorm ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u22a2 UniformIntegrable (fun i => \u03bc[g|\u2131 i]) 1 \u03bc ** have hg : Mem\u2112p g 1 \u03bc := mem\u2112p_one_iff_integrable.2 hint ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u22a2 UniformIntegrable (fun i => \u03bc[g|\u2131 i]) 1 \u03bc ** refine' uniformIntegrable_of le_rfl ENNReal.one_ne_top\n  (fun n => (stronglyMeasurable_condexp.mono (h\u2131 n)).aestronglyMeasurable) fun \u03b5 h\u03b5 => _ ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 ** by_cases hne : snorm g 1 \u03bc = 0 ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8\u03b4, h\u03b4, h\u27e9 := hg.snorm_indicator_le \u03bc le_rfl ENNReal.one_ne_top h\u03b5 ** case neg.intro.intro \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 ** set C : \u211d\u22650 := \u27e8\u03b4, h\u03b4.le\u27e9\u207b\u00b9 * (snorm g 1 \u03bc).toNNReal with hC ** case neg.intro.intro \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 ** have hCpos : 0 < C := mul_pos (inv_pos.2 h\u03b4) (ENNReal.toNNReal_pos hne hg.snorm_lt_top.ne) ** case neg.intro.intro \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 ** have : \u2200 n, \u03bc {x : \u03b1 | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 := by\n  intro n\n  have := mul_meas_ge_le_pow_snorm' \u03bc one_ne_zero ENNReal.one_ne_top\n    ((@stronglyMeasurable_condexp _ _ _ _ _ (\u2131 n) _ \u03bc g).mono (h\u2131 n)).aestronglyMeasurable C\n  rw [ENNReal.one_toReal, ENNReal.rpow_one, ENNReal.rpow_one, mul_comm, \u2190\n    ENNReal.le_div_iff_mul_le (Or.inl (ENNReal.coe_ne_zero.2 hCpos.ne.symm))\n      (Or.inl ENNReal.coe_lt_top.ne)] at this\n  simp_rw [ENNReal.coe_le_coe] at this\n  refine' this.trans _\n  rw [ENNReal.div_le_iff_le_mul (Or.inl (ENNReal.coe_ne_zero.2 hCpos.ne.symm))\n      (Or.inl ENNReal.coe_lt_top.ne),\n    hC, Nonneg.inv_mk, ENNReal.coe_mul, ENNReal.coe_toNNReal hg.snorm_lt_top.ne, \u2190 mul_assoc, \u2190\n    ENNReal.ofReal_eq_coe_nnreal, \u2190 ENNReal.ofReal_mul h\u03b4.le, mul_inv_cancel h\u03b4.ne.symm,\n    ENNReal.ofReal_one, one_mul]\n  exact snorm_one_condexp_le_snorm _ ** case neg.intro.intro \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C this : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e8C, fun n => le_trans _ (h {x : \u03b1 | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (hmeas n C) (this n))\u27e9 ** case neg.intro.intro \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C this : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 n : \u03b9 \u22a2 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n])) 1 \u03bc \u2264 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g) 1 \u03bc ** have hmeas\u2131 : MeasurableSet[\u2131 n] {x : \u03b1 | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} :=\n  @measurableSet_le _ _ _ _ _ (\u2131 n) _ _ _ _ _ measurable_const\n    (@Measurable.nnnorm _ _ _ _ _ (\u2131 n) _ stronglyMeasurable_condexp.measurable) ** case neg.intro.intro \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C this : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 n : \u03b9 hmeas\u2131 : MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u22a2 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n])) 1 \u03bc \u2264 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g) 1 \u03bc ** rw [\u2190 snorm_congr_ae (condexp_indicator hint hmeas\u2131)] ** case neg.intro.intro \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C this : \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 n : \u03b9 hmeas\u2131 : MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u22a2 snorm (\u03bc[Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g|\u2131 n]) 1 \u03bc \u2264 snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} g) 1 \u03bc ** exact snorm_one_condexp_le_snorm _ ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : snorm g 1 \u03bc = 0 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 ** rw [snorm_eq_zero_iff hg.1 one_ne_zero] at hne ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : g =\u1d50[\u03bc] 0 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (Set.indicator {x | C \u2264 \u2016(\u03bc[g|\u2131 i]) x\u2016\u208a} (\u03bc[g|\u2131 i])) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e80, fun n => (le_of_eq <|\n  (snorm_eq_zero_iff ((stronglyMeasurable_condexp.mono (h\u2131 n)).aestronglyMeasurable.indicator\n    (hmeas n 0)) one_ne_zero).2 _).trans (zero_le _)\u27e9 ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : g =\u1d50[\u03bc] 0 n : \u03b9 \u22a2 Set.indicator {x | 0 \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n]) =\u1d50[\u03bc] 0 ** filter_upwards [@condexp_congr_ae _ _ _ _ _ (\u2131 n) m0 \u03bc _ _ hne] with x hx ** case h \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : g =\u1d50[\u03bc] 0 n : \u03b9 x : \u03b1 hx : (\u03bc[g|\u2131 n]) x = (\u03bc[0|\u2131 n]) x \u22a2 Set.indicator {x | 0 \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} (\u03bc[g|\u2131 n]) x = OfNat.ofNat 0 x ** simp only [zero_le', Set.setOf_true, Set.indicator_univ, Pi.zero_apply, hx, condexp_zero] ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C \u22a2 \u2200 (n : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 ** intro n ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C n : \u03b9 \u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 ** have := mul_meas_ge_le_pow_snorm' \u03bc one_ne_zero ENNReal.one_ne_top\n  ((@stronglyMeasurable_condexp _ _ _ _ _ (\u2131 n) _ \u03bc g).mono (h\u2131 n)).aestronglyMeasurable C ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C n : \u03b9 this : \u2191C ^ ENNReal.toReal 1 * \u2191\u2191\u03bc {x | \u2191C \u2264 \u2191\u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 snorm (\u03bc[g|\u2131 n]) 1 \u03bc ^ ENNReal.toReal 1 \u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 ** rw [ENNReal.one_toReal, ENNReal.rpow_one, ENNReal.rpow_one, mul_comm, \u2190\n  ENNReal.le_div_iff_mul_le (Or.inl (ENNReal.coe_ne_zero.2 hCpos.ne.symm))\n    (Or.inl ENNReal.coe_lt_top.ne)] at this ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C n : \u03b9 this : \u2191\u2191\u03bc {x | \u2191C \u2264 \u2191\u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191C \u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 ** simp_rw [ENNReal.coe_le_coe] at this ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C n : \u03b9 this :   \u2191\u2191\u03bc {x | { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264     snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)) \u22a2 \u2191\u2191\u03bc {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 ** refine' this.trans _ ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C n : \u03b9 this :   \u2191\u2191\u03bc {x | { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264     snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)) \u22a2 snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)) \u2264 ENNReal.ofReal \u03b4 ** rw [ENNReal.div_le_iff_le_mul (Or.inl (ENNReal.coe_ne_zero.2 hCpos.ne.symm))\n    (Or.inl ENNReal.coe_lt_top.ne),\n  hC, Nonneg.inv_mk, ENNReal.coe_mul, ENNReal.coe_toNNReal hg.snorm_lt_top.ne, \u2190 mul_assoc, \u2190\n  ENNReal.ofReal_eq_coe_nnreal, \u2190 ENNReal.ofReal_mul h\u03b4.le, mul_inv_cancel h\u03b4.ne.symm,\n  ENNReal.ofReal_one, one_mul] ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_2 inst\u271d : IsFiniteMeasure \u03bc g : \u03b1 \u2192 \u211d hint : Integrable g \u2131 : \u03b9 \u2192 MeasurableSpace \u03b1 h\u2131 : \u2200 (i : \u03b9), \u2131 i \u2264 m0 hmeas : \u2200 (n : \u03b9) (C : \u211d\u22650), MeasurableSet {x | C \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} hg : Mem\u2112p g 1 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hne : \u00acsnorm g 1 \u03bc = 0 \u03b4 : \u211d h\u03b4 : 0 < \u03b4 h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s g) 1 \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 := { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hC : C = { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) hCpos : 0 < C n : \u03b9 this :   \u2191\u2191\u03bc {x | { val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc) \u2264 \u2016(\u03bc[g|\u2131 n]) x\u2016\u208a} \u2264     snorm (\u03bc[g|\u2131 n]) 1 \u03bc / \u2191({ val := \u03b4, property := (_ : 0 \u2264 \u03b4) }\u207b\u00b9 * ENNReal.toNNReal (snorm g 1 \u03bc)) \u22a2 snorm (\u03bc[g|\u2131 n]) 1 \u03bc \u2264 snorm g 1 \u03bc ** exact snorm_one_condexp_le_snorm _ ** Qed",
        "informal": ""
    },
    {
        "formal": "integrableOn_Iic_iff_integrableOn_Iio ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : PartialOrder \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E \u03bc : Measure \u03b1 a b : \u03b1 inst\u271d : NoAtoms \u03bc \u22a2 \u2191\u2191\u03bc {b} \u2260 \u22a4 ** rw [measure_singleton] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : PartialOrder \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E \u03bc : Measure \u03b1 a b : \u03b1 inst\u271d : NoAtoms \u03bc \u22a2 0 \u2260 \u22a4 ** exact ENNReal.zero_ne_top ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.neg_val ** n : \u2115 inst\u271d : NeZero n a : ZMod n \u22a2 val (-a) = if a = 0 then 0 else n - val a ** rw [neg_val'] ** n : \u2115 inst\u271d : NeZero n a : ZMod n \u22a2 (n - val a) % n = if a = 0 then 0 else n - val a ** by_cases h : a = 0 ** case neg n : \u2115 inst\u271d : NeZero n a : ZMod n h : \u00aca = 0 \u22a2 (n - val a) % n = if a = 0 then 0 else n - val a ** rw [if_neg h] ** case neg n : \u2115 inst\u271d : NeZero n a : ZMod n h : \u00aca = 0 \u22a2 (n - val a) % n = n - val a ** apply Nat.mod_eq_of_lt ** case neg.h n : \u2115 inst\u271d : NeZero n a : ZMod n h : \u00aca = 0 \u22a2 n - val a < n ** apply Nat.sub_lt (NeZero.pos n) ** case neg.h n : \u2115 inst\u271d : NeZero n a : ZMod n h : \u00aca = 0 \u22a2 0 < val a ** contrapose! h ** case neg.h n : \u2115 inst\u271d : NeZero n a : ZMod n h : val a \u2264 0 \u22a2 a = 0 ** rwa [le_zero_iff, val_eq_zero] at h ** case pos n : \u2115 inst\u271d : NeZero n a : ZMod n h : a = 0 \u22a2 (n - val a) % n = if a = 0 then 0 else n - val a ** rw [if_pos h, h, val_zero, tsub_zero, Nat.mod_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "not_primrec_ack_self ** \u22a2 \u00acPrimrec fun n => ack n n ** rw [Primrec.nat_iff] ** \u22a2 \u00acNat.Primrec fun n => ack n n ** exact not_nat_primrec_ack_self ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Ordered.memP_iff_lowerBound? ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t \u22a2 MemP cut t \u2194 \u2203 x, lowerBound? cut t none = some x \u2227 cut x = Ordering.eq ** refine memP_def.trans \u27e8fun \u27e8y, hy, ey\u27e9 => ?_, fun \u27e8x, hx, e\u27e9 => \u27e8_, lowerBound?_mem hx, e\u27e9\u27e9 ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t x\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq y : \u03b1 hy : y \u2208 t ey : cut y = Ordering.eq \u22a2 \u2203 x, lowerBound? cut t none = some x \u2227 cut x = Ordering.eq ** have \u27e8x, hx\u27e9 := ht.lowerBound?_exists.2 \u27e8_, hy, fun h => nomatch ey.symm.trans h\u27e9 ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t x\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq y : \u03b1 hy : y \u2208 t ey : cut y = Ordering.eq x : \u03b1 hx : lowerBound? cut t none = some x \u22a2 \u2203 x, lowerBound? cut t none = some x \u2227 cut x = Ordering.eq ** refine \u27e8x, hx, ?_\u27e9 ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t x\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq y : \u03b1 hy : y \u2208 t ey : cut y = Ordering.eq x : \u03b1 hx : lowerBound? cut t none = some x \u22a2 cut x = Ordering.eq ** cases ex : cut x ** case lt \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t x\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq y : \u03b1 hy : y \u2208 t ey : cut y = Ordering.eq x : \u03b1 hx : lowerBound? cut t none = some x ex : cut x = Ordering.lt \u22a2 Ordering.lt = Ordering.eq ** cases lowerBound?_le hx ex ** case eq \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t x\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq y : \u03b1 hy : y \u2208 t ey : cut y = Ordering.eq x : \u03b1 hx : lowerBound? cut t none = some x ex : cut x = Ordering.eq \u22a2 Ordering.eq = Ordering.eq ** rfl ** case gt \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t x\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq y : \u03b1 hy : y \u2208 t ey : cut y = Ordering.eq x : \u03b1 hx : lowerBound? cut t none = some x ex : cut x = Ordering.gt \u22a2 Ordering.gt = Ordering.eq ** cases e : cmp x y ** case gt.lt \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t x\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq y : \u03b1 hy : y \u2208 t ey : cut y = Ordering.eq x : \u03b1 hx : lowerBound? cut t none = some x ex : cut x = Ordering.gt e : cmp x y = Ordering.lt \u22a2 Ordering.gt = Ordering.eq ** cases ey.symm.trans <| ht.lowerBound?_least hx hy e ex ** case gt.eq \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t x\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq y : \u03b1 hy : y \u2208 t ey : cut y = Ordering.eq x : \u03b1 hx : lowerBound? cut t none = some x ex : cut x = Ordering.gt e : cmp x y = Ordering.eq \u22a2 Ordering.gt = Ordering.eq ** cases ey.symm.trans <| IsCut.congr e |>.symm.trans ex ** case gt.gt \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering cut : \u03b1 \u2192 Ordering t : RBNode \u03b1 inst\u271d\u00b9 : TransCmp cmp inst\u271d : IsCut cmp cut ht : Ordered cmp t x\u271d : \u2203 x, x \u2208 t \u2227 cut x = Ordering.eq y : \u03b1 hy : y \u2208 t ey : cut y = Ordering.eq x : \u03b1 hx : lowerBound? cut t none = some x ex : cut x = Ordering.gt e : cmp x y = Ordering.gt \u22a2 Ordering.gt = Ordering.eq ** cases ey.symm.trans <| IsCut.gt_trans (OrientedCmp.cmp_eq_gt.1 e) ex ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.cast_le ** \u03b1 : Type u_1 inst\u271d : LinearOrderedSemiring \u03b1 m n : Num \u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n ** rw [\u2190 not_lt] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedSemiring \u03b1 m n : Num \u22a2 \u00ac\u2191n < \u2191m \u2194 m \u2264 n ** exact not_congr cast_lt ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.setToL1_congr_left ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f : { x // x \u2208 Lp E 1 } \u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f ** suffices setToL1 hT = setToL1 hT' by rw [this] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f : { x // x \u2208 Lp E 1 } \u22a2 setToL1 hT = setToL1 hT' ** refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc hT) _ _ _ _ _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f : { x // x \u2208 Lp E 1 } \u22a2 ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT ** ext1 f ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** suffices setToL1 hT' f = setToL1SCLM \u03b1 E \u03bc hT f by rw [\u2190 this]; rfl ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** rw [setToL1_eq_setToL1SCLM] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(setToL1SCLM \u03b1 E \u03bc hT') f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** exact setToL1SCLM_congr_left hT' hT h.symm f ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f : { x // x \u2208 Lp E 1 } this : setToL1 hT = setToL1 hT' \u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f ** rw [this] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } this : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** rw [\u2190 this] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } this : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT') \u2191f ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "List.leftpad_length ** \u03b1 : Type u_1 n : Nat a : \u03b1 l : List \u03b1 \u22a2 length (leftpad n a l) = max n (length l) ** simp only [leftpad, length_append, length_replicate, Nat.sub_add_eq_max] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_Lp_of_tendstoInMeasure ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hg : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : TendstoInMeasure \u03bc f atTop g \u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) ** refine' tendsto_of_subseq_tendsto fun ns hns => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hg : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : TendstoInMeasure \u03bc f atTop g ns : \u2115 \u2192 \u2115 hns : Tendsto ns atTop atTop \u22a2 \u2203 ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p \u03bc) atTop (\ud835\udcdd 0) ** obtain \u27e8ms, _, hms'\u27e9 := TendstoInMeasure.exists_seq_tendsto_ae fun \u03b5 h\u03b5 => (hfg \u03b5 h\u03b5).comp hns ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hg : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : TendstoInMeasure \u03bc f atTop g ns : \u2115 \u2192 \u2115 hns : Tendsto ns atTop atTop ms : \u2115 \u2192 \u2115 left\u271d : StrictMono ms hms' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun i => f (ns (ms i)) x) atTop (\ud835\udcdd (g x)) \u22a2 \u2203 ms, Tendsto (fun n => snorm (f (ns (ms n)) - g) p \u03bc) atTop (\ud835\udcdd 0) ** exact \u27e8ms,\n  tendsto_Lp_of_tendsto_ae \u03bc hp hp' (fun _ => hf _) hg (fun \u03b5 h\u03b5 =>\n    let \u27e8\u03b4, h\u03b4, h\u03b4'\u27e9 := hui h\u03b5\n    \u27e8\u03b4, h\u03b4, fun i s hs h\u03bcs => h\u03b4' _ s hs h\u03bcs\u27e9)\n    hms'\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.scanl_head ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 v : Vector \u03b1 n \u22a2 head (scanl f b v) = b ** cases n ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 v : Vector \u03b1 Nat.zero \u22a2 head (scanl f b v) = b ** have : v = nil := by simp only [Nat.zero_eq, eq_iff_true_of_subsingleton] ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 v : Vector \u03b1 Nat.zero this : v = nil \u22a2 head (scanl f b v) = b ** simp only [this, scanl_nil, head_cons] ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 v : Vector \u03b1 Nat.zero \u22a2 v = nil ** simp only [Nat.zero_eq, eq_iff_true_of_subsingleton] ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 n\u271d : \u2115 v : Vector \u03b1 (Nat.succ n\u271d) \u22a2 head (scanl f b v) = b ** rw [\u2190 cons_head_tail v] ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b2 \u2192 \u03b1 \u2192 \u03b2 b : \u03b2 n\u271d : \u2115 v : Vector \u03b1 (Nat.succ n\u271d) \u22a2 head (scanl f b (head v ::\u1d65 tail v)) = b ** simp only [\u2190 get_zero, get_eq_get, toList_scanl, toList_cons, List.scanl, Fin.val_zero,\n  List.get] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u22a2 f \u2264\u1d50[\u03bc] g ** have A :\n  \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u03bc ({x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 spanningSets \u03bc p) = 0 := by\n  intro \u03b5 N p \u03b5pos\n  let s := {x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 spanningSets \u03bc p\n  have s_meas : MeasurableSet s := by\n    have A : MeasurableSet {x | g x + \u03b5 \u2264 f x} := measurableSet_le (hg.add measurable_const) hf\n    have B : MeasurableSet {x | g x \u2264 N} := measurableSet_le hg measurable_const\n    exact (A.inter B).inter (measurable_spanningSets \u03bc p)\n  have s_lt_top : \u03bc s < \u221e :=\n    (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top \u03bc p)\n  have A : (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 :=\n    calc\n      (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s = (\u222b\u207b x in s, g x \u2202\u03bc) + \u222b\u207b _ in s, \u03b5 \u2202\u03bc := by\n        simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n      _ = \u222b\u207b x in s, g x + \u03b5 \u2202\u03bc := (lintegral_add_right _ measurable_const).symm\n      _ \u2264 \u222b\u207b x in s, f x \u2202\u03bc :=\n        (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)\n      _ \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top\n  have B : (\u222b\u207b x in s, g x \u2202\u03bc) \u2260 \u221e := by\n    apply ne_of_lt\n    calc\n      (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc :=\n        set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n      _ = N * \u03bc s := by\n        simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n      _ < \u221e := by\n        simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n          ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff]\n  have : (\u03b5 : \u211d\u22650\u221e) * \u03bc s \u2264 0 := ENNReal.le_of_add_le_add_left B A\n  simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, \u03b5pos.ne', false_or_iff] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 \u22a2 f \u2264\u1d50[\u03bc] g ** obtain \u27e8u, _, u_pos, u_lim\u27e9 :\n  \u2203 u : \u2115 \u2192 \u211d\u22650, StrictAnti u \u2227 (\u2200 n, 0 < u n) \u2227 Tendsto u atTop (nhds 0) :=\n  exists_seq_strictAnti_tendsto (0 : \u211d\u22650) ** case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) \u22a2 f \u2264\u1d50[\u03bc] g ** let s := fun n : \u2115 => {x | g x + u n \u2264 f x \u2227 g x \u2264 (n : \u211d\u22650)} \u2229 spanningSets \u03bc n ** case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u22a2 f \u2264\u1d50[\u03bc] g ** have \u03bcs : \u2200 n, \u03bc (s n) = 0 := fun n => A _ _ _ (u_pos n) ** case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 \u22a2 f \u2264\u1d50[\u03bc] g ** have B : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n := by\n  intro x hx\n  simp only [Set.mem_compl_iff, Set.mem_setOf, not_le] at hx\n  have L1 : \u2200\u1da0 n in atTop, g x + u n \u2264 f x := by\n    have : Tendsto (fun n => g x + u n) atTop (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n      tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)\n    simp only [ENNReal.coe_zero, add_zero] at this\n    exact eventually_le_of_tendsto_lt hx this\n  have L2 : \u2200\u1da0 n : \u2115 in (atTop : Filter \u2115), g x \u2264 (n : \u211d\u22650) :=\n    haveI : Tendsto (fun n : \u2115 => ((n : \u211d\u22650) : \u211d\u22650\u221e)) atTop (\ud835\udcdd \u221e) := by\n      simp only [ENNReal.coe_nat]\n      exact ENNReal.tendsto_nat_nhds_top\n    eventually_ge_of_tendsto_gt (hx.trans_le le_top) this\n  apply Set.mem_iUnion.2\n  exact ((L1.and L2).and (eventually_mem_spanningSets \u03bc x)).exists ** case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 B : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n \u22a2 f \u2264\u1d50[\u03bc] g ** refine' le_antisymm _ bot_le ** case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 B : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n \u22a2 \u2191\u2191\u03bc {x | (fun x => f x \u2264 g x) x}\u1d9c \u2264 0 ** calc\n  \u03bc {x : \u03b1 | (fun x : \u03b1 => f x \u2264 g x) x}\u1d9c \u2264 \u03bc (\u22c3 n, s n) := measure_mono B\n  _ \u2264 \u2211' n, \u03bc (s n) := (measure_iUnion_le _)\n  _ = 0 := by simp only [\u03bcs, tsum_zero] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u22a2 \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 ** intro \u03b5 N p \u03b5pos ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 \u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 ** let s := {x | g x + \u03b5 \u2264 f x \u2227 g x \u2264 N} \u2229 spanningSets \u03bc p ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p \u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 ** have s_meas : MeasurableSet s := by\n  have A : MeasurableSet {x | g x + \u03b5 \u2264 f x} := measurableSet_le (hg.add measurable_const) hf\n  have B : MeasurableSet {x | g x \u2264 N} := measurableSet_le hg measurable_const\n  exact (A.inter B).inter (measurable_spanningSets \u03bc p) ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s \u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 ** have s_lt_top : \u03bc s < \u221e :=\n  (measure_mono (Set.inter_subset_right _ _)).trans_lt (measure_spanningSets_lt_top \u03bc p) ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 ** have A : (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 :=\n  calc\n    (\u222b\u207b x in s, g x \u2202\u03bc) + \u03b5 * \u03bc s = (\u222b\u207b x in s, g x \u2202\u03bc) + \u222b\u207b _ in s, \u03b5 \u2202\u03bc := by\n      simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n    _ = \u222b\u207b x in s, g x + \u03b5 \u2202\u03bc := (lintegral_add_right _ measurable_const).symm\n    _ \u2264 \u222b\u207b x in s, f x \u2202\u03bc :=\n      (set_lintegral_mono (hg.add measurable_const) hf fun x hx => hx.1.1)\n    _ \u2264 (\u222b\u207b x in s, g x \u2202\u03bc) + 0 := by rw [add_zero]; exact h s s_meas s_lt_top ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 A : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0 \u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 ** have B : (\u222b\u207b x in s, g x \u2202\u03bc) \u2260 \u221e := by\n  apply ne_of_lt\n  calc\n    (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc :=\n      set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n    _ = N * \u03bc s := by\n      simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n    _ < \u221e := by\n      simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n        ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 A : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0 B : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 ** have : (\u03b5 : \u211d\u22650\u221e) * \u03bc s \u2264 0 := ENNReal.le_of_add_le_add_left B A ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 A : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0 B : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4 this : \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 0 \u22a2 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 ** simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, \u03b5pos.ne', false_or_iff] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p \u22a2 MeasurableSet s ** have A : MeasurableSet {x | g x + \u03b5 \u2264 f x} := measurableSet_le (hg.add measurable_const) hf ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p A : MeasurableSet {x | g x + \u2191\u03b5 \u2264 f x} \u22a2 MeasurableSet s ** have B : MeasurableSet {x | g x \u2264 N} := measurableSet_le hg measurable_const ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p A : MeasurableSet {x | g x + \u2191\u03b5 \u2264 f x} B : MeasurableSet {x | g x \u2264 \u2191N} \u22a2 MeasurableSet s ** exact (A.inter B).inter (measurable_spanningSets \u03bc p) ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in s, \u2191\u03b5 \u2202\u03bc ** simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0 ** rw [add_zero] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc ** exact h s s_meas s_lt_top ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 A : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0 \u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2260 \u22a4 ** apply ne_of_lt ** case h \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 A : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0 \u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc < \u22a4 ** calc\n  (\u222b\u207b x in s, g x \u2202\u03bc) \u2264 \u222b\u207b _ in s, N \u2202\u03bc :=\n    set_lintegral_mono hg measurable_const fun x hx => hx.1.2\n  _ = N * \u03bc s := by\n    simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n  _ < \u221e := by\n    simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n      ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 A : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0 \u22a2 \u222b\u207b (x : \u03b1) in s, \u2191N \u2202\u03bc = \u2191N * \u2191\u2191\u03bc s ** simp only [lintegral_const, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p\u271d : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u03b5 N : \u211d\u22650 p : \u2115 \u03b5pos : 0 < \u03b5 s : Set \u03b1 := {x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p s_meas : MeasurableSet s s_lt_top : \u2191\u2191\u03bc s < \u22a4 A : \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + \u2191\u03b5 * \u2191\u2191\u03bc s \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc + 0 \u22a2 \u2191N * \u2191\u2191\u03bc s < \u22a4 ** simp only [lt_top_iff_ne_top, s_lt_top.ne, and_false_iff, ENNReal.coe_ne_top,\n  ENNReal.mul_eq_top, Ne.def, not_false_iff, false_and_iff, or_self_iff] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 \u22a2 {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n ** intro x hx ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : x \u2208 {x | f x \u2264 g x}\u1d9c \u22a2 x \u2208 \u22c3 n, s n ** simp only [Set.mem_compl_iff, Set.mem_setOf, not_le] at hx ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : g x < f x \u22a2 x \u2208 \u22c3 n, s n ** have L1 : \u2200\u1da0 n in atTop, g x + u n \u2264 f x := by\n  have : Tendsto (fun n => g x + u n) atTop (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n    tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)\n  simp only [ENNReal.coe_zero, add_zero] at this\n  exact eventually_le_of_tendsto_lt hx this ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : g x < f x L1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x \u22a2 x \u2208 \u22c3 n, s n ** have L2 : \u2200\u1da0 n : \u2115 in (atTop : Filter \u2115), g x \u2264 (n : \u211d\u22650) :=\n  haveI : Tendsto (fun n : \u2115 => ((n : \u211d\u22650) : \u211d\u22650\u221e)) atTop (\ud835\udcdd \u221e) := by\n    simp only [ENNReal.coe_nat]\n    exact ENNReal.tendsto_nat_nhds_top\n  eventually_ge_of_tendsto_gt (hx.trans_le le_top) this ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : g x < f x L1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x L2 : \u2200\u1da0 (n : \u2115) in atTop, g x \u2264 \u2191\u2191n \u22a2 x \u2208 \u22c3 n, s n ** apply Set.mem_iUnion.2 ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : g x < f x L1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x L2 : \u2200\u1da0 (n : \u2115) in atTop, g x \u2264 \u2191\u2191n \u22a2 \u2203 i, x \u2208 s i ** exact ((L1.and L2).and (eventually_mem_spanningSets \u03bc x)).exists ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : g x < f x \u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x ** have : Tendsto (fun n => g x + u n) atTop (\ud835\udcdd (g x + (0 : \u211d\u22650))) :=\n  tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim) ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : g x < f x this : Tendsto (fun n => g x + \u2191(u n)) atTop (\ud835\udcdd (g x + \u21910)) \u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x ** simp only [ENNReal.coe_zero, add_zero] at this ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : g x < f x this : Tendsto (fun n => g x + \u2191(u n)) atTop (\ud835\udcdd (g x)) \u22a2 \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x ** exact eventually_le_of_tendsto_lt hx this ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : g x < f x L1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x \u22a2 Tendsto (fun n => \u2191\u2191n) atTop (\ud835\udcdd \u22a4) ** simp only [ENNReal.coe_nat] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 x : \u03b1 hx : g x < f x L1 : \u2200\u1da0 (n : \u2115) in atTop, g x + \u2191(u n) \u2264 f x \u22a2 Tendsto (fun n => \u2191n) atTop (\ud835\udcdd \u22a4) ** exact ENNReal.tendsto_nat_nhds_top ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E p : \u211d\u22650\u221e inst\u271d : SigmaFinite \u03bc f g : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hg : Measurable g h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc A : \u2200 (\u03b5 N : \u211d\u22650) (p : \u2115), 0 < \u03b5 \u2192 \u2191\u2191\u03bc ({x | g x + \u2191\u03b5 \u2264 f x \u2227 g x \u2264 \u2191N} \u2229 spanningSets \u03bc p) = 0 u : \u2115 \u2192 \u211d\u22650 left\u271d : StrictAnti u u_pos : \u2200 (n : \u2115), 0 < u n u_lim : Tendsto u atTop (\ud835\udcdd 0) s : \u2115 \u2192 Set \u03b1 := fun n => {x | g x + \u2191(u n) \u2264 f x \u2227 g x \u2264 \u2191\u2191n} \u2229 spanningSets \u03bc n \u03bcs : \u2200 (n : \u2115), \u2191\u2191\u03bc (s n) = 0 B : {x | f x \u2264 g x}\u1d9c \u2286 \u22c3 n, s n \u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (s n) = 0 ** simp only [\u03bcs, tsum_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.support_mul_X ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R s : \u03c3 p : MvPolynomial \u03c3 R \u22a2 \u2200 (y : R), y * 1 = 0 \u2194 y = 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.norm_condexpIndL1_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G \u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** by_cases hs : MeasurableSet s ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : MeasurableSet s \u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : \u00acMeasurableSet s \u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : MeasurableSet s \u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** by_cases h\u03bcs : \u03bc s = \u221e ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : \u00acMeasurableSet s \u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** simp_rw [condexpIndL1_of_not_measurableSet hs] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : \u00acMeasurableSet s \u22a2 \u20160\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** rw [Lp.norm_zero] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : \u00acMeasurableSet s \u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = \u22a4 \u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** rw [condexpIndL1_of_measure_eq_top h\u03bcs x, Lp.norm_zero] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s = \u22a4 \u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : MeasurableSet s h\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4 \u22a2 \u2016condexpIndL1 hm \u03bc s x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs h\u03bcs x] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) x : G hs : MeasurableSet s h\u03bcs : \u00ac\u2191\u2191\u03bc s = \u22a4 \u22a2 \u2016condexpIndL1Fin hm hs h\u03bcs x\u2016 \u2264 ENNReal.toReal (\u2191\u2191\u03bc s) * \u2016x\u2016 ** exact norm_condexpIndL1Fin_le hs h\u03bcs x ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.restrict_sInf_eq_sInf_restrict ** \u03b1 : Type u_1 m : Set (OuterMeasure \u03b1) s : Set \u03b1 hm : Set.Nonempty m \u22a2 \u2191(restrict s) (sInf m) = sInf (\u2191(restrict s) '' m) ** simp only [sInf_eq_iInf, restrict_biInf, hm, iInf_image] ** Qed",
        "informal": ""
    },
    {
        "formal": "set_integral_withDensity_eq_set_integral_smul\u2080 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 s : Set \u03b1 hf : AEMeasurable f g : \u03b1 \u2192 E hs : MeasurableSet s \u22a2 (\u222b (a : \u03b1) in s, g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1) in s, f a \u2022 g a \u2202\u03bc ** rw [restrict_withDensity hs, integral_withDensity_eq_integral_smul\u2080 hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.disjoint_take_drop ** \u03b1 : Type u_1 m n : Nat x\u271d\u00b9 : Nodup [] x\u271d : m \u2264 n \u22a2 Disjoint (take m []) (drop n []) ** simp ** \u03b1 : Type u_1 m n : Nat x : \u03b1 xs : List \u03b1 hl : Nodup (x :: xs) h : m \u2264 n \u22a2 Disjoint (take m (x :: xs)) (drop n (x :: xs)) ** cases m <;> cases n <;> simp only [disjoint_cons_left, mem_cons, disjoint_cons_right,\n  drop, true_or, eq_self_iff_true, not_true, false_and, not_mem_nil, disjoint_nil_left, take] ** case succ.zero \u03b1 : Type u_1 x : \u03b1 xs : List \u03b1 hl : Nodup (x :: xs) n\u271d : Nat h : succ n\u271d \u2264 zero \u22a2 False ** case succ.zero => cases h ** \u03b1 : Type u_1 x : \u03b1 xs : List \u03b1 hl : Nodup (x :: xs) n\u271d : Nat h : succ n\u271d \u2264 zero \u22a2 False ** cases h ** case succ.succ \u03b1 : Type u_1 x : \u03b1 xs : List \u03b1 hl : Nodup (x :: xs) n\u271d\u00b9 n\u271d : Nat h : succ n\u271d\u00b9 \u2264 succ n\u271d \u22a2 \u00acx \u2208 drop n\u271d xs \u2227 Disjoint (take n\u271d\u00b9 xs) (drop n\u271d xs) ** cases hl with | cons h\u2080 h\u2081 =>\nrefine \u27e8fun h => h\u2080 _ (mem_of_mem_drop h) rfl, ?_\u27e9\nexact disjoint_take_drop h\u2081 (Nat.le_of_succ_le_succ h) ** case succ.succ.cons \u03b1 : Type u_1 x : \u03b1 xs : List \u03b1 n\u271d\u00b9 n\u271d : Nat h : succ n\u271d\u00b9 \u2264 succ n\u271d h\u2081 : Pairwise (fun x x_1 => x \u2260 x_1) xs h\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x \u2260 a' \u22a2 \u00acx \u2208 drop n\u271d xs \u2227 Disjoint (take n\u271d\u00b9 xs) (drop n\u271d xs) ** refine \u27e8fun h => h\u2080 _ (mem_of_mem_drop h) rfl, ?_\u27e9 ** case succ.succ.cons \u03b1 : Type u_1 x : \u03b1 xs : List \u03b1 n\u271d\u00b9 n\u271d : Nat h : succ n\u271d\u00b9 \u2264 succ n\u271d h\u2081 : Pairwise (fun x x_1 => x \u2260 x_1) xs h\u2080 : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x \u2260 a' \u22a2 Disjoint (take n\u271d\u00b9 xs) (drop n\u271d xs) ** exact disjoint_take_drop h\u2081 (Nat.le_of_succ_le_succ h) ** Qed",
        "informal": ""
    },
    {
        "formal": "ack_strictMono_right ** n\u2081 n\u2082 : \u2115 h : n\u2081 < n\u2082 \u22a2 ack 0 n\u2081 < ack 0 n\u2082 ** simpa using h ** m n : \u2115 _h : 0 < n + 1 \u22a2 ack (m + 1) 0 < ack (m + 1) (n + 1) ** rw [ack_succ_zero, ack_succ_succ] ** m n : \u2115 _h : 0 < n + 1 \u22a2 ack m 1 < ack m (ack (m + 1) n) ** exact ack_strictMono_right _ (one_lt_ack_succ_left m n) ** m n\u2081 n\u2082 : \u2115 h : n\u2081 + 1 < n\u2082 + 1 \u22a2 ack (m + 1) (n\u2081 + 1) < ack (m + 1) (n\u2082 + 1) ** rw [ack_succ_succ, ack_succ_succ] ** m n\u2081 n\u2082 : \u2115 h : n\u2081 + 1 < n\u2082 + 1 \u22a2 ack m (ack (m + 1) n\u2081) < ack m (ack (m + 1) n\u2082) ** apply ack_strictMono_right _ (ack_strictMono_right _ _) ** m n\u2081 n\u2082 : \u2115 h : n\u2081 + 1 < n\u2082 + 1 \u22a2 n\u2081 < n\u2082 ** rwa [add_lt_add_iff_right] at h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass ** \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f)) ** by_cases h_mass : \u03bc.mass = 0 ** case neg \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) f : \u03a9 \u2192\u1d47 \u211d\u22650 h_mass : \u00acmass \u03bc = 0 \u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f)) ** simp_rw [fun i => (\u03bcs i).testAgainstNN_eq_mass_mul f, \u03bc.testAgainstNN_eq_mass_mul f] ** case neg \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) f : \u03a9 \u2192\u1d47 \u211d\u22650 h_mass : \u00acmass \u03bc = 0 \u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F     (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) ** rw [ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds] at \u03bcs_lim ** case neg \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim :   Tendsto (ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) F     (\ud835\udcdd (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc))) mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) f : \u03a9 \u2192\u1d47 \u211d\u22650 h_mass : \u00acmass \u03bc = 0 \u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F     (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) ** rw [tendsto_iff_forall_testAgainstNN_tendsto] at \u03bcs_lim ** case neg \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim :   \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650),     Tendsto (fun i => testAgainstNN ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) i) f) F       (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) f : \u03a9 \u2192\u1d47 \u211d\u22650 h_mass : \u00acmass \u03bc = 0 \u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F     (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) ** have lim_pair :\n  Tendsto (fun i => (\u27e8(\u03bcs i).mass, (\u03bcs i).normalize.toFiniteMeasure.testAgainstNN f\u27e9 : \u211d\u22650 \u00d7 \u211d\u22650))\n    F (\ud835\udcdd \u27e8\u03bc.mass, \u03bc.normalize.toFiniteMeasure.testAgainstNN f\u27e9) :=\n  (Prod.tendsto_iff _ _).mpr \u27e8mass_lim, \u03bcs_lim f\u27e9 ** case neg \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim :   \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650),     Tendsto (fun i => testAgainstNN ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => normalize (\u03bcs i)) i) f) F       (\ud835\udcdd (testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) f : \u03a9 \u2192\u1d47 \u211d\u22650 h_mass : \u00acmass \u03bc = 0 lim_pair :   Tendsto (fun i => (mass (\u03bcs i), testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f)) F     (\ud835\udcdd (mass \u03bc, testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) \u22a2 Tendsto (fun i => mass (\u03bcs i) * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize (\u03bcs i))) f) F     (\ud835\udcdd (mass \u03bc * testAgainstNN (ProbabilityMeasure.toFiniteMeasure (normalize \u03bc)) f)) ** exact tendsto_mul.comp lim_pair ** case pos \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 \u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize \u03bc)) mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd (mass \u03bc)) f : \u03a9 \u2192\u1d47 \u211d\u22650 h_mass : mass \u03bc = 0 \u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f)) ** simp only [\u03bc.mass_zero_iff.mp h_mass, zero_testAgainstNN_apply, zero_mass,\n  eq_self_iff_true] at * ** case pos \u03a9 : Type u_1 inst\u271d\u00b2 : Nonempty \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03b3 : Type u_2 F : Filter \u03b3 \u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9 f : \u03a9 \u2192\u1d47 \u211d\u22650 \u03bcs_lim : Tendsto (fun i => normalize (\u03bcs i)) F (\ud835\udcdd (normalize 0)) mass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0) h_mass : True \u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd 0) ** exact tendsto_zero_testAgainstNN_of_tendsto_zero_mass mass_lim f ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Infinite.exists_ne_map_eq_of_mapsTo ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hs : Set.Infinite s hf : MapsTo f s t ht : Set.Finite t \u22a2 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y ** contrapose! ht ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s : Set \u03b1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hs : Set.Infinite s hf : MapsTo f s t ht : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2260 y \u2192 f x \u2260 f y \u22a2 \u00acSet.Finite t ** exact infinite_of_injOn_mapsTo (fun x hx y hy => not_imp_not.1 (ht x hx y hy)) hf hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_one_smul_measure ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F c : \u211d\u22650\u221e \u22a2 snorm f 1 (c \u2022 \u03bc) = c * snorm f 1 \u03bc ** rw [@snorm_smul_measure_of_ne_top _ _ _ \u03bc _ 1 (@ENNReal.coe_ne_top 1) f c] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F c : \u211d\u22650\u221e \u22a2 c ^ ENNReal.toReal (1 / 1) \u2022 snorm f 1 \u03bc = c * snorm f 1 \u03bc ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.dvd_iff_dvd_of_dvd_add ** a b c : Int H : a \u2223 b + c \u22a2 a \u2223 b \u2194 a \u2223 c ** rw [\u2190 Int.sub_neg] at H ** a b c : Int H : a \u2223 b - -c \u22a2 a \u2223 b \u2194 a \u2223 c ** rw [Int.dvd_iff_dvd_of_dvd_sub H, Int.dvd_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_restrict_uIoc_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d : LinearOrder \u03b1 a b : \u03b1 \u22a2 ae (Measure.restrict \u03bc (\u0399 a b)) = ae (Measure.restrict \u03bc (Ioc a b)) \u2294 ae (Measure.restrict \u03bc (Ioc b a)) ** simp only [uIoc_eq_union, ae_restrict_union_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "WType.cardinal_mk_le_of_le ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u \u03ba : Cardinal.{u} h\u03ba : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba \u22a2 #(WType \u03b2) \u2264 \u03ba ** induction' \u03ba using Cardinal.inductionOn with \u03b3 ** case h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u \u03ba : Cardinal.{u} h\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba \u03b3 : Type u h\u03ba : (sum fun a => #\u03b3 ^ #(\u03b2 a)) \u2264 #\u03b3 \u22a2 #(WType \u03b2) \u2264 #\u03b3 ** simp only [Cardinal.power_def, \u2190 Cardinal.mk_sigma, Cardinal.le_def] at h\u03ba ** case h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u \u03ba : Cardinal.{u} h\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba \u03b3 : Type u h\u03ba : Nonempty ((i : \u03b1) \u00d7 (\u03b2 i \u2192 \u03b3) \u21aa \u03b3) \u22a2 #(WType \u03b2) \u2264 #\u03b3 ** cases' h\u03ba with h\u03ba ** case h.intro \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u \u03ba : Cardinal.{u} h\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba \u03b3 : Type u h\u03ba : (i : \u03b1) \u00d7 (\u03b2 i \u2192 \u03b3) \u21aa \u03b3 \u22a2 #(WType \u03b2) \u2264 #\u03b3 ** exact Cardinal.mk_le_of_injective (elim_injective _ h\u03ba.1 h\u03ba.2) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.continuous_integral_integral ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' \u22a2 Continuous fun f => \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** rw [continuous_iff_continuousAt] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' \u22a2 \u2200 (x : { x // x \u2208 Lp E 1 }), ContinuousAt (fun f => \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) x ** intro g ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } \u22a2 ContinuousAt (fun f => \u222b (x : \u03b2), \u222b (y : \u03b3), \u2191\u2191f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) g ** refine'\n  tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd\n    (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) \u2202\u2191\u03b7 (a, x) - \u222b (y : \u03b3), \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) (\ud835\udcdd g)     (\ud835\udcdd 0) ** simp_rw [\u2190\n  kernel.lintegral_fn_integral_sub (fun x => (\u2016x\u2016\u208a : \u211d\u22650\u221e)) (L1.integrable_coeFn _)\n    (L1.integrable_coeFn g)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0) ** refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _ ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0)  case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } \u22a2 (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) \u2264 fun i =>     \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** swap ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0) ** have : \u2200 i : (MeasureTheory.Lp (\u03b1 := \u03b2 \u00d7 \u03b3) E 1 (((\u03ba \u2297\u2096 \u03b7) a) : Measure (\u03b2 \u00d7 \u03b3))),\n    Measurable fun z => (\u2016i z - g z\u2016\u208a : \u211d\u22650\u221e) := fun i =>\n  ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).ennnorm ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } this : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a \u22a2 Tendsto (fun i => \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a) (\ud835\udcdd g) (\ud835\udcdd 0) ** simp_rw [\u2190 kernel.lintegral_compProd _ _ _ (this _), \u2190 L1.ofReal_norm_sub_eq_lintegral, \u2190\n  ofReal_zero] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } this : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a \u22a2 Tendsto (fun i => ENNReal.ofReal \u2016i - g\u2016) (\ud835\udcdd g) (\ud835\udcdd (ENNReal.ofReal 0)) ** refine' (continuous_ofReal.tendsto 0).comp _ ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } this : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a \u22a2 Tendsto (fun i => \u2016i - g\u2016) (\ud835\udcdd g) (\ud835\udcdd 0) ** rw [\u2190 tendsto_iff_norm_sub_tendsto_zero] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } this : \u2200 (i : { x // x \u2208 Lp E 1 }), Measurable fun z => \u2191\u2016\u2191\u2191i z - \u2191\u2191g z\u2016\u208a \u22a2 Tendsto (fun i => i) (\ud835\udcdd g) (\ud835\udcdd g) ** exact tendsto_id ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } \u22a2 { x // x \u2208 Lp E 1 } \u2192 \u211d\u22650\u221e ** exact fun i => \u222b\u207b x, \u222b\u207b y, \u2016i (x, y) - g (x, y)\u2016\u208a \u2202\u03b7 (a, x) \u2202\u03ba a ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u2077 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u2076 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d\u2075 : IsSFiniteKernel \u03b7 a : \u03b1 inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E E' : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : CompleteSpace E' inst\u271d : NormedSpace \u211d E' g : { x // x \u2208 Lp E 1 } \u22a2 (fun i => \u222b\u207b (x : \u03b2), \u2191\u2016\u222b (y : \u03b3), \u2191\u2191i (x, y) - \u2191\u2191g (x, y) \u2202\u2191\u03b7 (a, x)\u2016\u208a \u2202\u2191\u03ba a) \u2264 fun i =>     \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3), \u2191\u2016\u2191\u2191i (x, y) - \u2191\u2191g (x, y)\u2016\u208a \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a ** exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Path.ordered_iff ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering p : Path \u03b1 \u22a2 Ordered cmp p \u2194     List.Pairwise (cmpLT cmp) (listL p) \u2227       List.Pairwise (cmpLT cmp) (listR p) \u2227 \u2200 (x : \u03b1), x \u2208 listL p \u2192 \u2200 (y : \u03b1), y \u2208 listR p \u2192 cmpLT cmp x y ** induction p with\n| root => simp\n| left _ _ x _ ih | right _ _ x _ ih => ?_ ** case left \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c\u271d : RBColor parent\u271d : Path \u03b1 x : \u03b1 r\u271d : RBNode \u03b1 ih :   Ordered cmp parent\u271d \u2194     List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227       List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227         \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y \u22a2 Ordered cmp (left c\u271d parent\u271d x r\u271d) \u2194     List.Pairwise (cmpLT cmp) (listL (left c\u271d parent\u271d x r\u271d)) \u2227       List.Pairwise (cmpLT cmp) (listR (left c\u271d parent\u271d x r\u271d)) \u2227         \u2200 (x_1 : \u03b1), x_1 \u2208 listL (left c\u271d parent\u271d x r\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (left c\u271d parent\u271d x r\u271d) \u2192 cmpLT cmp x_1 y  case right \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 x : \u03b1 parent\u271d : Path \u03b1 ih :   Ordered cmp parent\u271d \u2194     List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227       List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227         \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y \u22a2 Ordered cmp (right c\u271d l\u271d x parent\u271d) \u2194     List.Pairwise (cmpLT cmp) (listL (right c\u271d l\u271d x parent\u271d)) \u2227       List.Pairwise (cmpLT cmp) (listR (right c\u271d l\u271d x parent\u271d)) \u2227         \u2200 (x_1 : \u03b1),           x_1 \u2208 listL (right c\u271d l\u271d x parent\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (right c\u271d l\u271d x parent\u271d) \u2192 cmpLT cmp x_1 y ** all_goals\n  rw [Ordered, and_congr_right_eq fun h => by simp [All_def, rootOrdered_iff h]; rfl]\n  simp [List.pairwise_append, or_imp, forall_and, ih, RBNode.ordered_iff] ** case root \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering \u22a2 Ordered cmp root \u2194     List.Pairwise (cmpLT cmp) (listL root) \u2227       List.Pairwise (cmpLT cmp) (listR root) \u2227 \u2200 (x : \u03b1), x \u2208 listL root \u2192 \u2200 (y : \u03b1), y \u2208 listR root \u2192 cmpLT cmp x y ** simp ** case right \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 x : \u03b1 parent\u271d : Path \u03b1 ih :   Ordered cmp parent\u271d \u2194     List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227       List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227         \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y \u22a2 Ordered cmp (right c\u271d l\u271d x parent\u271d) \u2194     List.Pairwise (cmpLT cmp) (listL (right c\u271d l\u271d x parent\u271d)) \u2227       List.Pairwise (cmpLT cmp) (listR (right c\u271d l\u271d x parent\u271d)) \u2227         \u2200 (x_1 : \u03b1),           x_1 \u2208 listL (right c\u271d l\u271d x parent\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (right c\u271d l\u271d x parent\u271d) \u2192 cmpLT cmp x_1 y ** rw [Ordered, and_congr_right_eq fun h => by simp [All_def, rootOrdered_iff h]; rfl] ** case right \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 x : \u03b1 parent\u271d : Path \u03b1 ih :   Ordered cmp parent\u271d \u2194     List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227       List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227         \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y \u22a2 Ordered cmp parent\u271d \u2227       (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227         ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227           (\u2200 (x : \u03b1),               x \u2208 l\u271d \u2192 (\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227             RBNode.Ordered cmp l\u271d \u2194     List.Pairwise (cmpLT cmp) (listL (right c\u271d l\u271d x parent\u271d)) \u2227       List.Pairwise (cmpLT cmp) (listR (right c\u271d l\u271d x parent\u271d)) \u2227         \u2200 (x_1 : \u03b1),           x_1 \u2208 listL (right c\u271d l\u271d x parent\u271d) \u2192 \u2200 (y : \u03b1), y \u2208 listR (right c\u271d l\u271d x parent\u271d) \u2192 cmpLT cmp x_1 y ** simp [List.pairwise_append, or_imp, forall_and, ih, RBNode.ordered_iff] ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 x : \u03b1 parent\u271d : Path \u03b1 ih :   Ordered cmp parent\u271d \u2194     List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227       List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227         \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y h : Ordered cmp parent\u271d \u22a2 (All (fun x_1 => cmpLT cmp x_1 x) l\u271d \u2227       RootOrdered cmp parent\u271d x \u2227 All (RootOrdered cmp parent\u271d) l\u271d \u2227 RBNode.Ordered cmp l\u271d) =     ?m.118001 ** simp [All_def, rootOrdered_iff h] ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 x : \u03b1 parent\u271d : Path \u03b1 ih :   Ordered cmp parent\u271d \u2194     List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227       List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227         \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y h : Ordered cmp parent\u271d \u22a2 (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227       ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227         (\u2200 (x : \u03b1),             x \u2208 l\u271d \u2192 (\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227           RBNode.Ordered cmp l\u271d \u2194     ?m.118001 ** rfl ** case left \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c\u271d : RBColor parent\u271d : Path \u03b1 x : \u03b1 r\u271d : RBNode \u03b1 ih :   Ordered cmp parent\u271d \u2194     List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227       List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227         \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y \u22a2 (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227         List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227           \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227       (\u2200 (x_1 : \u03b1), x_1 \u2208 r\u271d \u2192 cmpLT cmp x x_1) \u2227         ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227           ((\u2200 (x : \u03b1), x \u2208 r\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227               \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227             List.Pairwise (cmpLT cmp) (toList r\u271d) \u2194     List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227       (((\u2200 (x_1 : \u03b1), x_1 \u2208 r\u271d \u2192 cmpLT cmp x x_1) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227           List.Pairwise (cmpLT cmp) (toList r\u271d) \u2227             List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227               \u2200 (x : \u03b1), x \u2208 r\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227         (\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227           (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 r\u271d \u2192 cmpLT cmp x x_2) \u2227             \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y ** exact \u27e8\n  fun \u27e8\u27e8hL, hR, LR\u27e9, xr, \u27e8Lx, xR\u27e9, \u27e8rL, rR\u27e9, hr\u27e9 =>\n    \u27e8hL, \u27e8\u27e8xr, xR\u27e9, hr, hR, rR\u27e9, Lx, fun _ ha _ hb => rL _ hb _ ha, LR\u27e9,\n  fun \u27e8hL, \u27e8\u27e8xr, xR\u27e9, hr, hR, rR\u27e9, Lx, Lr, LR\u27e9 =>\n    \u27e8\u27e8hL, hR, LR\u27e9, xr, \u27e8Lx, xR\u27e9, \u27e8fun _ ha _ hb => Lr _ hb _ ha, rR\u27e9, hr\u27e9\u27e9 ** case right \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c\u271d : RBColor l\u271d : RBNode \u03b1 x : \u03b1 parent\u271d : Path \u03b1 ih :   Ordered cmp parent\u271d \u2194     List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227       List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227         \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y \u22a2 (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227         List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227           \u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227       (\u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227         ((\u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227           ((\u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227               \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227             List.Pairwise (cmpLT cmp) (toList l\u271d) \u2194     (List.Pairwise (cmpLT cmp) (listL parent\u271d) \u2227         (List.Pairwise (cmpLT cmp) (toList l\u271d) \u2227 \u2200 (x_1 : \u03b1), x_1 \u2208 l\u271d \u2192 cmpLT cmp x_1 x) \u2227           (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 l\u271d \u2192 cmpLT cmp x x_2) \u2227             \u2200 (a : \u03b1), a \u2208 listL parent\u271d \u2192 cmpLT cmp a x) \u2227       List.Pairwise (cmpLT cmp) (listR parent\u271d) \u2227         (\u2200 (x : \u03b1), x \u2208 listL parent\u271d \u2192 \u2200 (y : \u03b1), y \u2208 listR parent\u271d \u2192 cmpLT cmp x y) \u2227           (\u2200 (x : \u03b1), x \u2208 l\u271d \u2192 \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a) \u2227             \u2200 (a : \u03b1), a \u2208 listR parent\u271d \u2192 cmpLT cmp x a ** exact \u27e8\n  fun \u27e8\u27e8hL, hR, LR\u27e9, lx, \u27e8Lx, xR\u27e9, \u27e8lL, lR\u27e9, hl\u27e9 =>\n    \u27e8\u27e8hL, \u27e8hl, lx\u27e9, fun _ ha _ hb => lL _ hb _ ha, Lx\u27e9, hR, LR, lR, xR\u27e9,\n  fun \u27e8\u27e8hL, \u27e8hl, lx\u27e9, Ll, Lx\u27e9, hR, LR, lR, xR\u27e9 =>\n   \u27e8\u27e8hL, hR, LR\u27e9, lx, \u27e8Lx, xR\u27e9, \u27e8fun _ ha _ hb => Ll _ hb _ ha, lR\u27e9, hl\u27e9\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.empty_card' ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x h : Fintype \u2191\u2205 \u22a2 Fintype.card \u2191\u2205 = Fintype.card \u2191\u2205 ** congr ** case h.e_2.h \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x h : Fintype \u2191\u2205 \u22a2 h = fintypeEmpty ** exact Subsingleton.elim _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.div_to_nat ** \u22a2 \u2191(0 / 0) = \u21910 / \u21910 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc t : Set \u03a9 h_indep : IndepSet t t \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 ** have h_0_1_top := measure_eq_zero_or_one_or_top_of_indepSet_self h_indep ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc t : Set \u03a9 h_indep : IndepSet t t h_0_1_top : \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 \u2228 \u2191\u2191\u03bc t = \u22a4 \u22a2 \u2191\u2191\u03bc t = 0 \u2228 \u2191\u2191\u03bc t = 1 ** simpa [measure_ne_top \u03bc] using h_0_1_top ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.map_linearMap_addHaar_eq_smul_addHaar ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** let \u03b9 := Fin (finrank \u211d E) ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** haveI : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) := by infer_instance ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) this : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** have : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) := by simp ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** have e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d := LinearEquiv.ofFinrankEq E (\u03b9 \u2192 \u211d) this ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** obtain \u27e8g, hg\u27e9 : \u2203 g, g = (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d).comp (f.comp (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E)) := \u27e8_, rfl\u27e9 ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** have gdet : LinearMap.det g = LinearMap.det f := by rw [hg]; exact LinearMap.det_conj f e ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** rw [\u2190 gdet] at hf \u22a2 ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc ** have fg : f = (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E).comp (g.comp (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d)) := by\n  ext x\n  simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp,\n    LinearEquiv.symm_apply_apply, hg] ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc ** simp only [fg, LinearEquiv.coe_coe, LinearMap.coe_comp] ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) \u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc ** have Ce : Continuous e := (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d).continuous_of_finiteDimensional ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) Ce : Continuous \u2191e \u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc ** have Cg : Continuous g := LinearMap.continuous_of_finiteDimensional g ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) Ce : Continuous \u2191e Cg : Continuous \u2191g \u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc ** have Cesymm : Continuous e.symm := (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E).continuous_of_finiteDimensional ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) Ce : Continuous \u2191e Cg : Continuous \u2191g Cesymm : Continuous \u2191(LinearEquiv.symm e) \u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc ** rw [\u2190 map_map Cesymm.measurable (Cg.comp Ce).measurable, \u2190 map_map Cg.measurable Ce.measurable] ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) Ce : Continuous \u2191e Cg : Continuous \u2191g Cesymm : Continuous \u2191(LinearEquiv.symm e) \u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc ** haveI : IsAddHaarMeasure (map e \u03bc) := (e : E \u2243+ (\u03b9 \u2192 \u211d)).isAddHaarMeasure_map \u03bc Ce Cesymm ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) Ce : Continuous \u2191e Cg : Continuous \u2191g Cesymm : Continuous \u2191(LinearEquiv.symm e) this : IsAddHaarMeasure (map (\u2191e) \u03bc) \u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc ** have ecomp : e.symm \u2218 e = id := by\n  ext x; simp only [id.def, Function.comp_apply, LinearEquiv.symm_apply_apply] ** case intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) Ce : Continuous \u2191e Cg : Continuous \u2191g Cesymm : Continuous \u2191(LinearEquiv.symm e) this : IsAddHaarMeasure (map (\u2191e) \u03bc) ecomp : \u2191(LinearEquiv.symm e) \u2218 \u2191e = id \u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc ** rw [map_linearMap_addHaar_pi_eq_smul_addHaar hf (map e \u03bc), Measure.map_smul,\n  map_map Cesymm.measurable Ce.measurable, ecomp, Measure.map_id] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) \u22a2 FiniteDimensional \u211d (\u03b9 \u2192 \u211d) ** infer_instance ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) this : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) \u22a2 finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) ** simp ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) \u22a2 \u2191LinearMap.det g = \u2191LinearMap.det f ** rw [hg] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E hf : \u2191LinearMap.det f \u2260 0 \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) \u22a2 \u2191LinearMap.det (LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f ** exact LinearMap.det_conj f e ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f \u22a2 f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) ** ext x ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f x : E \u22a2 \u2191f x = \u2191(LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)) x ** simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp,\n  LinearEquiv.symm_apply_apply, hg] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) Ce : Continuous \u2191e Cg : Continuous \u2191g Cesymm : Continuous \u2191(LinearEquiv.symm e) this : IsAddHaarMeasure (map (\u2191e) \u03bc) \u22a2 \u2191(LinearEquiv.symm e) \u2218 \u2191e = id ** ext x ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E \u03b9 : Type := Fin (finrank \u211d E) this\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) this\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d g : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det g \u2260 0 hg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e)) gdet : \u2191LinearMap.det g = \u2191LinearMap.det f fg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e) Ce : Continuous \u2191e Cg : Continuous \u2191g Cesymm : Continuous \u2191(LinearEquiv.symm e) this : IsAddHaarMeasure (map (\u2191e) \u03bc) x : E \u22a2 (\u2191(LinearEquiv.symm e) \u2218 \u2191e) x = id x ** simp only [id.def, Function.comp_apply, LinearEquiv.symm_apply_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "NNReal.count_const_le_le_of_tsum_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 a : \u03b1 \u2192 \u211d\u22650 a_mble : Measurable a a_summable : Summable a c : \u211d\u22650 tsum_le_c : \u2211' (i : \u03b1), a i \u2264 c \u03b5 : \u211d\u22650 \u03b5_ne_zero : \u03b5 \u2260 0 \u22a2 \u2191\u2191count {i | \u03b5 \u2264 a i} \u2264 \u2191c / \u2191\u03b5 ** rw [show (fun i => \u03b5 \u2264 a i) = fun i => (\u03b5 : \u211d\u22650\u221e) \u2264 ((\u2191) \u2218 a) i by\n    funext i\n    simp only [ENNReal.coe_le_coe, Function.comp]] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 a : \u03b1 \u2192 \u211d\u22650 a_mble : Measurable a a_summable : Summable a c : \u211d\u22650 tsum_le_c : \u2211' (i : \u03b1), a i \u2264 c \u03b5 : \u211d\u22650 \u03b5_ne_zero : \u03b5 \u2260 0 \u22a2 \u2191\u2191count {i | \u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i} \u2264 \u2191c / \u2191\u03b5 ** apply\n  ENNReal.count_const_le_le_of_tsum_le (measurable_coe_nnreal_ennreal.comp a_mble) _\n    (by exact_mod_cast \u03b5_ne_zero) (@ENNReal.coe_ne_top \u03b5) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 a : \u03b1 \u2192 \u211d\u22650 a_mble : Measurable a a_summable : Summable a c : \u211d\u22650 tsum_le_c : \u2211' (i : \u03b1), a i \u2264 c \u03b5 : \u211d\u22650 \u03b5_ne_zero : \u03b5 \u2260 0 \u22a2 \u2211' (i : \u03b1), (ENNReal.some \u2218 a) i \u2264 \u2191c ** convert ENNReal.coe_le_coe.mpr tsum_le_c ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 a : \u03b1 \u2192 \u211d\u22650 a_mble : Measurable a a_summable : Summable a c : \u211d\u22650 tsum_le_c : \u2211' (i : \u03b1), a i \u2264 c \u03b5 : \u211d\u22650 \u03b5_ne_zero : \u03b5 \u2260 0 \u22a2 \u2211' (i : \u03b1), (ENNReal.some \u2218 a) i = \u2191(\u2211' (i : \u03b1), a i) ** simp_rw [Function.comp_apply] ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 a : \u03b1 \u2192 \u211d\u22650 a_mble : Measurable a a_summable : Summable a c : \u211d\u22650 tsum_le_c : \u2211' (i : \u03b1), a i \u2264 c \u03b5 : \u211d\u22650 \u03b5_ne_zero : \u03b5 \u2260 0 \u22a2 \u2211' (i : \u03b1), \u2191(a i) = \u2191(\u2211' (i : \u03b1), a i) ** rw [ENNReal.tsum_coe_eq a_summable.hasSum] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 a : \u03b1 \u2192 \u211d\u22650 a_mble : Measurable a a_summable : Summable a c : \u211d\u22650 tsum_le_c : \u2211' (i : \u03b1), a i \u2264 c \u03b5 : \u211d\u22650 \u03b5_ne_zero : \u03b5 \u2260 0 \u22a2 (fun i => \u03b5 \u2264 a i) = fun i => \u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i ** funext i ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 a : \u03b1 \u2192 \u211d\u22650 a_mble : Measurable a a_summable : Summable a c : \u211d\u22650 tsum_le_c : \u2211' (i : \u03b1), a i \u2264 c \u03b5 : \u211d\u22650 \u03b5_ne_zero : \u03b5 \u2260 0 i : \u03b1 \u22a2 (\u03b5 \u2264 a i) = (\u2191\u03b5 \u2264 (ENNReal.some \u2218 a) i) ** simp only [ENNReal.coe_le_coe, Function.comp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 a : \u03b1 \u2192 \u211d\u22650 a_mble : Measurable a a_summable : Summable a c : \u211d\u22650 tsum_le_c : \u2211' (i : \u03b1), a i \u2264 c \u03b5 : \u211d\u22650 \u03b5_ne_zero : \u03b5 \u2260 0 \u22a2 \u2191\u03b5 \u2260 0 ** exact_mod_cast \u03b5_ne_zero ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.sdiff_union_erase_cancel ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t u v : Finset \u03b1 a b : \u03b1 hts : t \u2286 s ha : a \u2208 t \u22a2 s \\ t \u222a erase t a = erase s a ** simp_rw [erase_eq, sdiff_union_sdiff_cancel hts (singleton_subset_iff.2 ha)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.prehaar_mono ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G U : Set G hU : Set.Nonempty (interior U) K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 K\u2082.carrier \u22a2 prehaar (\u2191K\u2080) U K\u2081 \u2264 prehaar (\u2191K\u2080) U K\u2082 ** simp only [prehaar] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G U : Set G hU : Set.Nonempty (interior U) K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 K\u2082.carrier \u22a2 \u2191(index (\u2191K\u2081) U) / \u2191(index (\u2191K\u2080) U) \u2264 \u2191(index (\u2191K\u2082) U) / \u2191(index (\u2191K\u2080) U) ** rw [div_le_div_right] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G U : Set G hU : Set.Nonempty (interior U) K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 K\u2082.carrier \u22a2 \u2191(index (\u2191K\u2081) U) \u2264 \u2191(index (\u2191K\u2082) U)  G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G U : Set G hU : Set.Nonempty (interior U) K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 K\u2082.carrier \u22a2 0 < \u2191(index (\u2191K\u2080) U) ** exact_mod_cast index_mono K\u2082.2 h hU ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G U : Set G hU : Set.Nonempty (interior U) K\u2081 K\u2082 : Compacts G h : \u2191K\u2081 \u2286 K\u2082.carrier \u22a2 0 < \u2191(index (\u2191K\u2080) U) ** exact_mod_cast index_pos K\u2080 hU ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsStoppingTime.measurableSet_lt_of_pred ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i} ** by_cases hi_min : IsMin i ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : \u00acIsMin i \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i} ** have : {\u03c9 : \u03a9 | \u03c4 \u03c9 < i} = \u03c4 \u207b\u00b9' Set.Iio i := rfl ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : \u00acIsMin i this : {\u03c9 | \u03c4 \u03c9 < i} = \u03c4 \u207b\u00b9' Set.Iio i \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i} ** rw [this, \u2190 Iic_pred_of_not_isMin hi_min] ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : \u00acIsMin i this : {\u03c9 | \u03c4 \u03c9 < i} = \u03c4 \u207b\u00b9' Set.Iio i \u22a2 MeasurableSet (\u03c4 \u207b\u00b9' Set.Iic (pred i)) ** exact f.mono (pred_le i) _ (h\u03c4.measurableSet_le <| pred i) ** case pos \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : IsMin i \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i} ** suffices {\u03c9 : \u03a9 | \u03c4 \u03c9 < i} = \u2205 by rw [this]; exact @MeasurableSet.empty _ (f i) ** case pos \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : IsMin i \u22a2 {\u03c9 | \u03c4 \u03c9 < i} = \u2205 ** ext1 \u03c9 ** case pos.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : IsMin i \u03c9 : \u03a9 \u22a2 \u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 < i} \u2194 \u03c9 \u2208 \u2205 ** simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff] ** case pos.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : IsMin i \u03c9 : \u03a9 \u22a2 \u00ac\u03c4 \u03c9 < i ** rw [isMin_iff_forall_not_lt] at hi_min ** case pos.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : \u2200 (b : \u03b9), \u00acb < i \u03c9 : \u03a9 \u22a2 \u00ac\u03c4 \u03c9 < i ** exact hi_min (\u03c4 \u03c9) ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : IsMin i this : {\u03c9 | \u03c4 \u03c9 < i} = \u2205 \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 < i} ** rw [this] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b9 : Preorder \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : PredOrder \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 hi_min : IsMin i this : {\u03c9 | \u03c4 \u03c9 < i} = \u2205 \u22a2 MeasurableSet \u2205 ** exact @MeasurableSet.empty _ (f i) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image_add_right_Ioc ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1 inst\u271d\u00b2 : ExistsAddOfLE \u03b1 inst\u271d\u00b9 : LocallyFiniteOrder \u03b1 inst\u271d : DecidableEq \u03b1 a b c : \u03b1 \u22a2 image (fun x => x + c) (Ioc a b) = Ioc (a + c) (b + c) ** rw [\u2190 map_add_right_Ioc, map_eq_image, addRightEmbedding, Embedding.coeFn_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hasFiniteIntegral_prod_iff' ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u00d7 \u03b2 \u2192 E h1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd) \u22a2 HasFiniteIntegral f \u2194     (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)) \u2227 HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd ** rw [hasFiniteIntegral_congr h1f.ae_eq_mk,\n  hasFiniteIntegral_prod_iff h1f.stronglyMeasurable_mk] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u00d7 \u03b2 \u2192 E h1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd) \u22a2 ((\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2227       HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) \u2194     (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y)) \u2227 HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd ** apply and_congr ** case h\u2081 \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u00d7 \u03b2 \u2192 E h1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd) \u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194     \u2200\u1d50 (x : \u03b1) \u2202\u03bc, HasFiniteIntegral fun y => f (x, y) ** apply eventually_congr ** case h\u2081.h \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u00d7 \u03b2 \u2192 E h1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y) ** filter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm] ** case h \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u00d7 \u03b2 \u2192 E h1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd) \u22a2 \u2200 (a : \u03b1),     (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, AEStronglyMeasurable.mk f h1f (a, y) = f (a, y)) \u2192       ((HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (a, y)) \u2194 HasFiniteIntegral fun y => f (a, y)) ** intro x hx ** case h \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u00d7 \u03b2 \u2192 E h1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd) x : \u03b1 hx : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, AEStronglyMeasurable.mk f h1f (x, y) = f (x, y) \u22a2 (HasFiniteIntegral fun y => AEStronglyMeasurable.mk f h1f (x, y)) \u2194 HasFiniteIntegral fun y => f (x, y) ** exact hasFiniteIntegral_congr hx ** case h\u2082 \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u00d7 \u03b2 \u2192 E h1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd) \u22a2 (HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) \u2194     HasFiniteIntegral fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd ** apply hasFiniteIntegral_congr ** case h\u2082.h \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u00d7 \u03b2 \u2192 E h1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd) \u22a2 (fun x => \u222b (y : \u03b2), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03bd) =\u1da0[ae \u03bc] fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd ** filter_upwards [ae_ae_of_ae_prod h1f.ae_eq_mk.symm] with _ hx using\n  integral_congr_ae (EventuallyEq.fun_comp hx _) ** Qed",
        "informal": ""
    },
    {
        "formal": "VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator s 1 x)) ** haveI : IsLocallyFiniteMeasure (\u03bc.restrict s) :=\n  isLocallyFiniteMeasure_of_le restrict_le_self ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s this : IsLocallyFiniteMeasure (Measure.restrict \u03bc s) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator s 1 x)) ** filter_upwards [ae_tendsto_rnDeriv v (\u03bc.restrict s), rnDeriv_restrict \u03bc hs] ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s this : IsLocallyFiniteMeasure (Measure.restrict \u03bc s) \u22a2 \u2200 (a : \u03b1),     Tendsto (fun a => \u2191\u2191(Measure.restrict \u03bc s) a / \u2191\u2191\u03bc a) (filterAt v a) (\ud835\udcdd (rnDeriv (Measure.restrict \u03bc s) \u03bc a)) \u2192       rnDeriv (Measure.restrict \u03bc s) \u03bc a = indicator s 1 a \u2192         Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v a) (\ud835\udcdd (indicator s 1 a)) ** intro x hx h'x ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s this : IsLocallyFiniteMeasure (Measure.restrict \u03bc s) x : \u03b1 hx : Tendsto (fun a => \u2191\u2191(Measure.restrict \u03bc s) a / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (rnDeriv (Measure.restrict \u03bc s) \u03bc x)) h'x : rnDeriv (Measure.restrict \u03bc s) \u03bc x = indicator s 1 x \u22a2 Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator s 1 x)) ** simpa only [h'x, restrict_apply' hs, inter_comm] using hx ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.prob_compl_eq_zero_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d : IsProbabilityMeasure \u03bc hs : MeasurableSet s \u22a2 \u2191\u2191\u03bc s\u1d9c = 0 \u2194 \u2191\u2191\u03bc s = 1 ** rw [prob_compl_eq_one_sub hs, tsub_eq_zero_iff_le, one_le_prob_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.zero_smul_set ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : Zero \u03b1 inst\u271d\u00b9 : Zero \u03b2 inst\u271d : SMulWithZero \u03b1 \u03b2 s\u271d : Set \u03b1 t s : Set \u03b2 h : Set.Nonempty s \u22a2 0 \u2022 s = 0 ** simp only [\u2190 image_smul, image_eta, zero_smul, h.image_const, singleton_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ordConnected_image ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1 inst\u271d\u00b9 : Preorder \u03b2 s\u271d t : Set \u03b1 E : Type u_3 inst\u271d : OrderIsoClass E \u03b1 \u03b2 e : E s : Set \u03b1 hs : OrdConnected s \u22a2 OrdConnected (\u2191e '' s) ** erw [(e : \u03b1 \u2243o \u03b2).image_eq_preimage] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b1 inst\u271d\u00b9 : Preorder \u03b2 s\u271d t : Set \u03b1 E : Type u_3 inst\u271d : OrderIsoClass E \u03b1 \u03b2 e : E s : Set \u03b1 hs : OrdConnected s \u22a2 OrdConnected (\u2191(OrderIso.symm \u2191e) \u207b\u00b9' s) ** apply ordConnected_preimage (e : \u03b1 \u2243o \u03b2).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.coe_nnnorm_toLp ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hf : Mem\u2112p f p \u22a2 \u2191\u2016Mem\u2112p.toLp f hf\u2016\u208a = snorm f p \u03bc ** rw [nnnorm_toLp f hf, ENNReal.coe_toNNReal hf.2.ne] ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_signed_sum ** \u03b1\u271d \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 \u03b2 : Type u_1 t : Finset \u03b2 sgn : \u03b2 \u2192 SignType g : \u03b2 \u2192 \u03b1 hg : \u2200 (b : \u03b2), g b \u2208 s ht : card t = \u2211 a in s, Int.natAbs (f a) hf : \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b in t, if g b = a then \u2191(sgn b) else 0) = f a \u22a2 Fintype.card { x // x \u2208 t } = \u2211 a in s, Int.natAbs (f a) ** simp [ht] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.chaar_self ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G \u22a2 chaar K\u2080 K\u2080.toCompacts = 1 ** let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts \u22a2 chaar K\u2080 K\u2080.toCompacts = 1 ** have : Continuous eval := continuous_apply _ ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval \u22a2 chaar K\u2080 K\u2080.toCompacts = 1 ** show chaar K\u2080 \u2208 eval \u207b\u00b9' {(1 : \u211d)} ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval \u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' {1} ** apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval \u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' {1} ** unfold clPrehaar ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval \u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' {1} ** rw [IsClosed.closure_subset_iff] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval \u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' {1} ** rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9 ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' {1} ** apply prehaar_self ** case intro.intro.intro.intro.hU G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Set.Nonempty (interior U) ** rw [h2U.interior_eq] ** case intro.intro.intro.intro.hU G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Set.Nonempty U ** exact \u27e81, h3U\u27e9 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval \u22a2 IsClosed (eval \u207b\u00b9' {1}) ** apply continuous_iff_isClosed.mp this ** case a G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2080.toCompacts this : Continuous eval \u22a2 IsClosed {1} ** exact isClosed_singleton ** Qed",
        "informal": ""
    },
    {
        "formal": "ComputablePred.computable_iff_re_compl_re' ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 p : \u03b1 \u2192 Prop \u22a2 ComputablePred p \u2194 RePred p \u2227 RePred fun a => \u00acp a ** classical exact computable_iff_re_compl_re ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 p : \u03b1 \u2192 Prop \u22a2 ComputablePred p \u2194 RePred p \u2227 RePred fun a => \u00acp a ** exact computable_iff_re_compl_re ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.Cofix.bisim_aux ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y \u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y ** intro x ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x : Cofix F \u03b1 \u22a2 \u2200 (y : Cofix F \u03b1), r x y \u2192 x = y ** rcases x ** case mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x : Cofix F \u03b1 a\u271d : M (P F) \u03b1 \u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr a\u271d) y \u2192 Quot.mk Mcongr a\u271d = y ** clear x ** case mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y a\u271d : M (P F) \u03b1 \u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr a\u271d) y \u2192 Quot.mk Mcongr a\u271d = y ** rename M (P F) \u03b1 => x ** case mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x : M (P F) \u03b1 \u22a2 \u2200 (y : Cofix F \u03b1), r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y ** intro y ** case mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x : M (P F) \u03b1 y : Cofix F \u03b1 \u22a2 r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y ** rcases y ** case mk.mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x : M (P F) \u03b1 y : Cofix F \u03b1 a\u271d : M (P F) \u03b1 \u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr a\u271d) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\u271d ** clear y ** case mk.mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x a\u271d : M (P F) \u03b1 \u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr a\u271d) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\u271d ** rename M (P F) \u03b1 => y ** case mk.mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 \u22a2 r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr y ** intro rxy ** case mk.mk n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u22a2 Quot.mk Mcongr x = Quot.mk Mcongr y ** apply Quot.sound ** case mk.mk.a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u22a2 Mcongr x y ** let r' := fun x y => r (Quot.mk _ x) (Quot.mk _ y) ** case mk.mk.a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u22a2 Mcongr x y ** have hr' : r' = fun x y => r (Quot.mk _ x) (Quot.mk _ y) := rfl ** case mk.mk.a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u22a2 Mcongr x y ** have : IsPrecongr r' := by\n  intro a b r'ab\n  have h\u2080 :\n    appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P a) =\n      appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P b) :=\n    by rw [appendFun_comp_id, comp_map, comp_map]; exact h _ _ r'ab\n  have h\u2081 : \u2200 u v : q.P.M \u03b1, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v := by\n    intro u v cuv\n    apply Quot.sound\n    dsimp [hr']\n    rw [Quot.sound cuv]\n    apply h'\n  let f : Quot r \u2192 Quot r' :=\n    Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n      (by\n        intro c\n        apply Quot.inductionOn\n          (motive := fun c =>\n            \u2200b, r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) c\n        clear c\n        intro c d\n        apply Quot.inductionOn\n          (motive := fun d => r (Quot.mk Mcongr c) d \u2192\n            Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d) d\n        clear d\n        intro d rcd; apply Quot.sound; apply rcd)\n  have : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' := rfl\n  rw [\u2190 this, appendFun_comp_id, q.P.comp_map, q.P.comp_map, abs_map, abs_map, abs_map, abs_map,\n    h\u2080] ** case mk.mk.a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) this : IsPrecongr r' \u22a2 Mcongr x y ** refine' \u27e8r', this, rxy\u27e9 ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) \u22a2 IsPrecongr r' ** intro a b r'ab ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b \u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =     MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b) ** have h\u2080 :\n  appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P a) =\n    appendFun id (Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest q.P b) :=\n  by rw [appendFun_comp_id, comp_map, comp_map]; exact h _ _ r'ab ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) \u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =     MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b) ** have h\u2081 : \u2200 u v : q.P.M \u03b1, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v := by\n  intro u v cuv\n  apply Quot.sound\n  dsimp [hr']\n  rw [Quot.sound cuv]\n  apply h' ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v \u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =     MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b) ** let f : Quot r \u2192 Quot r' :=\n  Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n    (by\n      intro c\n      apply Quot.inductionOn\n        (motive := fun c =>\n          \u2200b, r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) c\n      clear c\n      intro c d\n      apply Quot.inductionOn\n        (motive := fun d => r (Quot.mk Mcongr c) d \u2192\n          Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d) d\n      clear d\n      intro d rcd; apply Quot.sound; apply rcd) ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v f : Quot r \u2192 Quot r' :=   Quot.lift (Quot.lift (Quot.mk r') h\u2081)     (_ : \u2200 (c b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) \u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =     MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b) ** have : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' := rfl ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v f : Quot r \u2192 Quot r' :=   Quot.lift (Quot.lift (Quot.mk r') h\u2081)     (_ : \u2200 (c b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) this : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' \u22a2 MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) a) =     MvQPF.abs ((TypeVec.id ::: Quot.mk r') <$$> M.dest (P F) b) ** rw [\u2190 this, appendFun_comp_id, q.P.comp_map, q.P.comp_map, abs_map, abs_map, abs_map, abs_map,\n  h\u2080] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b \u22a2 (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) ** rw [appendFun_comp_id, comp_map, comp_map] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b \u22a2 (TypeVec.id ::: Quot.mk r) <$$> (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r) <$$> (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) ** exact h _ _ r'ab ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) \u22a2 \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v ** intro u v cuv ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) u v : M (P F) \u03b1 cuv : Mcongr u v \u22a2 Quot.mk r' u = Quot.mk r' v ** apply Quot.sound ** case a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) u v : M (P F) \u03b1 cuv : Mcongr u v \u22a2 r' u v ** dsimp [hr'] ** case a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) u v : M (P F) \u03b1 cuv : Mcongr u v \u22a2 r (Quot.mk Mcongr u) (Quot.mk Mcongr v) ** rw [Quot.sound cuv] ** case a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) u v : M (P F) \u03b1 cuv : Mcongr u v \u22a2 r (Quot.mk Mcongr v) (Quot.mk Mcongr v) ** apply h' ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v \u22a2 \u2200 (a b : Cofix F \u03b1), r a b \u2192 Quot.lift (Quot.mk r') h\u2081 a = Quot.lift (Quot.mk r') h\u2081 b ** intro c ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : Cofix F \u03b1 \u22a2 \u2200 (b : Cofix F \u03b1), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b ** apply Quot.inductionOn\n  (motive := fun c =>\n    \u2200b, r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b) c ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : Cofix F \u03b1 \u22a2 \u2200 (a : M (P F) \u03b1) (b : Cofix F \u03b1),     r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b ** clear c ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v \u22a2 \u2200 (a : M (P F) \u03b1) (b : Cofix F \u03b1),     r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b ** intro c d ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : M (P F) \u03b1 d : Cofix F \u03b1 \u22a2 r (Quot.mk Mcongr c) d \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d ** apply Quot.inductionOn\n  (motive := fun d => r (Quot.mk Mcongr c) d \u2192\n    Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d) d ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : M (P F) \u03b1 d : Cofix F \u03b1 \u22a2 \u2200 (a : M (P F) \u03b1),     r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192       Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) ** clear d ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c : M (P F) \u03b1 \u22a2 \u2200 (a : M (P F) \u03b1),     r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192       Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) ** intro d rcd ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c d : M (P F) \u03b1 rcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d) \u22a2 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr d) ** apply Quot.sound ** case a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h' : \u2200 (x : Cofix F \u03b1), r x x h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y x y : M (P F) \u03b1 rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y) r' : M (P F) \u03b1 \u2192 M (P F) \u03b1 \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) hr' : r' = fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y) a b : M (P F) \u03b1 r'ab : r' a b h\u2080 :   (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a) =     (TypeVec.id ::: Quot.mk r \u2218 Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) b) h\u2081 : \u2200 (u v : M (P F) \u03b1), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v c d : M (P F) \u03b1 rcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d) \u22a2 r' c d ** apply rcd ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.bodd_add ** m n : \u2124 \u22a2 bodd (m + n) = xor (bodd m) (bodd n) ** cases' m with m m <;>\ncases' n with n n <;>\nsimp only [ofNat_eq_coe, ofNat_add_negSucc, negSucc_add_ofNat,\n           negSucc_add_negSucc, bodd_subNatNat] <;>\nsimp only [negSucc_coe, bodd_neg, bodd_coe, \u2190Nat.bodd_add, Bool.xor_comm, \u2190Nat.cast_add] ** case negSucc.negSucc m n : \u2115 \u22a2 Nat.bodd (Nat.succ (m + n) + 1) = Nat.bodd (m + 1 + (n + 1)) ** rw [\u2190Nat.succ_add, add_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_setToFun_of_dominated_convergence ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) \u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f)) ** have f_measurable : AEStronglyMeasurable f \u03bc :=\n  aestronglyMeasurable_of_tendsto_ae _ fs_measurable h_lim ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc \u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f)) ** have fs_int : \u2200 n, Integrable (fs n) \u03bc := fun n =>\n  bound_integrable.mono' (fs_measurable n) (h_bound _) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) \u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f)) ** have f_int : Integrable f \u03bc :=\n  \u27e8f_measurable,\n    hasFiniteIntegral_of_dominated_convergence bound_integrable.hasFiniteIntegral h_bound\n      h_lim\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f \u22a2 Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop     (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int))) ** refine' L1.tendsto_setToL1 hT _ _ _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f \u22a2 Tendsto (fun n => Integrable.toL1 (fs n) (_ : Integrable (fs n))) atTop (\ud835\udcdd (Integrable.toL1 f f_int)) ** rw [tendsto_iff_norm_sub_tendsto_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f \u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0) ** have lintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal <| \u222b\u207b a, ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) :=\n  (tendsto_toReal zero_ne_top).comp\n    (tendsto_lintegral_norm_of_dominated_convergence fs_measurable\n      bound_integrable.hasFiniteIntegral h_bound h_lim) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f lintegral_norm_tendsto_zero :   Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0) ** convert lintegral_norm_tendsto_zero with n ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f lintegral_norm_tendsto_zero :   Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 \u2016Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int\u2016 =     ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc) ** rw [L1.norm_def] ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f lintegral_norm_tendsto_zero :   Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc) =     ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc) ** congr 1 ** case h.e'_3.h.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f lintegral_norm_tendsto_zero :   Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc =     \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc ** refine' lintegral_congr_ae _ ** case h.e'_3.h.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f lintegral_norm_tendsto_zero :   Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a) =\u1d50[\u03bc] fun a =>     ENNReal.ofReal \u2016fs n a - f a\u2016 ** rw [\u2190 Integrable.toL1_sub] ** case h.e'_3.h.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f lintegral_norm_tendsto_zero :   Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) =\u1d50[\u03bc] fun a =>     ENNReal.ofReal \u2016fs n a - f a\u2016 ** refine' ((fs_int n).sub f_int).coeFn_toL1.mono fun x hx => _ ** case h.e'_3.h.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f lintegral_norm_tendsto_zero :   Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0) n : \u2115 x : \u03b1 hx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x \u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) x =     (fun a => ENNReal.ofReal \u2016fs n a - f a\u2016) x ** dsimp only ** case h.e'_3.h.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f lintegral_norm_tendsto_zero :   Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0) n : \u2115 x : \u03b1 hx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x \u22a2 \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x\u2016\u208a = ENNReal.ofReal \u2016fs n x - f x\u2016 ** rw [hx, ofReal_norm_eq_coe_nnnorm, Pi.sub_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f this :   Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop     (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int))) \u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f)) ** convert this with n ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f this :   Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop     (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int))) n : \u2115 \u22a2 setToFun \u03bc T hT (fs n) = \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n))) ** exact setToFun_eq hT (fs_int n) ** case h.e'_5.h.e'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C fs : \u2115 \u2192 \u03b1 \u2192 E f : \u03b1 \u2192 E bound : \u03b1 \u2192 \u211d fs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc bound_integrable : Integrable bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc fs_int : \u2200 (n : \u2115), Integrable (fs n) f_int : Integrable f this :   Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop     (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int))) \u22a2 setToFun \u03bc T hT f = \u2191(L1.setToL1 hT) (Integrable.toL1 f f_int) ** exact setToFun_eq hT f_int ** Qed",
        "informal": ""
    },
    {
        "formal": "List.length_erase_of_mem ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l : List \u03b1 h : a \u2208 l \u22a2 length (List.erase l a) = pred (length l) ** rw [erase_eq_eraseP] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l : List \u03b1 h : a \u2208 l \u22a2 length (eraseP (fun b => decide (a = b)) l) = pred (length l) ** exact length_eraseP_of_mem h (decide_eq_true rfl) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.powersetCard_nonempty ** \u03b1 : Type u_1 s\u271d t : Finset \u03b1 n : \u2115 s : Finset \u03b1 h : n \u2264 card s \u22a2 Finset.Nonempty (powersetCard n s) ** induction' s using Finset.induction_on with x s hx IH generalizing n ** case empty \u03b1 : Type u_1 s\u271d t : Finset \u03b1 n\u271d : \u2115 s : Finset \u03b1 h\u271d : n\u271d \u2264 card s n : \u2115 h : n \u2264 card \u2205 \u22a2 Finset.Nonempty (powersetCard n \u2205) ** rw [card_empty, le_zero_iff] at h ** case empty \u03b1 : Type u_1 s\u271d t : Finset \u03b1 n\u271d : \u2115 s : Finset \u03b1 h\u271d : n\u271d \u2264 card s n : \u2115 h : n = 0 \u22a2 Finset.Nonempty (powersetCard n \u2205) ** rw [h, powersetCard_zero] ** case empty \u03b1 : Type u_1 s\u271d t : Finset \u03b1 n\u271d : \u2115 s : Finset \u03b1 h\u271d : n\u271d \u2264 card s n : \u2115 h : n = 0 \u22a2 Finset.Nonempty {\u2205} ** exact Finset.singleton_nonempty _ ** case insert \u03b1 : Type u_1 s\u271d\u00b9 t : Finset \u03b1 n\u271d : \u2115 s\u271d : Finset \u03b1 h\u271d : n\u271d \u2264 card s\u271d x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s) n : \u2115 h : n \u2264 card (insert x s) \u22a2 Finset.Nonempty (powersetCard n (insert x s)) ** cases n ** case insert.zero \u03b1 : Type u_1 s\u271d\u00b9 t : Finset \u03b1 n : \u2115 s\u271d : Finset \u03b1 h\u271d : n \u2264 card s\u271d x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s) h : Nat.zero \u2264 card (insert x s) \u22a2 Finset.Nonempty (powersetCard Nat.zero (insert x s)) ** simp ** case insert.succ \u03b1 : Type u_1 s\u271d\u00b9 t : Finset \u03b1 n : \u2115 s\u271d : Finset \u03b1 h\u271d : n \u2264 card s\u271d x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s) n\u271d : \u2115 h : Nat.succ n\u271d \u2264 card (insert x s) \u22a2 Finset.Nonempty (powersetCard (Nat.succ n\u271d) (insert x s)) ** rw [card_insert_of_not_mem hx, Nat.succ_le_succ_iff] at h ** case insert.succ \u03b1 : Type u_1 s\u271d\u00b9 t : Finset \u03b1 n : \u2115 s\u271d : Finset \u03b1 h\u271d : n \u2264 card s\u271d x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s) n\u271d : \u2115 h : n\u271d \u2264 card s \u22a2 Finset.Nonempty (powersetCard (Nat.succ n\u271d) (insert x s)) ** rw [powersetCard_succ_insert hx] ** case insert.succ \u03b1 : Type u_1 s\u271d\u00b9 t : Finset \u03b1 n : \u2115 s\u271d : Finset \u03b1 h\u271d : n \u2264 card s\u271d x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s) n\u271d : \u2115 h : n\u271d \u2264 card s \u22a2 Finset.Nonempty (powersetCard (Nat.succ n\u271d) s \u222a image (insert x) (powersetCard n\u271d s)) ** refine' Nonempty.mono _ ((IH h).image (insert x)) ** case insert.succ \u03b1 : Type u_1 s\u271d\u00b9 t : Finset \u03b1 n : \u2115 s\u271d : Finset \u03b1 h\u271d : n \u2264 card s\u271d x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetCard n s) n\u271d : \u2115 h : n\u271d \u2264 card s \u22a2 image (insert x) (powersetCard n\u271d s) \u2286 powersetCard (Nat.succ n\u271d) s \u222a image (insert x) (powersetCard n\u271d s) ** exact subset_union_right _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.index_pos ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) \u22a2 0 < index (\u2191K) V ** unfold index ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) \u22a2 0 < sInf (Finset.card '' {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}) ** rw [Nat.sInf_def, Nat.find_pos, mem_image] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) \u22a2 \u00ac\u2203 x, x \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} \u2227 Finset.card x = 0 ** rintro \u27e8t, h1t, h2t\u27e9 ** case intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) t : Finset G h1t : t \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} h2t : Finset.card t = 0 \u22a2 False ** rw [Finset.card_eq_zero] at h2t ** case intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) t : Finset G h1t : t \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} h2t : t = \u2205 \u22a2 False ** subst h2t ** case intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) h1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} \u22a2 False ** obtain \u27e8g, hg\u27e9 := K.interior_nonempty ** case intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) h1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} g : G hg : g \u2208 interior \u2191K \u22a2 False ** show g \u2208 (\u2205 : Set G) ** case intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) h1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} g : G hg : g \u2208 interior \u2191K \u22a2 g \u2208 \u2205 ** convert h1t (interior_subset hg) ** case h.e'_5 G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) h1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} g : G hg : g \u2208 interior \u2191K \u22a2 \u2205 = \u22c3 g \u2208 \u2205, (fun h => g * h) \u207b\u00b9' V ** symm ** case h.e'_5 G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) h1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} g : G hg : g \u2208 interior \u2191K \u22a2 \u22c3 g \u2208 \u2205, (fun h => g * h) \u207b\u00b9' V = \u2205 ** simp only [Finset.not_mem_empty, iUnion_of_empty, iUnion_empty] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K : PositiveCompacts G V : Set G hV : Set.Nonempty (interior V) \u22a2 \u2203 n, n \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} ** exact index_defined K.isCompact hV ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.card_Iio_eq_card_Iic_sub_one ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrderBot \u03b1 a : \u03b1 \u22a2 card (Iio a) = card (Iic a) - 1 ** rw [Iic_eq_cons_Iio, card_cons, add_tsub_cancel_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.exists_code ** f : \u2115 \u2192. \u2115 h : Partrec f \u22a2 \u2203 c, eval c = f ** induction h ** case zero f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = pure 0  case succ f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191Nat.succ  case left f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1  case right f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2  case pair f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n  case comp f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d  case prec f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (f\u271d a)           (fun y IH => do             let i \u2190 IH             g\u271d (Nat.pair a (Nat.pair y i)))           n  case rfind f f\u271d : \u2115 \u2192. \u2115 a\u271d : Partrec f\u271d a_ih\u271d : \u2203 c, eval c = f\u271d \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n) ** case zero => exact \u27e8zero, rfl\u27e9 ** case succ f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191Nat.succ  case left f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1  case right f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2  case pair f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n  case comp f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d  case prec f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (f\u271d a)           (fun y IH => do             let i \u2190 IH             g\u271d (Nat.pair a (Nat.pair y i)))           n  case rfind f f\u271d : \u2115 \u2192. \u2115 a\u271d : Partrec f\u271d a_ih\u271d : \u2203 c, eval c = f\u271d \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n) ** case succ => exact \u27e8succ, rfl\u27e9 ** case left f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1  case right f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2  case pair f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n  case comp f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d  case prec f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (f\u271d a)           (fun y IH => do             let i \u2190 IH             g\u271d (Nat.pair a (Nat.pair y i)))           n  case rfind f f\u271d : \u2115 \u2192. \u2115 a\u271d : Partrec f\u271d a_ih\u271d : \u2203 c, eval c = f\u271d \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n) ** case left => exact \u27e8left, rfl\u27e9 ** case right f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2  case pair f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n  case comp f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d  case prec f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (f\u271d a)           (fun y IH => do             let i \u2190 IH             g\u271d (Nat.pair a (Nat.pair y i)))           n  case rfind f f\u271d : \u2115 \u2192. \u2115 a\u271d : Partrec f\u271d a_ih\u271d : \u2203 c, eval c = f\u271d \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n) ** case right => exact \u27e8right, rfl\u27e9 ** case pair f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n  case comp f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d  case prec f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (f\u271d a)           (fun y IH => do             let i \u2190 IH             g\u271d (Nat.pair a (Nat.pair y i)))           n  case rfind f f\u271d : \u2115 \u2192. \u2115 a\u271d : Partrec f\u271d a_ih\u271d : \u2203 c, eval c = f\u271d \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n) ** case pair f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8pair cf cg, rfl\u27e9 ** case comp f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d  case prec f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (f\u271d a)           (fun y IH => do             let i \u2190 IH             g\u271d (Nat.pair a (Nat.pair y i)))           n  case rfind f f\u271d : \u2115 \u2192. \u2115 a\u271d : Partrec f\u271d a_ih\u271d : \u2203 c, eval c = f\u271d \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n) ** case comp f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8comp cf cg, rfl\u27e9 ** case prec f f\u271d g\u271d : \u2115 \u2192. \u2115 a\u271d\u00b9 : Partrec f\u271d a\u271d : Partrec g\u271d a_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d a_ih\u271d : \u2203 c, eval c = g\u271d \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (f\u271d a)           (fun y IH => do             let i \u2190 IH             g\u271d (Nat.pair a (Nat.pair y i)))           n  case rfind f f\u271d : \u2115 \u2192. \u2115 a\u271d : Partrec f\u271d a_ih\u271d : \u2203 c, eval c = f\u271d \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n) ** case prec f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8prec cf cg, rfl\u27e9 ** case rfind f f\u271d : \u2115 \u2192. \u2115 a\u271d : Partrec f\u271d a_ih\u271d : \u2203 c, eval c = f\u271d \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n) ** case rfind f pf hf =>\n  rcases hf with \u27e8cf, rfl\u27e9\n  refine' \u27e8comp (rfind' cf) (pair Code.id zero), _\u27e9\n  simp [eval, Seq.seq, pure, PFun.pure, Part.map_id'] ** f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = pure 0 ** exact \u27e8zero, rfl\u27e9 ** f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191Nat.succ ** exact \u27e8succ, rfl\u27e9 ** f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).1 ** exact \u27e8left, rfl\u27e9 ** f : \u2115 \u2192. \u2115 \u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).2 ** exact \u27e8right, rfl\u27e9 ** f\u271d f g : \u2115 \u2192. \u2115 pf : Partrec f pg : Partrec g hf : \u2203 c, eval c = f hg : \u2203 c, eval c = g \u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f n) fun x => g n ** rcases hf with \u27e8cf, rfl\u27e9 ** case intro f g : \u2115 \u2192. \u2115 pg : Partrec g hg : \u2203 c, eval c = g cf : Code pf : Partrec (eval cf) \u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf n) fun x => g n ** rcases hg with \u27e8cg, rfl\u27e9 ** case intro.intro f : \u2115 \u2192. \u2115 cf : Code pf : Partrec (eval cf) cg : Code pg : Partrec (eval cg) \u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf n) fun x => eval cg n ** exact \u27e8pair cf cg, rfl\u27e9 ** f\u271d f g : \u2115 \u2192. \u2115 pf : Partrec f pg : Partrec g hf : \u2203 c, eval c = f hg : \u2203 c, eval c = g \u22a2 \u2203 c, eval c = fun n => g n >>= f ** rcases hf with \u27e8cf, rfl\u27e9 ** case intro f g : \u2115 \u2192. \u2115 pg : Partrec g hg : \u2203 c, eval c = g cf : Code pf : Partrec (eval cf) \u22a2 \u2203 c, eval c = fun n => g n >>= eval cf ** rcases hg with \u27e8cg, rfl\u27e9 ** case intro.intro f : \u2115 \u2192. \u2115 cf : Code pf : Partrec (eval cf) cg : Code pg : Partrec (eval cg) \u22a2 \u2203 c, eval c = fun n => eval cg n >>= eval cf ** exact \u27e8comp cf cg, rfl\u27e9 ** f\u271d f g : \u2115 \u2192. \u2115 pf : Partrec f pg : Partrec g hf : \u2203 c, eval c = f hg : \u2203 c, eval c = g \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (f a)           (fun y IH => do             let i \u2190 IH             g (Nat.pair a (Nat.pair y i)))           n ** rcases hf with \u27e8cf, rfl\u27e9 ** case intro f g : \u2115 \u2192. \u2115 pg : Partrec g hg : \u2203 c, eval c = g cf : Code pf : Partrec (eval cf) \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (eval cf a)           (fun y IH => do             let i \u2190 IH             g (Nat.pair a (Nat.pair y i)))           n ** rcases hg with \u27e8cg, rfl\u27e9 ** case intro.intro f : \u2115 \u2192. \u2115 cf : Code pf : Partrec (eval cf) cg : Code pg : Partrec (eval cg) \u22a2 \u2203 c,     eval c =       unpaired fun a n =>         Nat.rec (eval cf a)           (fun y IH => do             let i \u2190 IH             eval cg (Nat.pair a (Nat.pair y i)))           n ** exact \u27e8prec cf cg, rfl\u27e9 ** f\u271d f : \u2115 \u2192. \u2115 pf : Partrec f hf : \u2203 c, eval c = f \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a n) ** rcases hf with \u27e8cf, rfl\u27e9 ** case intro f : \u2115 \u2192. \u2115 cf : Code pf : Partrec (eval cf) \u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a n) ** refine' \u27e8comp (rfind' cf) (pair Code.id zero), _\u27e9 ** case intro f : \u2115 \u2192. \u2115 cf : Code pf : Partrec (eval cf) \u22a2 eval (comp (rfind' cf) (pair Code.id zero)) = fun a =>     Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a n) ** simp [eval, Seq.seq, pure, PFun.pure, Part.map_id'] ** f : \u2115 \u2192. \u2115 h : \u2203 c, eval c = f \u22a2 Partrec f ** rcases h with \u27e8c, rfl\u27e9 ** case intro c : Code \u22a2 Partrec (eval c) ** induction c ** case intro.zero  \u22a2 Partrec (eval zero)  case intro.succ  \u22a2 Partrec (eval succ)  case intro.left  \u22a2 Partrec (eval left)  case intro.right  \u22a2 Partrec (eval right)  case intro.pair a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))  case intro.comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))  case intro.prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))  case intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d)) ** case zero => exact Nat.Partrec.zero ** case intro.succ  \u22a2 Partrec (eval succ)  case intro.left  \u22a2 Partrec (eval left)  case intro.right  \u22a2 Partrec (eval right)  case intro.pair a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))  case intro.comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))  case intro.prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))  case intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d)) ** case succ => exact Nat.Partrec.succ ** case intro.left  \u22a2 Partrec (eval left)  case intro.right  \u22a2 Partrec (eval right)  case intro.pair a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))  case intro.comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))  case intro.prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))  case intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d)) ** case left => exact Nat.Partrec.left ** case intro.right  \u22a2 Partrec (eval right)  case intro.pair a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))  case intro.comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))  case intro.prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))  case intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d)) ** case right => exact Nat.Partrec.right ** case intro.pair a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))  case intro.comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))  case intro.prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))  case intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d)) ** case pair cf cg pf pg => exact pf.pair pg ** case intro.comp a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))  case intro.prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))  case intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d)) ** case comp cf cg pf pg => exact pf.comp pg ** case intro.prec a\u271d\u00b9 a\u271d : Code a_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9) a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))  case intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d)) ** case prec cf cg pf pg => exact pf.prec pg ** case intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d)) ** case rfind' cf pf => exact pf.rfind' ** \u22a2 Partrec (eval zero) ** exact Nat.Partrec.zero ** \u22a2 Partrec (eval succ) ** exact Nat.Partrec.succ ** \u22a2 Partrec (eval left) ** exact Nat.Partrec.left ** \u22a2 Partrec (eval right) ** exact Nat.Partrec.right ** cf cg : Code pf : Partrec (eval cf) pg : Partrec (eval cg) \u22a2 Partrec (eval (pair cf cg)) ** exact pf.pair pg ** cf cg : Code pf : Partrec (eval cf) pg : Partrec (eval cg) \u22a2 Partrec (eval (comp cf cg)) ** exact pf.comp pg ** cf cg : Code pf : Partrec (eval cf) pg : Partrec (eval cg) \u22a2 Partrec (eval (prec cf cg)) ** exact pf.prec pg ** cf : Code pf : Partrec (eval cf) \u22a2 Partrec (eval (rfind' cf)) ** exact pf.rfind' ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Icc_eq_cons_Ioc ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 h : a \u2264 b \u22a2 Icc a b = cons a (Ioc a b) (_ : \u00aca \u2208 Ioc a b) ** classical rw [cons_eq_insert, Ioc_insert_left h] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 h : a \u2264 b \u22a2 Icc a b = cons a (Ioc a b) (_ : \u00aca \u2208 Ioc a b) ** rw [cons_eq_insert, Ioc_insert_left h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.measurableSet_liminf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 s : \u2115 \u2192 Set \u03b1 hs : \u2200 (n : \u2115), MeasurableSet (s n) \u22a2 MeasurableSet (liminf s atTop) ** simpa only [\u2190 bliminf_true] using measurableSet_bliminf fun n _ => hs n ** Qed",
        "informal": ""
    },
    {
        "formal": "List.isPrefixOf_cons\u2082_self ** \u03b1 : Type u_1 as bs : List \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : LawfulBEq \u03b1 a : \u03b1 \u22a2 isPrefixOf (a :: as) (a :: bs) = isPrefixOf as bs ** simp [isPrefixOf_cons\u2082] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.sigma_insert ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2 : \u03b9 \u2192 Type u_4 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x : (i : \u03b9) \u00d7 \u03b1 i i j : \u03b9 a\u271d : \u03b1 i a : (i : \u03b9) \u2192 \u03b1 i \u22a2 (Set.Sigma s fun i => insert (a i) (t i)) = (fun i => { fst := i, snd := a i }) '' s \u222a Set.Sigma s t ** simp_rw [insert_eq, sigma_union, sigma_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "Monotone.locallyIntegrable ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u00b9 : MeasurableSpace X inst\u271d\u00b9\u2070 : TopologicalSpace X inst\u271d\u2079 : MeasurableSpace Y inst\u271d\u2078 : TopologicalSpace Y inst\u271d\u2077 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u2076 : BorelSpace X inst\u271d\u2075 : ConditionallyCompleteLinearOrder X inst\u271d\u2074 : ConditionallyCompleteLinearOrder E inst\u271d\u00b3 : OrderTopology X inst\u271d\u00b2 : OrderTopology E inst\u271d\u00b9 : SecondCountableTopology E inst\u271d : IsLocallyFiniteMeasure \u03bc hmono : Monotone f \u22a2 LocallyIntegrable f ** intro x ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u00b9 : MeasurableSpace X inst\u271d\u00b9\u2070 : TopologicalSpace X inst\u271d\u2079 : MeasurableSpace Y inst\u271d\u2078 : TopologicalSpace Y inst\u271d\u2077 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u2076 : BorelSpace X inst\u271d\u2075 : ConditionallyCompleteLinearOrder X inst\u271d\u2074 : ConditionallyCompleteLinearOrder E inst\u271d\u00b3 : OrderTopology X inst\u271d\u00b2 : OrderTopology E inst\u271d\u00b9 : SecondCountableTopology E inst\u271d : IsLocallyFiniteMeasure \u03bc hmono : Monotone f x : X \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** rcases \u03bc.finiteAt_nhds x with \u27e8U, hU, h'U\u27e9 ** case intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u00b9 : MeasurableSpace X inst\u271d\u00b9\u2070 : TopologicalSpace X inst\u271d\u2079 : MeasurableSpace Y inst\u271d\u2078 : TopologicalSpace Y inst\u271d\u2077 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u2076 : BorelSpace X inst\u271d\u2075 : ConditionallyCompleteLinearOrder X inst\u271d\u2074 : ConditionallyCompleteLinearOrder E inst\u271d\u00b3 : OrderTopology X inst\u271d\u00b2 : OrderTopology E inst\u271d\u00b9 : SecondCountableTopology E inst\u271d : IsLocallyFiniteMeasure \u03bc hmono : Monotone f x : X U : Set X hU : U \u2208 \ud835\udcdd x h'U : \u2191\u2191\u03bc U < \u22a4 \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** obtain \u27e8a, b, xab, hab, abU\u27e9 : \u2203 a b : X, x \u2208 Icc a b \u2227 Icc a b \u2208 \ud835\udcdd x \u2227 Icc a b \u2286 U ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u00b9 : MeasurableSpace X inst\u271d\u00b9\u2070 : TopologicalSpace X inst\u271d\u2079 : MeasurableSpace Y inst\u271d\u2078 : TopologicalSpace Y inst\u271d\u2077 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u2076 : BorelSpace X inst\u271d\u2075 : ConditionallyCompleteLinearOrder X inst\u271d\u2074 : ConditionallyCompleteLinearOrder E inst\u271d\u00b3 : OrderTopology X inst\u271d\u00b2 : OrderTopology E inst\u271d\u00b9 : SecondCountableTopology E inst\u271d : IsLocallyFiniteMeasure \u03bc hmono : Monotone f x : X U : Set X hU : U \u2208 \ud835\udcdd x h'U : \u2191\u2191\u03bc U < \u22a4 \u22a2 \u2203 a b, x \u2208 Icc a b \u2227 Icc a b \u2208 \ud835\udcdd x \u2227 Icc a b \u2286 U  case intro.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u00b9 : MeasurableSpace X inst\u271d\u00b9\u2070 : TopologicalSpace X inst\u271d\u2079 : MeasurableSpace Y inst\u271d\u2078 : TopologicalSpace Y inst\u271d\u2077 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u2076 : BorelSpace X inst\u271d\u2075 : ConditionallyCompleteLinearOrder X inst\u271d\u2074 : ConditionallyCompleteLinearOrder E inst\u271d\u00b3 : OrderTopology X inst\u271d\u00b2 : OrderTopology E inst\u271d\u00b9 : SecondCountableTopology E inst\u271d : IsLocallyFiniteMeasure \u03bc hmono : Monotone f x : X U : Set X hU : U \u2208 \ud835\udcdd x h'U : \u2191\u2191\u03bc U < \u22a4 a b : X xab : x \u2208 Icc a b hab : Icc a b \u2208 \ud835\udcdd x abU : Icc a b \u2286 U \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** exact exists_Icc_mem_subset_of_mem_nhds hU ** case intro.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u00b9 : MeasurableSpace X inst\u271d\u00b9\u2070 : TopologicalSpace X inst\u271d\u2079 : MeasurableSpace Y inst\u271d\u2078 : TopologicalSpace Y inst\u271d\u2077 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u2076 : BorelSpace X inst\u271d\u2075 : ConditionallyCompleteLinearOrder X inst\u271d\u2074 : ConditionallyCompleteLinearOrder E inst\u271d\u00b3 : OrderTopology X inst\u271d\u00b2 : OrderTopology E inst\u271d\u00b9 : SecondCountableTopology E inst\u271d : IsLocallyFiniteMeasure \u03bc hmono : Monotone f x : X U : Set X hU : U \u2208 \ud835\udcdd x h'U : \u2191\u2191\u03bc U < \u22a4 a b : X xab : x \u2208 Icc a b hab : Icc a b \u2208 \ud835\udcdd x abU : Icc a b \u2286 U \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** have ab : a \u2264 b := xab.1.trans xab.2 ** case intro.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u00b9 : MeasurableSpace X inst\u271d\u00b9\u2070 : TopologicalSpace X inst\u271d\u2079 : MeasurableSpace Y inst\u271d\u2078 : TopologicalSpace Y inst\u271d\u2077 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u2076 : BorelSpace X inst\u271d\u2075 : ConditionallyCompleteLinearOrder X inst\u271d\u2074 : ConditionallyCompleteLinearOrder E inst\u271d\u00b3 : OrderTopology X inst\u271d\u00b2 : OrderTopology E inst\u271d\u00b9 : SecondCountableTopology E inst\u271d : IsLocallyFiniteMeasure \u03bc hmono : Monotone f x : X U : Set X hU : U \u2208 \ud835\udcdd x h'U : \u2191\u2191\u03bc U < \u22a4 a b : X xab : x \u2208 Icc a b hab : Icc a b \u2208 \ud835\udcdd x abU : Icc a b \u2286 U ab : a \u2264 b \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** refine' \u27e8Icc a b, hab, _\u27e9 ** case intro.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u00b9\u00b9 : MeasurableSpace X inst\u271d\u00b9\u2070 : TopologicalSpace X inst\u271d\u2079 : MeasurableSpace Y inst\u271d\u2078 : TopologicalSpace Y inst\u271d\u2077 : NormedAddCommGroup E f g : X \u2192 E \u03bc : Measure X s : Set X inst\u271d\u2076 : BorelSpace X inst\u271d\u2075 : ConditionallyCompleteLinearOrder X inst\u271d\u2074 : ConditionallyCompleteLinearOrder E inst\u271d\u00b3 : OrderTopology X inst\u271d\u00b2 : OrderTopology E inst\u271d\u00b9 : SecondCountableTopology E inst\u271d : IsLocallyFiniteMeasure \u03bc hmono : Monotone f x : X U : Set X hU : U \u2208 \ud835\udcdd x h'U : \u2191\u2191\u03bc U < \u22a4 a b : X xab : x \u2208 Icc a b hab : Icc a b \u2208 \ud835\udcdd x abU : Icc a b \u2286 U ab : a \u2264 b \u22a2 IntegrableOn f (Icc a b) ** exact\n  (hmono.monotoneOn _).integrableOn_of_measure_ne_top (isLeast_Icc ab) (isGreatest_Icc ab)\n    ((measure_mono abU).trans_lt h'U).ne measurableSet_Icc ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Finite.finite_subsets ** \u03b1\u271d : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b1 : Type u a : Set \u03b1 h : Set.Finite a s : Set \u03b1 \u22a2 s \u2208 Finset.map Finset.coeEmb.toEmbedding (Finset.powerset (Finite.toFinset h)) \u2194 s \u2208 {b | b \u2286 a} ** simpa [\u2190 @exists_finite_iff_finset \u03b1 fun t => t \u2286 a \u2227 t = s, Finite.subset_toFinset, \u2190\n  and_assoc, Finset.coeEmb] using h.subset ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Finite.injOn_of_encard_image_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t\u271d s : Set \u03b1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hs : Set.Finite s h : encard (f '' s) = encard s \u22a2 InjOn f s ** obtain (h' | hne) := isEmpty_or_nonempty \u03b1 ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t\u271d s : Set \u03b1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hs : Set.Finite s h : encard (f '' s) = encard s hne : Nonempty \u03b1 \u22a2 InjOn f s ** rw [\u2190(f.invFunOn_injOn_image s).encard_image] at h ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t\u271d s : Set \u03b1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hs : Set.Finite s hne : Nonempty \u03b1 h : encard (Function.invFunOn f s '' (f '' s)) = encard s \u22a2 InjOn f s ** rw [injOn_iff_invFunOn_image_image_eq_self] ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t\u271d s : Set \u03b1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hs : Set.Finite s hne : Nonempty \u03b1 h : encard (Function.invFunOn f s '' (f '' s)) = encard s \u22a2 Function.invFunOn f s '' (f '' s) = s ** exact hs.eq_of_subset_of_encard_le (f.invFunOn_image_image_subset s) h.symm.le ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t\u271d s : Set \u03b1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hs : Set.Finite s h : encard (f '' s) = encard s h' : IsEmpty \u03b1 \u22a2 InjOn f s ** rw [s.eq_empty_of_isEmpty] ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t\u271d s : Set \u03b1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hs : Set.Finite s h : encard (f '' s) = encard s h' : IsEmpty \u03b1 \u22a2 InjOn f \u2205 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image\u2082_insert_right ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 \u03b6 : Type u_11 \u03b6' : Type u_12 \u03bd : Type u_13 inst\u271d\u2078 : DecidableEq \u03b1' inst\u271d\u2077 : DecidableEq \u03b2' inst\u271d\u2076 : DecidableEq \u03b3 inst\u271d\u2075 : DecidableEq \u03b3' inst\u271d\u2074 : DecidableEq \u03b4 inst\u271d\u00b3 : DecidableEq \u03b4' inst\u271d\u00b2 : DecidableEq \u03b5 inst\u271d\u00b9 : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 inst\u271d : DecidableEq \u03b2 \u22a2 \u2191(image\u2082 f s (insert b t)) = \u2191(image (fun a => f a b) s \u222a image\u2082 f s t) ** push_cast ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 \u03b6 : Type u_11 \u03b6' : Type u_12 \u03bd : Type u_13 inst\u271d\u2078 : DecidableEq \u03b1' inst\u271d\u2077 : DecidableEq \u03b2' inst\u271d\u2076 : DecidableEq \u03b3 inst\u271d\u2075 : DecidableEq \u03b3' inst\u271d\u2074 : DecidableEq \u03b4 inst\u271d\u00b3 : DecidableEq \u03b4' inst\u271d\u00b2 : DecidableEq \u03b5 inst\u271d\u00b9 : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 inst\u271d : DecidableEq \u03b2 \u22a2 image2 f (\u2191s) (insert b \u2191t) = (fun a => f a b) '' \u2191s \u222a image2 f \u2191s \u2191t ** exact image2_insert_right ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_trim ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hm : m \u2264 m0 f : \u03b1 \u2192 E hf : StronglyMeasurable f \u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd ** by_cases h0 : p = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hm : m \u2264 m0 f : \u03b1 \u2192 E hf : StronglyMeasurable f h0 : \u00acp = 0 \u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd ** by_cases h_top : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hm : m \u2264 m0 f : \u03b1 \u2192 E hf : StronglyMeasurable f h0 : \u00acp = 0 h_top : \u00acp = \u22a4 \u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd ** simpa only [snorm_eq_snorm' h0 h_top] using snorm'_trim hm hf ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hm : m \u2264 m0 f : \u03b1 \u2192 E hf : StronglyMeasurable f h0 : p = 0 \u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd ** simp [h0] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hm : m \u2264 m0 f : \u03b1 \u2192 E hf : StronglyMeasurable f h0 : \u00acp = 0 h_top : p = \u22a4 \u22a2 snorm f p (Measure.trim \u03bd hm) = snorm f p \u03bd ** simpa only [h_top, snorm_exponent_top] using snormEssSup_trim hm hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l\u271d l' l : Filter \u03b1 inst\u271d : IsMeasurablyGenerated l hfm : StronglyMeasurableAtFilter f l h\u03bc : FiniteAtFilter \u03bc l hf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f) \u22a2 IntegrableAtFilter f l ** obtain \u27e8C, hC\u27e9 : \u2203 C, \u2200\u1da0 s in l.smallSets, \u2200 x \u2208 s, \u2016f x\u2016 \u2264 C :=\n  hf.imp fun C hC => eventually_smallSets.2 \u27e8_, hC, fun t => id\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l\u271d l' l : Filter \u03b1 inst\u271d : IsMeasurablyGenerated l hfm : StronglyMeasurableAtFilter f l h\u03bc : FiniteAtFilter \u03bc l hf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f) C : \u211d hC : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C \u22a2 IntegrableAtFilter f l ** rcases (hfm.eventually.and (h\u03bc.eventually.and hC)).exists_measurable_mem_of_smallSets with\n  \u27e8s, hsl, hsm, hfm, h\u03bc, hC\u27e9 ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l\u271d l' l : Filter \u03b1 inst\u271d : IsMeasurablyGenerated l hfm\u271d : StronglyMeasurableAtFilter f l h\u03bc\u271d : FiniteAtFilter \u03bc l hf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f) C : \u211d hC\u271d : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C s : Set \u03b1 hsl : s \u2208 l hsm : MeasurableSet s hfm : AEStronglyMeasurable f (restrict \u03bc s) h\u03bc : \u2191\u2191\u03bc s < \u22a4 hC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C \u22a2 IntegrableAtFilter f l ** refine' \u27e8s, hsl, \u27e8hfm, hasFiniteIntegral_restrict_of_bounded h\u03bc (C := C) _\u27e9\u27e9 ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l\u271d l' l : Filter \u03b1 inst\u271d : IsMeasurablyGenerated l hfm\u271d : StronglyMeasurableAtFilter f l h\u03bc\u271d : FiniteAtFilter \u03bc l hf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f) C : \u211d hC\u271d : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C s : Set \u03b1 hsl : s \u2208 l hsm : MeasurableSet s hfm : AEStronglyMeasurable f (restrict \u03bc s) h\u03bc : \u2191\u2191\u03bc s < \u22a4 hC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C \u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc s, \u2016f x\u2016 \u2264 C ** rw [ae_restrict_eq hsm, eventually_inf_principal] ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l\u271d l' l : Filter \u03b1 inst\u271d : IsMeasurablyGenerated l hfm\u271d : StronglyMeasurableAtFilter f l h\u03bc\u271d : FiniteAtFilter \u03bc l hf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f) C : \u211d hC\u271d : \u2200\u1da0 (s : Set \u03b1) in smallSets l, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C s : Set \u03b1 hsl : s \u2208 l hsm : MeasurableSet s hfm : AEStronglyMeasurable f (restrict \u03bc s) h\u03bc : \u2191\u2191\u03bc s < \u22a4 hC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2016f x\u2016 \u2264 C \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 \u2016f x\u2016 \u2264 C ** exact eventually_of_forall hC ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.UniformIntegrable.spec' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) hfu : UniformIntegrable f p \u03bc \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8-, hfu, M, hM\u27e9 := hfu ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := hfu h\u03b5 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8C, hC\u27e9 : \u2203 C : \u211d\u22650, \u2200 i, \u03bc { x | C \u2264 \u2016f i x\u2016\u208a } \u2264 ENNReal.ofReal \u03b4 := by\n  by_contra hcon; push_neg at hcon\n  choose \u2110 h\u2110 using hcon\n  lift \u03b4 to \u211d\u22650 using h\u03b4pos.le\n  have : \u2200 C : \u211d\u22650, C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 snorm (f (\u2110 C)) p \u03bc := by\n    intro C\n    calc\n      C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 C \u2022 \u03bc { x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a } ^ (1 / p.toReal) := by\n        rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]\n        simp_rw [ENNReal.ofReal_coe_nnreal] at h\u2110\n        refine' mul_le_mul' le_rfl\n          (ENNReal.rpow_le_rpow (h\u2110 C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))\n      _ \u2264 snorm ({ x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a }.indicator (f (\u2110 C))) p \u03bc := by\n        refine' snorm_indicator_ge_of_bdd_below hp hp' _\n          (measurableSet_le measurable_const (hf _).nnnorm.measurable)\n          (eventually_of_forall fun x hx => _)\n        rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]\n      _ \u2264 snorm (f (\u2110 C)) p \u03bc := snorm_indicator_le _\n  specialize this (2 * max M 1 * HPow.hPow \u03b4\u207b\u00b9 (1 / p.toReal))\n  rw [ENNReal.coe_rpow_of_nonneg _ (one_div_nonneg.2 ENNReal.toReal_nonneg), \u2190 ENNReal.coe_smul,\n    smul_eq_mul, mul_assoc, NNReal.inv_rpow,\n    inv_mul_cancel (NNReal.rpow_pos (NNReal.coe_pos.1 h\u03b4pos)).ne.symm, mul_one, ENNReal.coe_mul,\n    \u2190 NNReal.inv_rpow] at this\n  refine' (lt_of_le_of_lt (le_trans\n    (hM <| \u2110 <| 2 * max M 1 * HPow.hPow \u03b4\u207b\u00b9 (1 / p.toReal)) (le_max_left (M : \u211d\u22650\u221e) 1))\n      (lt_of_lt_of_le _ this)).ne rfl\n  rw [\u2190 ENNReal.coe_one, \u2190 ENNReal.coe_max, \u2190 ENNReal.coe_mul, ENNReal.coe_lt_coe]\n  exact lt_two_mul_self (lt_max_of_lt_right one_pos) ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** exact \u27e8C, fun i => h\u03b4 i _ (measurableSet_le measurable_const (hf i).nnnorm.measurable) (hC i)\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 C, \u2200 (i : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 ** by_contra hcon ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hcon : \u00ac\u2203 C, \u2200 (i : \u03b9), \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a} \u2264 ENNReal.ofReal \u03b4 \u22a2 False ** push_neg at hcon ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hcon : \u2200 (C : \u211d\u22650), \u2203 i, ENNReal.ofReal \u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f i x\u2016\u208a} \u22a2 False ** choose \u2110 h\u2110 using hcon ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u2110 : \u211d\u22650 \u2192 \u03b9 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} \u22a2 False ** lift \u03b4 to \u211d\u22650 using h\u03b4pos.le ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} \u22a2 False ** have : \u2200 C : \u211d\u22650, C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 snorm (f (\u2110 C)) p \u03bc := by\n  intro C\n  calc\n    C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 C \u2022 \u03bc { x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a } ^ (1 / p.toReal) := by\n      rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]\n      simp_rw [ENNReal.ofReal_coe_nnreal] at h\u2110\n      refine' mul_le_mul' le_rfl\n        (ENNReal.rpow_le_rpow (h\u2110 C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))\n    _ \u2264 snorm ({ x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a }.indicator (f (\u2110 C))) p \u03bc := by\n      refine' snorm_indicator_ge_of_bdd_below hp hp' _\n        (measurableSet_le measurable_const (hf _).nnnorm.measurable)\n        (eventually_of_forall fun x hx => _)\n      rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]\n    _ \u2264 snorm (f (\u2110 C)) p \u03bc := snorm_indicator_le _ ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} this : \u2200 (C : \u211d\u22650), C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 snorm (f (\u2110 C)) p \u03bc \u22a2 False ** specialize this (2 * max M 1 * HPow.hPow \u03b4\u207b\u00b9 (1 / p.toReal)) ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} this :   (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)) \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264     snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc \u22a2 False ** rw [ENNReal.coe_rpow_of_nonneg _ (one_div_nonneg.2 ENNReal.toReal_nonneg), \u2190 ENNReal.coe_smul,\n  smul_eq_mul, mul_assoc, NNReal.inv_rpow,\n  inv_mul_cancel (NNReal.rpow_pos (NNReal.coe_pos.1 h\u03b4pos)).ne.symm, mul_one, ENNReal.coe_mul,\n  \u2190 NNReal.inv_rpow] at this ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} this : \u21912 * \u2191(max M 1) \u2264 snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc \u22a2 False ** refine' (lt_of_le_of_lt (le_trans\n  (hM <| \u2110 <| 2 * max M 1 * HPow.hPow \u03b4\u207b\u00b9 (1 / p.toReal)) (le_max_left (M : \u211d\u22650\u221e) 1))\n    (lt_of_lt_of_le _ this)).ne rfl ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} this : \u21912 * \u2191(max M 1) \u2264 snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc \u22a2 max (\u2191M) 1 < \u21912 * \u2191(max M 1) ** rw [\u2190 ENNReal.coe_one, \u2190 ENNReal.coe_max, \u2190 ENNReal.coe_mul, ENNReal.coe_lt_coe] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} this : \u21912 * \u2191(max M 1) \u2264 snorm (f (\u2110 (2 * max M 1 * \u03b4\u207b\u00b9 ^ (1 / ENNReal.toReal p)))) p \u03bc \u22a2 max M 1 < 2 * max M 1 ** exact lt_two_mul_self (lt_max_of_lt_right one_pos) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} \u22a2 \u2200 (C : \u211d\u22650), C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 snorm (f (\u2110 C)) p \u03bc ** intro C ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} C : \u211d\u22650 \u22a2 C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 snorm (f (\u2110 C)) p \u03bc ** calc\n  C \u2022 (\u03b4 : \u211d\u22650\u221e) ^ (1 / p.toReal) \u2264 C \u2022 \u03bc { x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a } ^ (1 / p.toReal) := by\n    rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]\n    simp_rw [ENNReal.ofReal_coe_nnreal] at h\u2110\n    refine' mul_le_mul' le_rfl\n      (ENNReal.rpow_le_rpow (h\u2110 C).le (one_div_nonneg.2 ENNReal.toReal_nonneg))\n  _ \u2264 snorm ({ x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a }.indicator (f (\u2110 C))) p \u03bc := by\n    refine' snorm_indicator_ge_of_bdd_below hp hp' _\n      (measurableSet_le measurable_const (hf _).nnnorm.measurable)\n      (eventually_of_forall fun x hx => _)\n    rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]\n  _ \u2264 snorm (f (\u2110 C)) p \u03bc := snorm_indicator_le _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} C : \u211d\u22650 \u22a2 C \u2022 \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 C \u2022 \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p) ** rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} C : \u211d\u22650 \u22a2 \u2191C * \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 \u2191C * \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p) ** simp_rw [ENNReal.ofReal_coe_nnreal] at h\u2110 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 C : \u211d\u22650 h\u2110 : \u2200 (C : \u211d\u22650), \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} \u22a2 \u2191C * \u2191\u03b4 ^ (1 / ENNReal.toReal p) \u2264 \u2191C * \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p) ** refine' mul_le_mul' le_rfl\n  (ENNReal.rpow_le_rpow (h\u2110 C).le (one_div_nonneg.2 ENNReal.toReal_nonneg)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} C : \u211d\u22650 \u22a2 C \u2022 \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} ^ (1 / ENNReal.toReal p) \u2264 snorm (indicator {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} (f (\u2110 C))) p \u03bc ** refine' snorm_indicator_ge_of_bdd_below hp hp' _\n  (measurableSet_le measurable_const (hf _).nnnorm.measurable)\n  (eventually_of_forall fun x hx => _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hp : p \u2260 0 hp' : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfu : UnifIntegrable f p \u03bc M : \u211d\u22650 hM : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191M \u2110 : \u211d\u22650 \u2192 \u03b9 \u03b4 : \u211d\u22650 h\u03b4pos : 0 < \u2191\u03b4 h\u03b4 :   \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u2191\u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 h\u2110 : \u2200 (C : \u211d\u22650), ENNReal.ofReal \u2191\u03b4 < \u2191\u2191\u03bc {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} C : \u211d\u22650 x : \u03b1 hx : x \u2208 {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} \u22a2 C \u2264 \u2016indicator {x | C \u2264 \u2016f (\u2110 C) x\u2016\u208a} (f (\u2110 C)) x\u2016\u208a ** rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx] ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.fin_val_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 n : \u2115 f : \u03b1 \u2192 Fin n \u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec f ** letI : Primcodable { a // id a < n } := Primcodable.subtype (nat_lt.comp .id (const _)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 n : \u2115 f : \u03b1 \u2192 Fin n this : Primcodable { a // id a < n } := Primcodable.subtype (_ : PrimrecPred fun a => id a < n) \u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec f ** exact (Iff.trans (by rfl) subtype_val_iff).trans (of_equiv_iff _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 n : \u2115 f : \u03b1 \u2192 Fin n this : Primcodable { a // id a < n } := Primcodable.subtype (_ : PrimrecPred fun a => id a < n) \u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec fun a => \u2191(\u2191Fin.equivSubtype (f a)) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pairwise_disjoint_Ioo_zpow ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 \u22a2 Pairwise (Disjoint on fun n => Ioo (b ^ n) (b ^ (n + 1))) ** simpa only [one_mul] using pairwise_disjoint_Ioo_mul_zpow 1 b ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.dvd_add_right ** a b c : Int H : a \u2223 b \u22a2 a \u2223 b + c \u2194 a \u2223 c ** rw [Int.add_comm, Int.dvd_add_left H] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_integral_condexpL1Clm ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** let S := spanningSets (\u03bc.trim hm) ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** have hS_meas : \u2200 i, MeasurableSet[m] (S i) := measurable_spanningSets (\u03bc.trim hm) ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** have hS_meas0 : \u2200 i, MeasurableSet (S i) := fun i => hm _ (hS_meas i) ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** have hs_eq : s = \u22c3 i, S i \u2229 s := by\n  simp_rw [Set.inter_comm]\n  rw [\u2190 Set.inter_iUnion, iUnion_spanningSets (\u03bc.trim hm), Set.inter_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** have hS_finite : \u2200 i, \u03bc (S i \u2229 s) < \u221e := by\n  refine' fun i => (measure_mono (Set.inter_subset_left _ _)).trans_lt _\n  have hS_finite_trim := measure_spanningSets_lt_top (\u03bc.trim hm) i\n  rwa [trim_measurableSet_eq hm (hS_meas i)] at hS_finite_trim ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** have h_mono : Monotone fun i => S i \u2229 s := by\n  intro i j hij x\n  simp_rw [Set.mem_inter_iff]\n  exact fun h => \u27e8monotone_spanningSets (\u03bc.trim hm) hij h.1, h.2\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 h_mono : Monotone fun i => S i \u2229 s \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** have h_eq_forall :\n  (fun i => \u222b x in S i \u2229 s, condexpL1Clm F' hm \u03bc f x \u2202\u03bc) = fun i => \u222b x in S i \u2229 s, f x \u2202\u03bc :=\n  funext fun i =>\n    set_integral_condexpL1Clm_of_measure_ne_top f (@MeasurableSet.inter \u03b1 m _ _ (hS_meas i) hs)\n      (hS_finite i).ne ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 h_mono : Monotone fun i => S i \u2229 s h_eq_forall :   (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** have h_right : Tendsto (fun i => \u222b x in S i \u2229 s, f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b x in s, f x \u2202\u03bc)) := by\n  have h :=\n    tendsto_set_integral_of_monotone (fun i => (hS_meas0 i).inter (hm s hs)) h_mono\n      (L1.integrable_coeFn f).integrableOn\n  rwa [\u2190 hs_eq] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 h_mono : Monotone fun i => S i \u2229 s h_eq_forall :   (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc h_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc)) \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** have h_left : Tendsto (fun i => \u222b x in S i \u2229 s, condexpL1Clm F' hm \u03bc f x \u2202\u03bc) atTop\n    (\ud835\udcdd (\u222b x in s, condexpL1Clm F' hm \u03bc f x \u2202\u03bc)) := by\n  have h := tendsto_set_integral_of_monotone (fun i => (hS_meas0 i).inter (hm s hs)) h_mono\n    (L1.integrable_coeFn (condexpL1Clm F' hm \u03bc f)).integrableOn\n  rwa [\u2190 hs_eq] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 h_mono : Monotone fun i => S i \u2229 s h_eq_forall :   (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc h_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc)) h_left :   Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) atTop     (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc)) \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** rw [h_eq_forall] at h_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 h_mono : Monotone fun i => S i \u2229 s h_eq_forall :   (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc h_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc)) h_left :   Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc)) \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** exact tendsto_nhds_unique h_left h_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) \u22a2 s = \u22c3 i, S i \u2229 s ** simp_rw [Set.inter_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) \u22a2 s = \u22c3 i, s \u2229 spanningSets (Measure.trim \u03bc hm) i ** rw [\u2190 Set.inter_iUnion, iUnion_spanningSets (\u03bc.trim hm), Set.inter_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s \u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 ** refine' fun i => (measure_mono (Set.inter_subset_left _ _)).trans_lt _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s i : \u2115 \u22a2 \u2191\u2191\u03bc (S i) < \u22a4 ** have hS_finite_trim := measure_spanningSets_lt_top (\u03bc.trim hm) i ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s i : \u2115 hS_finite_trim : \u2191\u2191(Measure.trim \u03bc hm) (spanningSets (Measure.trim \u03bc hm) i) < \u22a4 \u22a2 \u2191\u2191\u03bc (S i) < \u22a4 ** rwa [trim_measurableSet_eq hm (hS_meas i)] at hS_finite_trim ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 \u22a2 Monotone fun i => S i \u2229 s ** intro i j hij x ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 i j : \u2115 hij : i \u2264 j x : \u03b1 \u22a2 x \u2208 (fun i => S i \u2229 s) i \u2192 x \u2208 (fun i => S i \u2229 s) j ** simp_rw [Set.mem_inter_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 i j : \u2115 hij : i \u2264 j x : \u03b1 \u22a2 x \u2208 spanningSets (Measure.trim \u03bc hm) i \u2227 x \u2208 s \u2192 x \u2208 spanningSets (Measure.trim \u03bc hm) j \u2227 x \u2208 s ** exact fun h => \u27e8monotone_spanningSets (\u03bc.trim hm) hij h.1, h.2\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 h_mono : Monotone fun i => S i \u2229 s h_eq_forall :   (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc)) ** have h :=\n  tendsto_set_integral_of_monotone (fun i => (hS_meas0 i).inter (hm s hs)) h_mono\n    (L1.integrable_coeFn f).integrableOn ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 h_mono : Monotone fun i => S i \u2229 s h_eq_forall :   (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc h : Tendsto (fun i => \u222b (a : \u03b1) in S i \u2229 s, \u2191\u2191f a \u2202\u03bc) atTop (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, S n \u2229 s, \u2191\u2191f a \u2202\u03bc)) \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc)) ** rwa [\u2190 hs_eq] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 h_mono : Monotone fun i => S i \u2229 s h_eq_forall :   (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc h_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc)) \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) atTop     (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc)) ** have h := tendsto_set_integral_of_monotone (fun i => (hS_meas0 i).inter (hm s hs)) h_mono\n  (L1.integrable_coeFn (condexpL1Clm F' hm \u03bc f)).integrableOn ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } hs : MeasurableSet s S : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm) hS_meas : \u2200 (i : \u2115), MeasurableSet (S i) hS_meas0 : \u2200 (i : \u2115), MeasurableSet (S i) hs_eq : s = \u22c3 i, S i \u2229 s hS_finite : \u2200 (i : \u2115), \u2191\u2191\u03bc (S i \u2229 s) < \u22a4 h_mono : Monotone fun i => S i \u2229 s h_eq_forall :   (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) = fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc h_right : Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc)) h :   Tendsto (fun i => \u222b (a : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) a \u2202\u03bc) atTop     (\ud835\udcdd (\u222b (a : \u03b1) in \u22c3 n, S n \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) a \u2202\u03bc)) \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in S i \u2229 s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc) atTop     (\ud835\udcdd (\u222b (x : \u03b1) in s, \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f) x \u2202\u03bc)) ** rwa [\u2190 hs_eq] at h ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.le_total ** a b : Int H : NonNeg (-(b - a)) \u22a2 b \u2264 a ** rwa [show -(b - a) = a - b by simp [Int.add_comm, Int.sub_eq_add_neg]] at H ** a b : Int H : NonNeg (-(b - a)) \u22a2 -(b - a) = a - b ** simp [Int.add_comm, Int.sub_eq_add_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "Ctop.Realizer.ofEquiv_F ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 \u03c4 : Type u_4 inst\u271d : TopologicalSpace \u03b1 F : Realizer \u03b1 E : F.\u03c3 \u2243 \u03c4 s : \u03c4 \u22a2 f (ofEquiv F E).F s = f F.F (\u2191E.symm s) ** delta ofEquiv ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 \u03c4 : Type u_4 inst\u271d : TopologicalSpace \u03b1 F : Realizer \u03b1 E : F.\u03c3 \u2243 \u03c4 s : \u03c4 \u22a2 f { \u03c3 := \u03c4, F := Ctop.ofEquiv E F.F, eq := (_ : toTopsp (Ctop.ofEquiv E F.F) = inst\u271d) }.F s = f F.F (\u2191E.symm s) ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrable_withDensity_iff_integrable_smul' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : NormedAddCommGroup \u03b2 inst\u271d\u00b2 : NormedAddCommGroup \u03b3 E : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 E \u22a2 Integrable g \u2194 Integrable fun x => ENNReal.toReal (f x) \u2022 g x ** rw [\u2190 withDensity_congr_ae (coe_toNNReal_ae_eq hflt),\n  integrable_withDensity_iff_integrable_smul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : NormedAddCommGroup \u03b2 inst\u271d\u00b2 : NormedAddCommGroup \u03b3 E : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 E \u22a2 (Integrable fun x => ENNReal.toNNReal (f x) \u2022 g x) \u2194 Integrable fun x => ENNReal.toReal (f x) \u2022 g x ** simp_rw [NNReal.smul_def, ENNReal.toReal] ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : NormedAddCommGroup \u03b2 inst\u271d\u00b2 : NormedAddCommGroup \u03b3 E : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f hflt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 E \u22a2 Measurable fun x => ENNReal.toNNReal (f x) ** exact hf.ennreal_toNNReal ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.isUnit_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 \u22a2 IsUnit s \u2194 \u2203 a, s = {a} \u2227 IsUnit a ** constructor ** case mp F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 \u22a2 IsUnit s \u2192 \u2203 a, s = {a} \u2227 IsUnit a ** rintro \u27e8u, rfl\u27e9 ** case mp.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 t : Set \u03b1 u : (Set \u03b1)\u02e3 \u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a ** obtain \u27e8a, b, ha, hb, h\u27e9 := Set.mul_eq_one_iff.1 u.mul_inv ** case mp.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 t : Set \u03b1 u : (Set \u03b1)\u02e3 a b : \u03b1 ha : \u2191u = {a} hb : \u2191u\u207b\u00b9 = {b} h : a * b = 1 \u22a2 \u2203 a, \u2191u = {a} \u2227 IsUnit a ** refine' \u27e8a, ha, \u27e8a, b, h, singleton_injective _\u27e9, rfl\u27e9 ** case mp.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 t : Set \u03b1 u : (Set \u03b1)\u02e3 a b : \u03b1 ha : \u2191u = {a} hb : \u2191u\u207b\u00b9 = {b} h : a * b = 1 \u22a2 {b * a} = {1} ** rw [\u2190 singleton_mul_singleton, \u2190 ha, \u2190 hb] ** case mp.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 t : Set \u03b1 u : (Set \u03b1)\u02e3 a b : \u03b1 ha : \u2191u = {a} hb : \u2191u\u207b\u00b9 = {b} h : a * b = 1 \u22a2 \u2191u\u207b\u00b9 * \u2191u = {1} ** exact u.inv_mul ** case mpr F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 s t : Set \u03b1 \u22a2 (\u2203 a, s = {a} \u2227 IsUnit a) \u2192 IsUnit s ** rintro \u27e8a, rfl, ha\u27e9 ** case mpr.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : DivisionMonoid \u03b1 t : Set \u03b1 a : \u03b1 ha : IsUnit a \u22a2 IsUnit {a} ** exact ha.set ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.smul_le_stoppedValue_hitting ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 \u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264     ENNReal.ofReal       (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},         stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) ** have hn : Set.Icc 0 n = {k | k \u2264 n} := by ext x; simp ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} \u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264     ENNReal.ofReal       (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},         stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) ** have : \u2200 \u03c9, ((\u03b5 : \u211d) \u2264 (range (n + 1)).sup' nonempty_range_succ fun k => f k \u03c9) \u2192\n    (\u03b5 : \u211d) \u2264 stoppedValue f (hitting f {y : \u211d | \u2191\u03b5 \u2264 y} 0 n) \u03c9 := by\n  intro x hx\n  simp_rw [le_sup'_iff, mem_range, Nat.lt_succ_iff] at hx\n  refine' stoppedValue_hitting_mem _\n  simp only [Set.mem_setOf_eq, exists_prop, hn]\n  exact\n    let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n    \u27e8j, hj\u2081, hj\u2082\u27e9 ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} this :   \u2200 (\u03c9 : \u03a9),     (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192       \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264     ENNReal.ofReal       (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},         stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) ** have h := set_integral_ge_of_const_le (measurableSet_le measurable_const\n  (Finset.measurable_range_sup'' fun n _ => (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)))\n    (measure_ne_top _ _) this (Integrable.integrableOn (hsub.integrable_stoppedValue\n      (hitting_isStoppingTime hsub.adapted measurableSet_Ici) hitting_le)) ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} this :   \u2200 (\u03c9 : \u03a9),     (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192       \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 h :   \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264     \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},       stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc \u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264     ENNReal.ofReal       (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},         stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) ** rw [ENNReal.le_ofReal_iff_toReal_le, ENNReal.toReal_smul] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 \u22a2 Set.Icc 0 n = {k | k \u2264 n} ** ext x ** case h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n x : \u2115 \u22a2 x \u2208 Set.Icc 0 n \u2194 x \u2208 {k | k \u2264 n} ** simp ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} \u22a2 \u2200 (\u03c9 : \u03a9),     (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192       \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 ** intro x hx ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} x : \u03a9 hx : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x \u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x ** simp_rw [le_sup'_iff, mem_range, Nat.lt_succ_iff] at hx ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} x : \u03a9 hx : \u2203 b, b \u2264 n \u2227 \u2191\u03b5 \u2264 f b x \u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x ** refine' stoppedValue_hitting_mem _ ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} x : \u03a9 hx : \u2203 b, b \u2264 n \u2227 \u2191\u03b5 \u2264 f b x \u22a2 \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j x \u2208 {y | \u2191\u03b5 \u2264 y} ** simp only [Set.mem_setOf_eq, exists_prop, hn] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} x : \u03a9 hx : \u2203 b, b \u2264 n \u2227 \u2191\u03b5 \u2264 f b x \u22a2 \u2203 j, j \u2264 n \u2227 \u2191\u03b5 \u2264 f j x ** exact\n  let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n  \u27e8j, hj\u2081, hj\u2082\u27e9 ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} this :   \u2200 (\u03c9 : \u03a9),     (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192       \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 h :   \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264     \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},       stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc \u22a2 \u03b5 \u2022 ENNReal.toReal (\u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}) \u2264     \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},       stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc ** exact h ** case ha \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} this :   \u2200 (\u03c9 : \u03a9),     (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192       \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 h :   \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264     \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},       stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc \u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2260 \u22a4 ** exact ENNReal.mul_ne_top (by simp) (measure_ne_top _ _) ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} this :   \u2200 (\u03c9 : \u03a9),     (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192       \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 h :   \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264     \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},       stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc \u22a2 \u2191(RingHom.id \u211d\u22650\u221e) (\u2191\u2191(RingHom.toMonoidWithZeroHom ENNReal.ofNNRealHom) \u03b5) \u2260 \u22a4 ** simp ** case hb \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc \u03b5 : \u211d\u22650 n : \u2115 hn : Set.Icc 0 n = {k | k \u2264 n} this :   \u2200 (\u03c9 : \u03a9),     (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2192       \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 h :   \u2191\u03b5 * ENNReal.toReal (\u2191\u2191\u03bc {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a}) \u2264     \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k a},       stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc \u22a2 0 \u2264     \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},       stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc ** exact le_trans (mul_nonneg \u03b5.coe_nonneg ENNReal.toReal_nonneg) h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.measurableSet_bliminf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 s : \u2115 \u2192 Set \u03b1 p : \u2115 \u2192 Prop h : \u2200 (n : \u2115), p n \u2192 MeasurableSet (s n) \u22a2 MeasurableSet (bliminf s atTop p) ** simp only [Filter.bliminf_eq_iSup_biInf_of_nat, iInf_eq_iInter, iSup_eq_iUnion] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 s : \u2115 \u2192 Set \u03b1 p : \u2115 \u2192 Prop h : \u2200 (n : \u2115), p n \u2192 MeasurableSet (s n) \u22a2 MeasurableSet (\u22c3 i, \u22c2 j, \u22c2 (_ : p j \u2227 i \u2264 j), s j) ** exact .iUnion fun n => .iInter fun m => .iInter fun hm => h m hm.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "DomMulAct.edist_smul_Lp ** M : Type u_1 N : Type u_2 \u03b1 : Type u_3 E : Type u_4 inst\u271d\u2076 : MeasurableSpace M inst\u271d\u2075 : MeasurableSpace N inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : SMul M \u03b1 inst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc inst\u271d : MeasurableSMul M \u03b1 c : M\u1d48\u1d50\u1d43 f g : { x // x \u2208 Lp E p } \u22a2 edist (c \u2022 f) (c \u2022 g) = edist f g ** simp only [Lp.edist_dist, dist_smul_Lp] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Iterator.ValidFor.remainingBytes ** l r : List Char \u22a2 Iterator.remainingBytes { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } } = utf8Len r ** simp [Iterator.remainingBytes, Nat.add_sub_cancel_left] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_exchange' ** \u03b1 : Type u_1 s t : Set \u03b1 a b : \u03b1 ha : \u00aca \u2208 s hb : b \u2208 s \u22a2 encard (insert a s \\ {b}) = encard s ** rw [\u2190insert_diff_singleton_comm (by rintro rfl; exact ha hb), encard_exchange ha hb] ** \u03b1 : Type u_1 s t : Set \u03b1 a b : \u03b1 ha : \u00aca \u2208 s hb : b \u2208 s \u22a2 a \u2260 b ** rintro rfl ** \u03b1 : Type u_1 s t : Set \u03b1 a : \u03b1 ha : \u00aca \u2208 s hb : a \u2208 s \u22a2 False ** exact ha hb ** Qed",
        "informal": ""
    },
    {
        "formal": "Decidable.not_imp_self ** a : Prop inst\u271d : Decidable a \u22a2 \u00aca \u2192 a \u2194 a ** have := @imp_not_self (\u00aca) ** a : Prop inst\u271d : Decidable a this : \u00aca \u2192 \u00ac\u00aca \u2194 \u00ac\u00aca \u22a2 \u00aca \u2192 a \u2194 a ** rwa [not_not] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_surjective ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 \u22a2 Surjective (preimage f) \u2194 Injective f ** refine' \u27e8fun h x x' hx => _, Injective.preimage_surjective\u27e9 ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Surjective (preimage f) x x' : \u03b1 hx : f x = f x' \u22a2 x = x' ** cases' h {x} with s hs ** case intro \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Surjective (preimage f) x x' : \u03b1 hx : f x = f x' s : Set \u03b2 hs : f \u207b\u00b9' s = {x} \u22a2 x = x' ** have := mem_singleton x ** case intro \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Surjective (preimage f) x x' : \u03b1 hx : f x = f x' s : Set \u03b2 hs : f \u207b\u00b9' s = {x} this : x \u2208 {x} \u22a2 x = x' ** rwa [\u2190 hs, mem_preimage, hx, \u2190 mem_preimage, hs, mem_singleton_iff, eq_comm] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pi_eq_empty_iff ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 \u22a2 pi s t = \u2205 \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205 ** rw [\u2190 not_nonempty_iff_eq_empty, pi_nonempty_iff] ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 \u22a2 (\u00ac\u2200 (i : \u03b9), \u2203 x, i \u2208 s \u2192 x \u2208 t i) \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205 ** push_neg ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 \u22a2 (\u2203 i, \u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 \u2203 i, IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205 ** refine' exists_congr fun i => _ ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i\u271d i : \u03b9 \u22a2 (\u2200 (x : \u03b1 i), i \u2208 s \u2227 \u00acx \u2208 t i) \u2194 IsEmpty (\u03b1 i) \u2228 i \u2208 s \u2227 t i = \u2205 ** cases isEmpty_or_nonempty (\u03b1 i) <;> simp [*, forall_and, eq_empty_iff_forall_not_mem] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableEquiv.measurableSet_preimage ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u : Set \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 e : \u03b1 \u2243\u1d50 \u03b2 s : Set \u03b2 h : MeasurableSet (\u2191e \u207b\u00b9' s) \u22a2 MeasurableSet s ** simpa only [symm_preimage_preimage] using e.symm.measurable h ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.monotone_preCdf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a ** simp_rw [Monotone, ae_all_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 \u22a2 \u2200 (i i_1 : \u211a), i \u2264 i_1 \u2192 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, preCdf \u03c1 i a \u2264 preCdf \u03c1 i_1 a ** refine' fun r r' hrr' =>\n  ae_le_of_forall_set_lintegral_le_of_sigmaFinite measurable_preCdf measurable_preCdf\n    fun s hs _ => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r r' : \u211a hrr' : r \u2264 r' s : Set \u03b1 hs : MeasurableSet s x\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 r' x \u2202Measure.fst \u03c1 ** rw [set_lintegral_preCdf_fst \u03c1 r hs, set_lintegral_preCdf_fst \u03c1 r' hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r r' : \u211a hrr' : r \u2264 r' s : Set \u03b1 hs : MeasurableSet s x\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4 \u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r) s \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191r') s ** refine' Measure.IicSnd_mono \u03c1 _ s hs ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 r r' : \u211a hrr' : r \u2264 r' s : Set \u03b1 hs : MeasurableSet s x\u271d : \u2191\u2191(Measure.fst \u03c1) s < \u22a4 \u22a2 \u2191r \u2264 \u2191r' ** exact_mod_cast hrr' ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.nontrivial_coe_sort ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1 \u22a2 Nontrivial \u2191s \u2194 Set.Nontrivial s ** rw [\u2190 nontrivial_univ_iff, Set.Nontrivial, Set.Nontrivial] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1 \u22a2 (\u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y) \u2194 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y ** apply Iff.intro ** case mp \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1 \u22a2 (\u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y) \u2192 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y ** rintro \u27e8x, _, y, _, hxy\u27e9 ** case mp.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1 x : \u2191s left\u271d\u00b9 : x \u2208 univ y : \u2191s left\u271d : y \u2208 univ hxy : x \u2260 y \u22a2 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y ** exact \u27e8x, Subtype.prop x, y, Subtype.prop y, fun h => hxy (Subtype.coe_injective h)\u27e9 ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1 \u22a2 (\u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y) \u2192 \u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y ** rintro \u27e8x, hx, y, hy, hxy\u27e9 ** case mpr.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1 x : \u03b1 hx : x \u2208 s y : \u03b1 hy : y \u2208 s hxy : x \u2260 y \u22a2 \u2203 x, x \u2208 univ \u2227 \u2203 y, y \u2208 univ \u2227 x \u2260 y ** exact \u27e8\u27e8x, hx\u27e9, mem_univ _, \u27e8y, hy\u27e9, mem_univ _, Subtype.mk_eq_mk.not.mpr hxy\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.encode_ofNatCode ** \u22a2 encodeCode (ofNatCode 0) = 0 ** simp [ofNatCode, encodeCode] ** \u22a2 encodeCode (ofNatCode 1) = 1 ** simp [ofNatCode, encodeCode] ** \u22a2 encodeCode (ofNatCode 2) = 2 ** simp [ofNatCode, encodeCode] ** \u22a2 encodeCode (ofNatCode 3) = 3 ** simp [ofNatCode, encodeCode] ** n : \u2115 \u22a2 encodeCode (ofNatCode (n + 4)) = n + 4 ** let m := n.div2.div2 ** n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 encodeCode (ofNatCode (n + 4)) = n + 4 ** have hm : m < n + 4 := by\n  simp only [div2_val]\n  exact\n    lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n      (Nat.succ_le_succ (Nat.le_add_right _ _)) ** n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 \u22a2 encodeCode (ofNatCode (n + 4)) = n + 4 ** have _m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm ** n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 _m1 : (unpair m).1 < n + 4 \u22a2 encodeCode (ofNatCode (n + 4)) = n + 4 ** have _m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm ** n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 _m1 : (unpair m).1 < n + 4 _m2 : (unpair m).2 < n + 4 \u22a2 encodeCode (ofNatCode (n + 4)) = n + 4 ** have IH := encode_ofNatCode m ** n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 _m1 : (unpair m).1 < n + 4 _m2 : (unpair m).2 < n + 4 IH : encodeCode (ofNatCode m) = m \u22a2 encodeCode (ofNatCode (n + 4)) = n + 4 ** have IH1 := encode_ofNatCode m.unpair.1 ** n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 _m1 : (unpair m).1 < n + 4 _m2 : (unpair m).2 < n + 4 IH : encodeCode (ofNatCode m) = m IH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1 \u22a2 encodeCode (ofNatCode (n + 4)) = n + 4 ** have IH2 := encode_ofNatCode m.unpair.2 ** n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 _m1 : (unpair m).1 < n + 4 _m2 : (unpair m).2 < n + 4 IH : encodeCode (ofNatCode m) = m IH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1 IH2 : encodeCode (ofNatCode (unpair m).2) = (unpair m).2 \u22a2 encodeCode (ofNatCode (n + 4)) = n + 4 ** conv_rhs => rw [\u2190 Nat.bit_decomp n, \u2190 Nat.bit_decomp n.div2] ** n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 _m1 : (unpair m).1 < n + 4 _m2 : (unpair m).2 < n + 4 IH : encodeCode (ofNatCode m) = m IH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1 IH2 : encodeCode (ofNatCode (unpair m).2) = (unpair m).2 \u22a2 encodeCode (ofNatCode (n + 4)) = bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4 ** simp only [ofNatCode._eq_5] ** n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 _m1 : (unpair m).1 < n + 4 _m2 : (unpair m).2 < n + 4 IH : encodeCode (ofNatCode m) = m IH1 : encodeCode (ofNatCode (unpair m).1) = (unpair m).1 IH2 : encodeCode (ofNatCode (unpair m).2) = (unpair m).2 \u22a2 encodeCode       (match bodd n, bodd (div2 n) with       | false, false => pair (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2)       | false, true => comp (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2)       | true, false => prec (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2)       | true, true => rfind' (ofNatCode (div2 (div2 n)))) =     bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4 ** cases n.bodd <;> cases n.div2.bodd <;>\n  simp [encodeCode, ofNatCode, IH, IH1, IH2, Nat.bit_val] ** n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 m < n + 4 ** simp only [div2_val] ** n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 n / 2 / 2 < n + 4 ** exact\n  lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n    (Nat.succ_le_succ (Nat.le_add_right _ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsFundamentalDomain.essSup_measure_restrict ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x \u22a2 essSup f (Measure.restrict \u03bc s) = essSup f \u03bc ** refine' le_antisymm (essSup_mono_measure' Measure.restrict_le_self) _ ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x \u22a2 essSup f \u03bc \u2264 essSup f (Measure.restrict \u03bc s) ** rw [essSup_eq_sInf (\u03bc.restrict s) f, essSup_eq_sInf \u03bc f] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x \u22a2 sInf {a | \u2191\u2191\u03bc {x | a < f x} = 0} \u2264 sInf {a | \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0} ** refine' sInf_le_sInf _ ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x \u22a2 {a | \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0} \u2286 {a | \u2191\u2191\u03bc {x | a < f x} = 0} ** rintro a (ha : (\u03bc.restrict s) {x : \u03b1 | a < f x} = 0) ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x a : \u211d\u22650\u221e ha : \u2191\u2191(Measure.restrict \u03bc s) {x | a < f x} = 0 \u22a2 a \u2208 {a | \u2191\u2191\u03bc {x | a < f x} = 0} ** rw [Measure.restrict_apply\u2080' hs.nullMeasurableSet] at ha ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x a : \u211d\u22650\u221e ha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0 \u22a2 a \u2208 {a | \u2191\u2191\u03bc {x | a < f x} = 0} ** refine' measure_zero_of_invariant hs _ _ ha ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x a : \u211d\u22650\u221e ha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0 \u22a2 \u2200 (g : G), g \u2022 {x | a < f x} = {x | a < f x} ** intro \u03b3 ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x a : \u211d\u22650\u221e ha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0 \u03b3 : G \u22a2 \u03b3 \u2022 {x | a < f x} = {x | a < f x} ** ext x ** case h G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x a : \u211d\u22650\u221e ha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0 \u03b3 : G x : \u03b1 \u22a2 x \u2208 \u03b3 \u2022 {x | a < f x} \u2194 x \u2208 {x | a < f x} ** rw [mem_smul_set_iff_inv_smul_mem] ** case h G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (\u03b3 : G) (x : \u03b1), f (\u03b3 \u2022 x) = f x a : \u211d\u22650\u221e ha : \u2191\u2191\u03bc ({x | a < f x} \u2229 s) = 0 \u03b3 : G x : \u03b1 \u22a2 \u03b3\u207b\u00b9 \u2022 x \u2208 {x | a < f x} \u2194 x \u2208 {x | a < f x} ** simp only [mem_setOf_eq, hf \u03b3\u207b\u00b9 x] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.offDiag_inter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 x : \u03b1 \u00d7 \u03b1 \u22a2 \u2191(offDiag (s \u2229 t)) = \u2191(offDiag s \u2229 offDiag t) ** push_cast ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 x : \u03b1 \u00d7 \u03b1 \u22a2 Set.offDiag (\u2191s \u2229 \u2191t) = Set.offDiag \u2191s \u2229 Set.offDiag \u2191t ** exact Set.offDiag_inter _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** set g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** have hg : Submartingale g \u2131 \u03bc :=\n  hf.sub_martingale (martingale_const_fun _ _ (hf.adapted 0) (hf.integrable 0)) ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** have hg0 : g 0 = 0 := by\n  ext \u03c9\n  simp only [sub_self, Pi.zero_apply] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** have hgbdd : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 i : \u2115, |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R := by\n  simpa only [sub_sub_sub_cancel_right] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** filter_upwards [hg.bddAbove_iff_exists_tendsto_aux hg0 hgbdd] with \u03c9 h\u03c9 ** case h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) \u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** convert h\u03c9 using 1 ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc \u22a2 g 0 = 0 ** ext \u03c9 ** case h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc \u03c9 : \u03a9 \u22a2 g 0 \u03c9 = OfNat.ofNat 0 \u03c9 ** simp only [sub_self, Pi.zero_apply] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R ** simpa only [sub_sub_sub_cancel_right] ** case h.e'_1.a \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) \u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 BddAbove (Set.range fun n => g n \u03c9) ** refine' \u27e8fun h => _, fun h => _\u27e9 <;> obtain \u27e8b, hb\u27e9 := h <;>\nrefine' \u27e8b + |f 0 \u03c9|, fun y hy => _\u27e9 <;> obtain \u27e8n, rfl\u27e9 := hy ** case h.e'_1.a.refine'_1.intro.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) b : \u211d hb : b \u2208 upperBounds (Set.range fun n => f n \u03c9) n : \u2115 \u22a2 (fun n => g n \u03c9) n \u2264 b + |f 0 \u03c9| ** simp_rw [sub_eq_add_neg] ** case h.e'_1.a.refine'_1.intro.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) b : \u211d hb : b \u2208 upperBounds (Set.range fun n => f n \u03c9) n : \u2115 \u22a2 f n \u03c9 + -f 0 \u03c9 \u2264 b + |f 0 \u03c9| ** exact add_le_add (hb \u27e8n, rfl\u27e9) (neg_le_abs_self _) ** case h.e'_1.a.refine'_2.intro.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) b : \u211d hb : b \u2208 upperBounds (Set.range fun n => g n \u03c9) n : \u2115 \u22a2 (fun n => f n \u03c9) n \u2264 b + |f 0 \u03c9| ** exact sub_le_iff_le_add.1 (le_trans (sub_le_sub_left (le_abs_self _) _) (hb \u27e8n, rfl\u27e9)) ** case h.e'_2.a \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) \u22a2 (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) ** refine' \u27e8fun h => _, fun h => _\u27e9 <;> obtain \u27e8c, hc\u27e9 := h ** case h.e'_2.a.refine'_1.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) c : \u211d hc : Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) \u22a2 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) ** exact \u27e8c - f 0 \u03c9, hc.sub_const _\u27e9 ** case h.e'_2.a.refine'_2.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) c : \u211d hc : Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) \u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** refine' \u27e8c + f 0 \u03c9, _\u27e9 ** case h.e'_2.a.refine'_2.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) c : \u211d hc : Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) \u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9)) ** have := hc.add_const (f 0 \u03c9) ** case h.e'_2.a.refine'_2.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 r : \u211d R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9 hg : Submartingale g \u2131 \u03bc hg0 : g 0 = 0 hgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9 : BddAbove (Set.range fun n => g n \u03c9) \u2194 \u2203 c, Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) c : \u211d hc : Tendsto (fun n => g n \u03c9) atTop (\ud835\udcdd c) this : Tendsto (fun k => g k \u03c9 + f 0 \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9)) \u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9)) ** simpa only [sub_add_cancel] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_subset_real_measurableEquiv ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 \u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s) ** by_cases h\u03b1 : Countable \u03b1 ** case pos \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 \u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s) ** cases finite_or_infinite \u03b1 ** case pos.inl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Finite \u03b1 \u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s) ** obtain \u27e8n, h_nonempty_equiv\u27e9 := exists_nat_measurableEquiv_range_coe_fin_of_finite \u03b1 ** case pos.inl.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Finite \u03b1 n : \u2115 h_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x)) \u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s) ** refine' \u27e8_, _, h_nonempty_equiv\u27e9 ** case pos.inl.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Finite \u03b1 n : \u2115 h_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x)) \u22a2 MeasurableSet (range fun x => \u2191\u2191x) ** letI : MeasurableSpace (Fin n) := borel (Fin n) ** case pos.inl.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Finite \u03b1 n : \u2115 h_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x)) this : MeasurableSpace (Fin n) := borel (Fin n) \u22a2 MeasurableSet (range fun x => \u2191\u2191x) ** haveI : BorelSpace (Fin n) := \u27e8rfl\u27e9 ** case pos.inl.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Finite \u03b1 n : \u2115 h_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x)) this\u271d : MeasurableSpace (Fin n) := borel (Fin n) this : BorelSpace (Fin n) \u22a2 MeasurableSet (range fun x => \u2191\u2191x) ** refine' MeasurableEmbedding.measurableSet_range _ ** case pos.inl.intro.refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Finite \u03b1 n : \u2115 h_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x)) this\u271d : MeasurableSpace (Fin n) := borel (Fin n) this : BorelSpace (Fin n) \u22a2 MeasurableSpace (Fin n) ** infer_instance ** case pos.inl.intro.refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Finite \u03b1 n : \u2115 h_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x)) this\u271d : MeasurableSpace (Fin n) := borel (Fin n) this : BorelSpace (Fin n) \u22a2 MeasurableEmbedding fun x => \u2191\u2191x ** exact\n  continuous_of_discreteTopology.measurableEmbedding\n    (Nat.cast_injective.comp Fin.val_injective) ** case pos.inr \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Infinite \u03b1 \u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s) ** refine' \u27e8_, _, measurableEquiv_range_coe_nat_of_infinite_of_countable \u03b1\u27e9 ** case pos.inr \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Infinite \u03b1 \u22a2 MeasurableSet (range Nat.cast) ** refine' MeasurableEmbedding.measurableSet_range _ ** case pos.inr.refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Infinite \u03b1 \u22a2 MeasurableSpace \u2115 ** infer_instance ** case pos.inr.refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : Countable \u03b1 h\u271d : Infinite \u03b1 \u22a2 MeasurableEmbedding Nat.cast ** exact continuous_of_discreteTopology.measurableEmbedding Nat.cast_injective ** case neg \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : \u00acCountable \u03b1 \u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s) ** refine'\n  \u27e8univ, MeasurableSet.univ,\n    \u27e8(PolishSpace.measurableEquivOfNotCountable h\u03b1 _ : \u03b1 \u2243\u1d50 (univ : Set \u211d))\u27e9\u27e9 ** case neg \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : \u00acCountable \u03b1 \u22a2 \u00acCountable \u2191univ ** rw [countable_coe_iff] ** case neg \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : StandardBorelSpace \u03b1 h\u03b1 : \u00acCountable \u03b1 \u22a2 \u00acSet.Countable univ ** exact Cardinal.not_countable_real ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.not_mem_erase_self ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 s : Finmap \u03b2 \u22a2 \u00aca \u2208 erase a s ** rw [mem_erase, not_and_or, not_not] ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 s : Finmap \u03b2 \u22a2 a = a \u2228 \u00aca \u2208 s ** left ** case h \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 s : Finmap \u03b2 \u22a2 a = a ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.continuous_of_dominated ** \u03b1 : Type u_1 E : Type u_2 F\u271d : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F\u271d inst\u271d\u2075 : NormedSpace \u211d F\u271d inst\u271d\u2074 : CompleteSpace F\u271d G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X F : X \u2192 \u03b1 \u2192 G bound : \u03b1 \u2192 \u211d hF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc h_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a bound_integrable : Integrable bound h_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a \u22a2 Continuous fun x => \u222b (a : \u03b1), F x a \u2202\u03bc ** by_cases hG : CompleteSpace G ** case pos \u03b1 : Type u_1 E : Type u_2 F\u271d : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F\u271d inst\u271d\u2075 : NormedSpace \u211d F\u271d inst\u271d\u2074 : CompleteSpace F\u271d G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X F : X \u2192 \u03b1 \u2192 G bound : \u03b1 \u2192 \u211d hF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc h_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a bound_integrable : Integrable bound h_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a hG : CompleteSpace G \u22a2 Continuous fun x => \u222b (a : \u03b1), F x a \u2202\u03bc ** simp only [integral, hG, L1.integral] ** case pos \u03b1 : Type u_1 E : Type u_2 F\u271d : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F\u271d inst\u271d\u2075 : NormedSpace \u211d F\u271d inst\u271d\u2074 : CompleteSpace F\u271d G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X F : X \u2192 \u03b1 \u2192 G bound : \u03b1 \u2192 \u211d hF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc h_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a bound_integrable : Integrable bound h_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a hG : CompleteSpace G \u22a2 Continuous fun x =>     if h : True then if hf : Integrable fun a => F x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0     else 0 ** exact continuous_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul \u03bc)\n  hF_meas h_bound bound_integrable h_cont ** case neg \u03b1 : Type u_1 E : Type u_2 F\u271d : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F\u271d inst\u271d\u2075 : NormedSpace \u211d F\u271d inst\u271d\u2074 : CompleteSpace F\u271d G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X F : X \u2192 \u03b1 \u2192 G bound : \u03b1 \u2192 \u211d hF_meas : \u2200 (x : X), AEStronglyMeasurable (F x) \u03bc h_bound : \u2200 (x : X), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a bound_integrable : Integrable bound h_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Continuous fun x => F x a hG : \u00acCompleteSpace G \u22a2 Continuous fun x => \u222b (a : \u03b1), F x a \u2202\u03bc ** simp [integral, hG, continuous_const] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.mem_supported ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 \u22a2 p \u2208 supported R s \u2194 \u2191(vars p) \u2286 s ** rw [supported_eq_range_rename, AlgHom.mem_range] ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 \u22a2 (\u2203 x, \u2191(rename Subtype.val) x = p) \u2194 \u2191(vars p) \u2286 s ** constructor ** case mp \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 \u22a2 (\u2203 x, \u2191(rename Subtype.val) x = p) \u2192 \u2191(vars p) \u2286 s ** rintro \u27e8p, rfl\u27e9 ** case mp.intro \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R q : MvPolynomial \u03c3 R s t : Set \u03c3 p : MvPolynomial { x // x \u2208 s } R \u22a2 \u2191(vars (\u2191(rename Subtype.val) p)) \u2286 s ** refine' _root_.trans (Finset.coe_subset.2 (vars_rename _ _)) _ ** case mp.intro \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R q : MvPolynomial \u03c3 R s t : Set \u03c3 p : MvPolynomial { x // x \u2208 s } R \u22a2 \u2191(Finset.image Subtype.val (vars p)) \u2286 s ** simp ** case mpr \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 \u22a2 \u2191(vars p) \u2286 s \u2192 \u2203 x, \u2191(rename Subtype.val) x = p ** intro hs ** case mpr \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 hs : \u2191(vars p) \u2286 s \u22a2 \u2203 x, \u2191(rename Subtype.val) x = p ** exact exists_rename_eq_of_vars_subset_range p ((\u2191) : s \u2192 \u03c3) Subtype.val_injective (by simpa) ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 hs : \u2191(vars p) \u2286 s \u22a2 \u2191(vars p) \u2286 Set.range Subtype.val ** simpa ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.restrict_lintegral_eq_lintegral_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192\u209b \u211d\u22650\u221e s : Set \u03b1 hs : MeasurableSet s \u22a2 lintegral (restrict f s) \u03bc = lintegral f (Measure.restrict \u03bc s) ** rw [f.restrict_lintegral hs, lintegral_restrict] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.filter_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s : Set \u03b1 h : \u2203 a, a \u2208 s \u2227 a \u2208 support p a : \u03b1 \u22a2 \u2191(filter p s h) a = Set.indicator s (\u2191p) a * (\u2211' (a' : \u03b1), Set.indicator s (\u2191p) a')\u207b\u00b9 ** rw [filter, normalize_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.coeff_X' ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m\u271d : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R inst\u271d : DecidableEq \u03c3 i : \u03c3 m : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff m (X i) = if (fun\u2080 | i => 1) = m then 1 else 0 ** rw [\u2190 coeff_X_pow, pow_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Content.measure_eq_content_of_regular ** G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K = (fun s => \u2191(toFun \u03bc s)) K ** refine' le_antisymm _ _ ** case refine'_1 G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K ** apply ENNReal.le_of_forall_pos_le_add ** case refine'_1.h G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 (fun s => \u2191(toFun \u03bc s)) K < \u22a4 \u2192 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 ** intro \u03b5 \u03b5pos _ ** case refine'_1.h G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 a\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4 \u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 ** obtain \u27e8K', K'_hyp\u27e9 := contentRegular_exists_compact \u03bc H K (ne_bot_of_gt \u03b5pos) ** case refine'_1.h.intro G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 a\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4 K' : Compacts G K'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 ** calc\n  \u03bc.measure \u2191K \u2264 \u03bc.measure (interior \u2191K') := by\n    rw [\u03bc.measure_apply isOpen_interior.measurableSet,\n      \u03bc.measure_apply K.isCompact.measurableSet]\n    exact \u03bc.outerMeasure.mono K'_hyp.left\n  _ \u2264 \u03bc K' := by\n    rw [\u03bc.measure_apply (IsOpen.measurableSet isOpen_interior)]\n    exact \u03bc.outerMeasure_interior_compacts K'\n  _ \u2264 \u03bc K + \u03b5 := K'_hyp.right ** G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 a\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4 K' : Compacts G K'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u22a2 \u2191\u2191(Content.measure \u03bc) \u2191K \u2264 \u2191\u2191(Content.measure \u03bc) (interior \u2191K') ** rw [\u03bc.measure_apply isOpen_interior.measurableSet,\n  \u03bc.measure_apply K.isCompact.measurableSet] ** G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 a\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4 K' : Compacts G K'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u22a2 \u2191(Content.outerMeasure \u03bc) \u2191K \u2264 \u2191(Content.outerMeasure \u03bc) (interior \u2191K') ** exact \u03bc.outerMeasure.mono K'_hyp.left ** G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 a\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4 K' : Compacts G K'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u22a2 \u2191\u2191(Content.measure \u03bc) (interior \u2191K') \u2264 (fun s => \u2191(toFun \u03bc s)) K' ** rw [\u03bc.measure_apply (IsOpen.measurableSet isOpen_interior)] ** G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 a\u271d : (fun s => \u2191(toFun \u03bc s)) K < \u22a4 K' : Compacts G K'_hyp : K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u22a2 \u2191(Content.outerMeasure \u03bc) (interior \u2191K') \u2264 (fun s => \u2191(toFun \u03bc s)) K' ** exact \u03bc.outerMeasure_interior_compacts K' ** case refine'_2 G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191\u2191(Content.measure \u03bc) \u2191K ** rw [\u03bc.measure_apply (IsCompact.measurableSet K.isCompact)] ** case refine'_2 G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : MeasurableSpace G inst\u271d\u00b9 : T2Space G inst\u271d : BorelSpace G H : ContentRegular \u03bc K : Compacts G \u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Content.outerMeasure \u03bc) \u2191K ** exact \u03bc.le_outerMeasure_compacts K ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.lintegral_coeFn ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 inst\u271d : TopologicalSpace \u03b4 f : \u03b1 \u2192\u2098[\u03bc] \u211d\u22650\u221e \u22a2 \u222b\u207b (a : \u03b1), \u2191f a \u2202\u03bc = lintegral f ** rw [\u2190 lintegral_mk, mk_coeFn] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Iterator.ValidFor.hasPrev ** l r : List Char it : Iterator h : ValidFor l r it \u22a2 Iterator.hasPrev it = true \u2194 l \u2260 [] ** simp [Iterator.hasPrev, h.pos, Nat.pos_iff_ne_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul ** c d a b : Nat cop : Coprime c d h : a * b = c * d \u22a2 gcd a c * gcd b c = c ** apply Nat.dvd_antisymm ** case a c d a b : Nat cop : Coprime c d h : a * b = c * d \u22a2 gcd a c * gcd b c \u2223 c ** apply ((cop.gcd_left _).mul (cop.gcd_left _)).dvd_of_dvd_mul_right ** case a c d a b : Nat cop : Coprime c d h : a * b = c * d \u22a2 gcd a c * gcd b c \u2223 c * d ** rw [\u2190 h] ** case a c d a b : Nat cop : Coprime c d h : a * b = c * d \u22a2 gcd a c * gcd b c \u2223 a * b ** apply Nat.mul_dvd_mul (gcd_dvd ..).1 (gcd_dvd ..).1 ** case a c d a b : Nat cop : Coprime c d h : a * b = c * d \u22a2 c \u2223 gcd a c * gcd b c ** rw [gcd_comm a, gcd_comm b] ** case a c d a b : Nat cop : Coprime c d h : a * b = c * d \u22a2 c \u2223 gcd c a * gcd c b ** refine Nat.dvd_trans ?_ (gcd_mul_dvd_mul_gcd ..) ** case a c d a b : Nat cop : Coprime c d h : a * b = c * d \u22a2 c \u2223 gcd c (a * b) ** rw [h, gcd_mul_right_right d c] ** case a c d a b : Nat cop : Coprime c d h : a * b = c * d \u22a2 c \u2223 c ** apply Nat.dvd_refl ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.trCfg_init ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) \u22a2 TrCfg (TM2.init k L) (TM1.init (trInit k L)) ** rw [(_ : TM1.init _ = _)] ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) \u22a2 TrCfg (TM2.init k L) ?m.742315 ** refine' \u27e8ListBlank.mk (L.reverse.map fun a \u21a6 update default k (some a)), fun k' \u21a6 _\u27e9 ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) k' : K \u22a2 ListBlank.map (proj k') (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L))) =     ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k'))) ** refine' ListBlank.ext fun i \u21a6 _ ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) k' : K i : \u2115 \u22a2 ListBlank.nth       (ListBlank.map (proj k') (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) i =     ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i ** rw [ListBlank.map_mk, ListBlank.nth_mk, List.getI_eq_iget_get?, List.map_map] ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) k' : K i : \u2115 \u22a2 Option.iget (List.get? (List.map ((proj k').f \u2218 fun a => update default k (some a)) (List.reverse L)) i) =     ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i ** have : ((proj k').f \u2218 fun a => update (\u03b2 := fun k => Option (\u0393 k)) default k (some a))\n  = fun a => (proj k').f (update (\u03b2 := fun k => Option (\u0393 k)) default k (some a)) := rfl ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) k' : K i : \u2115 this : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a)) \u22a2 Option.iget (List.get? (List.map ((proj k').f \u2218 fun a => update default k (some a)) (List.reverse L)) i) =     ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i ** rw [this, List.get?_map, proj, PointedMap.mk_val] ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) k' : K i : \u2115 this : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a)) \u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =     ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i ** by_cases h : k' = k ** case pos K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) k' : K i : \u2115 this : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a)) h : k' = k \u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =     ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i ** subst k' ** case pos K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) i : \u2115 this : ((proj k).f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k) (update default k (some a)) \u22a2 Option.iget (Option.map (fun a => update default k (some a) k) (List.get? (List.reverse L) i)) =     ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k)))) i ** simp only [Function.update_same] ** case pos K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) i : \u2115 this : ((proj k).f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k) (update default k (some a)) \u22a2 Option.iget (Option.map (fun a => some a) (List.get? (List.reverse L) i)) =     ListBlank.nth (ListBlank.mk (List.reverse (List.map some L))) i ** rw [ListBlank.nth_mk, List.getI_eq_iget_get?, \u2190 List.map_reverse, List.get?_map] ** case neg K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) k' : K i : \u2115 this : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a)) h : \u00ack' = k \u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =     ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i ** simp only [Function.update_noteq h] ** case neg K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) k' : K i : \u2115 this : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a)) h : \u00ack' = k \u22a2 Option.iget (Option.map (fun a => default k') (List.get? (List.reverse L) i)) =     ListBlank.nth (ListBlank.mk (List.reverse (List.map some []))) i ** rw [ListBlank.nth_mk, List.getI_eq_iget_get?, List.map, List.reverse_nil] ** case neg K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) k' : K i : \u2115 this : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a)) h : \u00ack' = k \u22a2 Option.iget (Option.map (fun a => default k') (List.get? (List.reverse L) i)) = Option.iget (List.get? [] i) ** cases L.reverse.get? i <;> rfl ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) \u22a2 TM1.init (trInit k L) =     { l := Option.map normal (some default), var := default,       Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) } ** rw [trInit, TM1.init] ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) \u22a2 { l := some default, var := default,       Tape :=         Tape.mk\u2081           (let L' := List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L);           (true, (List.headI L').2) :: List.tail L') } =     { l := Option.map normal (some default), var := default,       Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) } ** dsimp only ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) \u22a2 { l := some default, var := default,       Tape :=         Tape.mk\u2081           ((true, (List.headI (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L))).2) ::             List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L))) } =     { l := Option.map normal (some default), var := default,       Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) } ** congr <;> cases L.reverse <;> try rfl ** case e_Tape.e_l.e_tail.cons K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) head\u271d : \u0393 k tail\u271d : List (\u0393 k) \u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (head\u271d :: tail\u271d)) =     List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f       (List.tail (List.map (fun a => update default k (some a)) (head\u271d :: tail\u271d))) ** simp only [List.map_map, List.tail_cons, List.map] ** case e_Tape.e_l.e_tail.cons K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) head\u271d : \u0393 k tail\u271d : List (\u0393 k) \u22a2 List.map (fun a => (false, update (fun x => none) k (some a))) tail\u271d =     List.map (Prod.mk false \u2218 fun a => update default k (some a)) tail\u271d ** rfl ** case e_Tape.e_l.e_tail.nil K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 k : K L : List (\u0393 k) \u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) []) =     List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f       (List.tail (List.map (fun a => update default k (some a)) [])) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_affineSubspace ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : AffineSubspace \u211d E hs : s \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc \u2191s = 0 ** rcases s.eq_bot_or_nonempty with (rfl | hne) ** case inr E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : AffineSubspace \u211d E hs : s \u2260 \u22a4 hne : Set.Nonempty \u2191s \u22a2 \u2191\u2191\u03bc \u2191s = 0 ** rw [Ne.def, \u2190 AffineSubspace.direction_eq_top_iff_of_nonempty hne] at hs ** case inr E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : AffineSubspace \u211d E hs : \u00acAffineSubspace.direction s = \u22a4 hne : Set.Nonempty \u2191s \u22a2 \u2191\u2191\u03bc \u2191s = 0 ** rcases hne with \u27e8x, hx : x \u2208 s\u27e9 ** case inr.intro E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : AffineSubspace \u211d E hs : \u00acAffineSubspace.direction s = \u22a4 x : E hx : x \u2208 s \u22a2 \u2191\u2191\u03bc \u2191s = 0 ** simpa only [AffineSubspace.coe_direction_eq_vsub_set_right hx, vsub_eq_sub, sub_eq_add_neg,\n  image_add_right, neg_neg, measure_preimage_add_right] using addHaar_submodule \u03bc s.direction hs ** case inl E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : \u22a5 \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc \u2191\u22a5 = 0 ** rw [AffineSubspace.bot_coe, measure_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "IsClosed.measurableSet_image_of_continuousOn_injOn ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 \u03b2 : Type u_4 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 s : Set \u03b3 hs : IsClosed s f : \u03b3 \u2192 \u03b2 f_cont : ContinuousOn f s f_inj : InjOn f s \u22a2 MeasurableSet (f '' s) ** rw [image_eq_range] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 \u03b2 : Type u_4 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 s : Set \u03b3 hs : IsClosed s f : \u03b3 \u2192 \u03b2 f_cont : ContinuousOn f s f_inj : InjOn f s \u22a2 MeasurableSet (range fun x => f \u2191x) ** haveI : PolishSpace s := IsClosed.polishSpace hs ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 \u03b2 : Type u_4 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 s : Set \u03b3 hs : IsClosed s f : \u03b3 \u2192 \u03b2 f_cont : ContinuousOn f s f_inj : InjOn f s this : PolishSpace \u2191s \u22a2 MeasurableSet (range fun x => f \u2191x) ** apply measurableSet_range_of_continuous_injective ** case f_cont \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 \u03b2 : Type u_4 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 s : Set \u03b3 hs : IsClosed s f : \u03b3 \u2192 \u03b2 f_cont : ContinuousOn f s f_inj : InjOn f s this : PolishSpace \u2191s \u22a2 Continuous fun x => f \u2191x ** rwa [continuousOn_iff_continuous_restrict] at f_cont ** case f_inj \u03b1 : Type u_1 \u03b9 : Type u_2 \u03b3 : Type u_3 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : PolishSpace \u03b3 \u03b2 : Type u_4 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : OpensMeasurableSpace \u03b2 s : Set \u03b3 hs : IsClosed s f : \u03b3 \u2192 \u03b2 f_cont : ContinuousOn f s f_inj : InjOn f s this : PolishSpace \u2191s \u22a2 Injective fun x => f \u2191x ** rwa [injOn_iff_injective] at f_inj ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.card_fintype_Ico_of_le ** a b : \u2124 h : a \u2264 b \u22a2 \u2191(Fintype.card \u2191(Set.Ico a b)) = b - a ** rw [card_fintype_Ico, toNat_sub_of_le h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.two_lt_ncard ** \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 2 < ncard s \u2194 \u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 s \u2227 \u2203 c, c \u2208 s \u2227 a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c ** simp only [two_lt_ncard_iff hs, exists_and_left, exists_prop] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.tail_eq_tail? ** \u03b1 : Type u_1 l : List \u03b1 \u22a2 tail l = Option.getD (tail? l) [] ** simp [tail_eq_tailD] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condCount_inter' ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hs : Set.Finite s \u22a2 \u2191\u2191(condCount s) (t \u2229 u) = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u ** rw [\u2190 Set.inter_comm] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hs : Set.Finite s \u22a2 \u2191\u2191(condCount s) (u \u2229 t) = \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u ** exact condCount_inter hs ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.comapRight_apply' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 f : \u03b3 \u2192 \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } hf : MeasurableEmbedding f a : \u03b1 t : Set \u03b3 ht : MeasurableSet t \u22a2 \u2191\u2191(\u2191(comapRight \u03ba hf) a) t = \u2191\u2191(\u2191\u03ba a) (f '' t) ** rw [comapRight_apply,\n  Measure.comap_apply _ hf.injective (fun s => hf.measurableSet_image.mpr) _ ht] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_union_eq_top_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 \u22a2 \u00ac\u2191\u2191\u03bc (s \u222a t) = \u22a4 \u2194 \u00ac(\u2191\u2191\u03bc s = \u22a4 \u2228 \u2191\u2191\u03bc t = \u22a4) ** simp only [\u2190 lt_top_iff_ne_top, \u2190 Ne.def, not_or, measure_union_lt_top_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_inter_add_diff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 B : Set \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e A : Set \u03b1 hB : MeasurableSet B \u22a2 \u222b\u207b (x : \u03b1) in A \u2229 B, f x \u2202\u03bc + \u222b\u207b (x : \u03b1) in A \\ B, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in A, f x \u2202\u03bc ** rw [\u2190 lintegral_add_measure, restrict_inter_add_diff _ hB] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_ae_eq_restrict_zero ** \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** by_cases hm : m \u2264 m0 ** case pos \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0  case neg \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** swap ** case pos \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm) ** case pos \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0  case neg \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** swap ** case pos \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm ** case pos \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** have : SigmaFinite ((\u03bc.restrict s).trim hm) := by\n  rw [\u2190 restrict_trim hm _ hs]\n  exact Restrict.sigmaFinite _ s ** case pos \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** by_cases hf_int : Integrable f \u03bc ** case pos \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0  case neg \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : \u00acIntegrable f \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** swap ** case pos \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** refine' ae_eq_of_forall_set_integral_eq_of_sigmaFinite' hm _ _ _ _ _ ** case neg \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** simp_rw [condexp_of_not_le hm] ** case neg \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : \u00acm \u2264 m0 \u22a2 0 =\u1d50[Measure.restrict \u03bc s] 0 ** rfl ** case neg \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm] ** case neg \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 0 =\u1d50[Measure.restrict \u03bc s] 0 ** rfl ** \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) ** rw [\u2190 restrict_trim hm _ hs] ** \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 SigmaFinite (Measure.restrict (Measure.trim \u03bc hm) s) ** exact Restrict.sigmaFinite _ s ** case neg \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : \u00acIntegrable f \u22a2 \u03bc[f|m] =\u1d50[Measure.restrict \u03bc s] 0 ** rw [condexp_undef hf_int] ** case pos.refine'_1 \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f \u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191(Measure.restrict \u03bc s) s_1 < \u22a4 \u2192 IntegrableOn (\u03bc[f|m]) s_1 ** exact fun t _ _ => integrable_condexp.integrableOn.integrableOn ** case pos.refine'_2 \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f \u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191(Measure.restrict \u03bc s) s_1 < \u22a4 \u2192 IntegrableOn 0 s_1 ** exact fun t _ _ => (integrable_zero _ _ _).integrableOn ** case pos.refine'_3 \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f \u22a2 \u2200 (s_1 : Set \u03b1),     MeasurableSet s_1 \u2192       \u2191\u2191(Measure.restrict \u03bc s) s_1 < \u22a4 \u2192         \u222b (x : \u03b1) in s_1, (\u03bc[f|m]) x \u2202Measure.restrict \u03bc s = \u222b (x : \u03b1) in s_1, OfNat.ofNat 0 x \u2202Measure.restrict \u03bc s ** intro t ht _ ** case pos.refine'_3 \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f t : Set \u03b1 ht : MeasurableSet t a\u271d : \u2191\u2191(Measure.restrict \u03bc s) t < \u22a4 \u22a2 \u222b (x : \u03b1) in t, (\u03bc[f|m]) x \u2202Measure.restrict \u03bc s = \u222b (x : \u03b1) in t, OfNat.ofNat 0 x \u2202Measure.restrict \u03bc s ** rw [Measure.restrict_restrict (hm _ ht), set_integral_condexp hm hf_int (ht.inter hs), \u2190\n  Measure.restrict_restrict (hm _ ht)] ** case pos.refine'_3 \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f t : Set \u03b1 ht : MeasurableSet t a\u271d : \u2191\u2191(Measure.restrict \u03bc s) t < \u22a4 \u22a2 \u222b (x : \u03b1) in t, f x \u2202Measure.restrict \u03bc s = \u222b (x : \u03b1) in t, OfNat.ofNat 0 x \u2202Measure.restrict \u03bc s ** refine' set_integral_congr_ae (hm _ ht) _ ** case pos.refine'_3 \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f t : Set \u03b1 ht : MeasurableSet t a\u271d : \u2191\u2191(Measure.restrict \u03bc s) t < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, x \u2208 t \u2192 f x = OfNat.ofNat 0 x ** filter_upwards [hf] with x hx _ using hx ** case pos.refine'_4 \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f \u22a2 AEStronglyMeasurable' m (\u03bc[f|m]) (Measure.restrict \u03bc s) ** exact stronglyMeasurable_condexp.aeStronglyMeasurable' ** case pos.refine'_5 \u03b1 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 m m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E \u03bc : Measure \u03b1 f : \u03b1 \u2192 E s : Set \u03b1 hs : MeasurableSet s hf : f =\u1d50[Measure.restrict \u03bc s] 0 hm : m \u2264 m0 h\u03bcm this\u271d : SigmaFinite (Measure.trim \u03bc hm) this : SigmaFinite (Measure.trim (Measure.restrict \u03bc s) hm) hf_int : Integrable f \u22a2 AEStronglyMeasurable' m 0 (Measure.restrict \u03bc s) ** exact stronglyMeasurable_zero.aeStronglyMeasurable' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.meas_ge_le_mul_pow_snorm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : AEStronglyMeasurable f \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p ** by_cases \u03b5 = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : AEStronglyMeasurable f \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h : \u00ac\u03b5 = \u22a4 \u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p ** have h\u03b5pow : \u03b5 ^ p.toReal \u2260 0 := (ENNReal.rpow_pos (pos_iff_ne_zero.2 h\u03b5) h).ne.symm ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : AEStronglyMeasurable f \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h : \u00ac\u03b5 = \u22a4 h\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0 \u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p ** have h\u03b5pow' : \u03b5 ^ p.toReal \u2260 \u221e := ENNReal.rpow_ne_top_of_nonneg ENNReal.toReal_nonneg h ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : AEStronglyMeasurable f \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h : \u00ac\u03b5 = \u22a4 h\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0 h\u03b5pow' : \u03b5 ^ ENNReal.toReal p \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p ** rw [ENNReal.inv_rpow, \u2190 ENNReal.mul_le_mul_left h\u03b5pow h\u03b5pow', \u2190 mul_assoc,\n  ENNReal.mul_inv_cancel h\u03b5pow h\u03b5pow', one_mul] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : AEStronglyMeasurable f \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h : \u00ac\u03b5 = \u22a4 h\u03b5pow : \u03b5 ^ ENNReal.toReal p \u2260 0 h\u03b5pow' : \u03b5 ^ ENNReal.toReal p \u2260 \u22a4 \u22a2 \u03b5 ^ ENNReal.toReal p * \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 snorm f p \u03bc ^ ENNReal.toReal p ** exact mul_meas_ge_le_pow_snorm' \u03bc hp_ne_zero hp_ne_top hf \u03b5 ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : AEStronglyMeasurable f \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 h : \u03b5 = \u22a4 \u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 \u2191\u2016f x\u2016\u208a} \u2264 \u03b5\u207b\u00b9 ^ ENNReal.toReal p * snorm f p \u03bc ^ ENNReal.toReal p ** simp [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Setoid.eqv_classes_of_disjoint_union ** \u03b1 : Type u_1 c : Set (Set \u03b1) hu : \u22c3\u2080 c = Set.univ H : Set.PairwiseDisjoint c id a : \u03b1 \u22a2 a \u2208 \u22c3\u2080 c ** rw [hu] ** \u03b1 : Type u_1 c : Set (Set \u03b1) hu : \u22c3\u2080 c = Set.univ H : Set.PairwiseDisjoint c id a : \u03b1 \u22a2 a \u2208 Set.univ ** exact Set.mem_univ a ** Qed",
        "informal": ""
    },
    {
        "formal": "String.offsetOfPos_of_valid ** l r : List Char \u22a2 offsetOfPos { data := l ++ r } { byteIdx := utf8Len l } = List.length l ** simpa using offsetOfPosAux_of_valid [] l r 0 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.erase_cons_of_ne ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t u v : Finset \u03b1 a\u271d b\u271d a b : \u03b1 s : Finset \u03b1 ha : \u00aca \u2208 s hb : a \u2260 b \u22a2 erase (cons a s ha) b = cons a (erase s b) (_ : a \u2208 erase s b \u2192 False) ** simp only [cons_eq_insert, erase_insert_of_ne hb] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_finset_sum ** \u03b9\u271d : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d \u03b9 : Type u_6 s : Finset \u03b9 f : \u03b9 \u2192 \u211d \u2192 E h : \u2200 (i : \u03b9), i \u2208 s \u2192 IntervalIntegrable (f i) \u03bc a b \u22a2 \u222b (x : \u211d) in a..b, \u2211 i in s, f i x \u2202\u03bc = \u2211 i in s, \u222b (x : \u211d) in a..b, f i x \u2202\u03bc ** simp only [intervalIntegral_eq_integral_uIoc, integral_finset_sum s fun i hi => (h i hi).def,\n  Finset.smul_sum] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Ici_eq_cons_Ioi ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrderTop \u03b1 a : \u03b1 \u22a2 Ici a = cons a (Ioi a) (_ : \u00aca \u2208 Ioi a) ** classical rw [cons_eq_insert, Ioi_insert] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrderTop \u03b1 a : \u03b1 \u22a2 Ici a = cons a (Ioi a) (_ : \u00aca \u2208 Ioi a) ** rw [cons_eq_insert, Ioi_insert] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.iUnion_accumulate ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : \u03b1 \u2192 Set \u03b2 t : \u03b1 \u2192 Set \u03b3 inst\u271d : Preorder \u03b1 \u22a2 \u22c3 x, Accumulate s x = \u22c3 x, s x ** apply Subset.antisymm ** case h\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : \u03b1 \u2192 Set \u03b2 t : \u03b1 \u2192 Set \u03b3 inst\u271d : Preorder \u03b1 \u22a2 \u22c3 x, Accumulate s x \u2286 \u22c3 x, s x ** simp only [subset_def, mem_iUnion, exists_imp, mem_accumulate] ** case h\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : \u03b1 \u2192 Set \u03b2 t : \u03b1 \u2192 Set \u03b3 inst\u271d : Preorder \u03b1 \u22a2 \u2200 (x : \u03b2) (x_1 x_2 : \u03b1), x_2 \u2264 x_1 \u2227 x \u2208 s x_2 \u2192 \u2203 i, x \u2208 s i ** intro z x x' \u27e8_, hz\u27e9 ** case h\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : \u03b1 \u2192 Set \u03b2 t : \u03b1 \u2192 Set \u03b3 inst\u271d : Preorder \u03b1 z : \u03b2 x x' : \u03b1 left\u271d : x' \u2264 x hz : z \u2208 s x' \u22a2 \u2203 i, z \u2208 s i ** exact \u27e8x', hz\u27e9 ** case h\u2082 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s : \u03b1 \u2192 Set \u03b2 t : \u03b1 \u2192 Set \u03b3 inst\u271d : Preorder \u03b1 \u22a2 \u22c3 x, s x \u2286 \u22c3 x, Accumulate s x ** exact iUnion_mono fun i => subset_accumulate ** Qed",
        "informal": ""
    },
    {
        "formal": "Computability.decode_encodeNat ** \u22a2 \u2200 (n : \u2115), decodeNat (encodeNat n) = n ** intro n ** n : \u2115 \u22a2 decodeNat (encodeNat n) = n ** conv_rhs => rw [\u2190 Num.to_of_nat n] ** n : \u2115 \u22a2 decodeNat (encodeNat n) = \u2191\u2191n ** exact congr_arg ((\u2191) : Num \u2192 \u2115) (decode_encodeNum n) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.noncommProd_insert_of_not_mem' ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b) ha : \u00aca \u2208 s \u22a2 Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute ** convert noncommProd_lemma _ f comm using 3 ** case h.e'_2.h.e'_2.h.a F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b) ha : \u00aca \u2208 s x\u271d : \u03b2 \u22a2 x\u271d \u2208 f a ::\u2098 Multiset.map f s.val \u2194 x\u271d \u2208 Multiset.map f (insert a s).val ** simp [@eq_comm _ (f a)] ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b) ha : \u00aca \u2208 s \u22a2 Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)       (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute) =     noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) * f a ** rw [Multiset.noncommProd_cons', noncommProd] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.mul_pos ** a b : Int ha : 0 < a hb : 0 < b \u22a2 0 < a * b ** let \u27e8n, hn\u27e9 := eq_succ_of_zero_lt ha ** a b : Int ha : 0 < a hb : 0 < b n : Nat hn : a = \u2191(succ n) \u22a2 0 < a * b ** let \u27e8m, hm\u27e9 := eq_succ_of_zero_lt hb ** a b : Int ha : 0 < a hb : 0 < b n : Nat hn : a = \u2191(succ n) m : Nat hm : b = \u2191(succ m) \u22a2 0 < a * b ** rw [hn, hm, \u2190 ofNat_mul] ** a b : Int ha : 0 < a hb : 0 < b n : Nat hn : a = \u2191(succ n) m : Nat hm : b = \u2191(succ m) \u22a2 0 < \u2191(succ n * succ m) ** apply ofNat_succ_pos ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.Fix.dest_mk ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : F (\u03b1 ::: Fix F \u03b1) \u22a2 dest (mk x) = x ** unfold Fix.dest ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : F (\u03b1 ::: Fix F \u03b1) \u22a2 rec (MvFunctor.map (TypeVec.id ::: mk)) (mk x) = x ** rw [Fix.rec_eq, \u2190 Fix.dest, \u2190 comp_map] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : F (\u03b1 ::: Fix F \u03b1) \u22a2 ((TypeVec.id ::: mk) \u229a (TypeVec.id ::: dest)) <$$> x = x ** conv =>\n  rhs\n  rw [\u2190 MvFunctor.id_map x] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : F (\u03b1 ::: Fix F \u03b1) \u22a2 ((TypeVec.id ::: mk) \u229a (TypeVec.id ::: dest)) <$$> x = TypeVec.id <$$> x ** rw [\u2190 appendFun_comp, id_comp] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : F (\u03b1 ::: Fix F \u03b1) \u22a2 (TypeVec.id ::: mk \u2218 dest) <$$> x = TypeVec.id <$$> x ** have : Fix.mk \u2218 Fix.dest = _root_.id := by\n  ext (x : Fix F \u03b1)\n  apply Fix.mk_dest ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : F (\u03b1 ::: Fix F \u03b1) this : mk \u2218 dest = _root_.id \u22a2 (TypeVec.id ::: mk \u2218 dest) <$$> x = TypeVec.id <$$> x ** rw [this, appendFun_id_id] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : F (\u03b1 ::: Fix F \u03b1) \u22a2 mk \u2218 dest = _root_.id ** ext (x : Fix F \u03b1) ** case h n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x\u271d : F (\u03b1 ::: Fix F \u03b1) x : Fix F \u03b1 \u22a2 (mk \u2218 dest) x = _root_.id x ** apply Fix.mk_dest ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurePreserving_piFinTwo ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b1 : Fin 2 \u2192 Type u m : (i : Fin 2) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : Fin 2) \u2192 Measure (\u03b1 i) inst\u271d : \u2200 (i : Fin 2), SigmaFinite (\u03bc i) \u22a2 MeasurePreserving \u2191(MeasurableEquiv.piFinTwo \u03b1) ** refine' \u27e8MeasurableEquiv.measurable _, (Measure.prod_eq fun s t _ _ => _).symm\u27e9 ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b1 : Fin 2 \u2192 Type u m : (i : Fin 2) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : Fin 2) \u2192 Measure (\u03b1 i) inst\u271d : \u2200 (i : Fin 2), SigmaFinite (\u03bc i) s : Set (\u03b1 0) t : Set (\u03b1 1) x\u271d\u00b9 : MeasurableSet s x\u271d : MeasurableSet t \u22a2 \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.piFinTwo \u03b1)) (Measure.pi \u03bc)) (s \u00d7\u02e2 t) = \u2191\u2191(\u03bc 0) s * \u2191\u2191(\u03bc 1) t ** rw [MeasurableEquiv.map_apply, MeasurableEquiv.piFinTwo_apply, Fin.preimage_apply_01_prod,\n  Measure.pi_pi, Fin.prod_univ_two] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b1 : Fin 2 \u2192 Type u m : (i : Fin 2) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : Fin 2) \u2192 Measure (\u03b1 i) inst\u271d : \u2200 (i : Fin 2), SigmaFinite (\u03bc i) s : Set (\u03b1 0) t : Set (\u03b1 1) x\u271d\u00b9 : MeasurableSet s x\u271d : MeasurableSet t \u22a2 \u2191\u2191(\u03bc 0) (Fin.cons s (Fin.cons t finZeroElim) 0) * \u2191\u2191(\u03bc 1) (Fin.cons s (Fin.cons t finZeroElim) 1) =     \u2191\u2191(\u03bc 0) s * \u2191\u2191(\u03bc 1) t ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_image_linearMap ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E s : Set E \u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s ** rcases ne_or_eq (LinearMap.det f) 0 with (hf | hf) ** case inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E s : Set E hf : \u2191LinearMap.det f \u2260 0 \u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s ** let g := (f.equivOfDetNeZero hf).toContinuousLinearEquiv ** case inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E s : Set E hf : \u2191LinearMap.det f \u2260 0 g : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf) \u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s ** change \u03bc (g '' s) = _ ** case inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E s : Set E hf : \u2191LinearMap.det f \u2260 0 g : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf) \u22a2 \u2191\u2191\u03bc (\u2191g '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s ** rw [ContinuousLinearEquiv.image_eq_preimage g s, addHaar_preimage_continuousLinearEquiv] ** case inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E s : Set E hf : \u2191LinearMap.det f \u2260 0 g : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf) \u22a2 ENNReal.ofReal |\u2191LinearMap.det \u2191\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm g))| * \u2191\u2191\u03bc s =     ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s ** congr ** case inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E s : Set E hf : \u2191LinearMap.det f = 0 \u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s ** simp only [hf, zero_mul, ENNReal.ofReal_zero, abs_zero] ** case inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E s : Set E hf : \u2191LinearMap.det f = 0 \u22a2 \u2191\u2191\u03bc ((fun a => \u2191f a) '' s) = 0 ** have : \u03bc (LinearMap.range f) = 0 :=\n  addHaar_submodule \u03bc _ (LinearMap.range_lt_top_of_det_eq_zero hf).ne ** case inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2192\u2097[\u211d] E s : Set E hf : \u2191LinearMap.det f = 0 this : \u2191\u2191\u03bc \u2191(LinearMap.range f) = 0 \u22a2 \u2191\u2191\u03bc ((fun a => \u2191f a) '' s) = 0 ** exact le_antisymm (le_trans (measure_mono (image_subset_range _ _)) this.le) (zero_le _) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L2.mem_L1_inner ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } \u22a2 AEEqFun.mk (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) (_ : AEStronglyMeasurable (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) \u03bc) \u2208 Lp \ud835\udd5c 1 ** simp_rw [mem_Lp_iff_snorm_lt_top, snorm_aeeqFun] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } \u22a2 snorm (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) 1 \u03bc < \u22a4 ** exact snorm_inner_lt_top f g ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.lt_iff_le_and_ne ** a b : Int \u22a2 a < b \u2194 a \u2264 b \u2227 a \u2260 b ** refine \u27e8fun h => \u27e8Int.le_of_lt h, Int.ne_of_lt h\u27e9, fun \u27e8aleb, aneb\u27e9 => ?_\u27e9 ** a b : Int x\u271d : a \u2264 b \u2227 a \u2260 b aleb : a \u2264 b aneb : a \u2260 b \u22a2 a < b ** let \u27e8n, hn\u27e9 := le.dest aleb ** a b : Int x\u271d : a \u2264 b \u2227 a \u2260 b aleb : a \u2264 b aneb : a \u2260 b n : Nat hn : a + \u2191n = b \u22a2 a < b ** have : n \u2260 0 := aneb.imp fun eq => by rw [\u2190 hn, eq, ofNat_zero, Int.add_zero] ** a b : Int x\u271d : a \u2264 b \u2227 a \u2260 b aleb : a \u2264 b aneb : a \u2260 b n : Nat hn : a + \u2191n = b this : n \u2260 0 \u22a2 a < b ** apply lt.intro ** case h a b : Int x\u271d : a \u2264 b \u2227 a \u2260 b aleb : a \u2264 b aneb : a \u2260 b n : Nat hn : a + \u2191n = b this : n \u2260 0 \u22a2 a + \u2191(succ ?n) = b  case n a b : Int x\u271d : a \u2264 b \u2227 a \u2260 b aleb : a \u2264 b aneb : a \u2260 b n : Nat hn : a + \u2191n = b this : n \u2260 0 \u22a2 Nat ** rwa [\u2190 Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero this)] at hn ** a b : Int x\u271d : a \u2264 b \u2227 a \u2260 b aleb : a \u2264 b aneb : a \u2260 b n : Nat hn : a + \u2191n = b eq : n = 0 \u22a2 a = b ** rw [\u2190 hn, eq, ofNat_zero, Int.add_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_biUnion_toMeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 I : Set \u03b2 hc : Set.Countable I s : \u03b2 \u2192 Set \u03b1 \u22a2 \u2191\u2191\u03bc (\u22c3 b \u2208 I, toMeasurable \u03bc (s b)) = \u2191\u2191\u03bc (\u22c3 b \u2208 I, s b) ** haveI := hc.toEncodable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 I : Set \u03b2 hc : Set.Countable I s : \u03b2 \u2192 Set \u03b1 this : Encodable \u2191I \u22a2 \u2191\u2191\u03bc (\u22c3 b \u2208 I, toMeasurable \u03bc (s b)) = \u2191\u2191\u03bc (\u22c3 b \u2208 I, s b) ** simp only [biUnion_eq_iUnion, measure_iUnion_toMeasurable] ** Qed",
        "informal": ""
    },
    {
        "formal": "Option.pbind_eq_bind ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 p : \u03b1 \u2192 Prop f\u271d : (a : \u03b1) \u2192 p a \u2192 \u03b2 x\u271d : Option \u03b1 f : \u03b1 \u2192 Option \u03b2 x : Option \u03b1 \u22a2 (pbind x fun a x => f a) = Option.bind x f ** cases x <;> simp only [pbind, none_bind', some_bind'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.self_mem_ordConnectedComponent ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x y z : \u03b1 \u22a2 x \u2208 ordConnectedComponent s x \u2194 x \u2208 s ** rw [mem_ordConnectedComponent, uIcc_self, singleton_subset_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** let \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b x in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** have hn\u03c6 : \u2200 n, \u2200 x \u2208 s, 0 \u2264 \u03c6 n x := by\n  intro n x hx\n  apply mul_nonneg (inv_nonneg.2 _) (pow_nonneg (hnc x hx) _)\n  exact set_integral_nonneg hs.measurableSet fun x hx => pow_nonneg (hnc x hx) _ ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** have I : \u2200 n, IntegrableOn (fun x => c x ^ n) s \u03bc := fun n =>\n  ContinuousOn.integrableOn_compact hs (hc.pow n) ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** have J : \u2200 n, 0 \u2264\u1d50[\u03bc.restrict s] fun x : \u03b1 => c x ^ n := by\n  intro n\n  filter_upwards [ae_restrict_mem hs.measurableSet] with x hx\n  exact pow_nonneg (hnc x hx) n ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** have P : \u2200 n, (0 : \u211d) < \u222b x in s, c x ^ n \u2202\u03bc := by\n  intro n\n  refine' (set_integral_pos_iff_support_of_nonneg_ae (J n) (I n)).2 _\n  obtain \u27e8u, u_open, x\u2080_u, hu\u27e9 : \u2203 u : Set \u03b1, IsOpen u \u2227 x\u2080 \u2208 u \u2227 u \u2229 s \u2286 c \u207b\u00b9' Ioi 0 :=\n    _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi (0 : \u211d)) isOpen_Ioi hnc\u2080\n  apply (h\u03bc u u_open x\u2080_u).trans_le\n  exact measure_mono fun x hx => \u27e8ne_of_gt (pow_pos (a := c x) (hu hx) _), hx.2\u27e9 ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** have hi\u03c6 : \u2200 n, \u222b x in s, \u03c6 n x \u2202\u03bc = 1 := fun n => by\n  rw [integral_mul_left, inv_mul_cancel (P n).ne'] ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 A : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** have : Tendsto (fun i : \u2115 => \u222b x : \u03b1 in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) :=\n  tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt hs.measurableSet\n    hs.measure_lt_top.ne (eventually_of_forall hn\u03c6) A (eventually_of_forall hi\u03c6) hmg hcg ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 A : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) this : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** convert this ** case h.e'_3.h \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 A : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) this : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) x\u271d : \u2115 \u22a2 (\u222b (x : \u03b1) in s, c x ^ x\u271d \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ x\u271d \u2022 g x \u2202\u03bc = \u222b (x : \u03b1) in s, \u03c6 x\u271d x \u2022 g x \u2202\u03bc ** simp_rw [\u2190 smul_smul, integral_smul] ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n \u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x ** intro n x hx ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n n : \u2115 x : \u03b1 hx : x \u2208 s \u22a2 0 \u2264 \u03c6 n x ** apply mul_nonneg (inv_nonneg.2 _) (pow_nonneg (hnc x hx) _) ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n n : \u2115 x : \u03b1 hx : x \u2208 s \u22a2 0 \u2264 \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc ** exact set_integral_nonneg hs.measurableSet fun x hx => pow_nonneg (hnc x hx) _ ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s \u22a2 \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n ** intro n ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s n : \u2115 \u22a2 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n ** filter_upwards [ae_restrict_mem hs.measurableSet] with x hx ** case h \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s n : \u2115 x : \u03b1 hx : x \u2208 s \u22a2 OfNat.ofNat 0 x \u2264 c x ^ n ** exact pow_nonneg (hnc x hx) n ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n \u22a2 \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc ** intro n ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n n : \u2115 \u22a2 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc ** refine' (set_integral_pos_iff_support_of_nonneg_ae (J n) (I n)).2 _ ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n n : \u2115 \u22a2 0 < \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s) ** obtain \u27e8u, u_open, x\u2080_u, hu\u27e9 : \u2203 u : Set \u03b1, IsOpen u \u2227 x\u2080 \u2208 u \u2227 u \u2229 s \u2286 c \u207b\u00b9' Ioi 0 :=\n  _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi (0 : \u211d)) isOpen_Ioi hnc\u2080 ** case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n n : \u2115 u : Set \u03b1 u_open : IsOpen u x\u2080_u : x\u2080 \u2208 u hu : u \u2229 s \u2286 c \u207b\u00b9' Ioi 0 \u22a2 0 < \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s) ** apply (h\u03bc u u_open x\u2080_u).trans_le ** case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n n : \u2115 u : Set \u03b1 u_open : IsOpen u x\u2080_u : x\u2080 \u2208 u hu : u \u2229 s \u2286 c \u207b\u00b9' Ioi 0 \u22a2 \u2191\u2191\u03bc (u \u2229 s) \u2264 \u2191\u2191\u03bc ((Function.support fun x => c x ^ n) \u2229 s) ** exact measure_mono fun x hx => \u27e8ne_of_gt (pow_pos (a := c x) (hu hx) _), hx.2\u27e9 ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc n : \u2115 \u22a2 \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 ** rw [integral_mul_left, inv_mul_cancel (P n).ne'] ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 \u22a2 \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) ** intro u u_open x\u2080u ** case intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t \u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) ** obtain \u27e8t', tt', t'x\u2080\u27e9 : \u2203 t', t < t' \u2227 t' < c x\u2080 := exists_between tx\u2080 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 \u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) ** have t'_pos : 0 < t' := t_pos.trans_lt tt' ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' \u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) ** obtain \u27e8v, v_open, x\u2080_v, hv\u27e9 : \u2203 v : Set \u03b1, IsOpen v \u2227 x\u2080 \u2208 v \u2227 v \u2229 s \u2286 c \u207b\u00b9' Ioi t' :=\n  _root_.continuousOn_iff.1 hc x\u2080 h\u2080 (Ioi t') isOpen_Ioi t'x\u2080 ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' M : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n \u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) ** have N :\n  Tendsto (fun n => (\u03bc (v \u2229 s)).toReal\u207b\u00b9 * (t / t') ^ n) atTop\n    (\ud835\udcdd ((\u03bc (v \u2229 s)).toReal\u207b\u00b9 * 0)) := by\n  apply Tendsto.mul tendsto_const_nhds _\n  apply tendsto_pow_atTop_nhds_0_of_lt_1 (div_nonneg t_pos t'_pos.le)\n  exact (div_lt_one t'_pos).2 tt' ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' M : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n N : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd ((ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * 0)) \u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) ** rw [mul_zero] at N ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' M : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n N : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0) \u22a2 TendstoUniformlyOn \u03c6 0 atTop (s \\ u) ** refine' tendstoUniformlyOn_iff.2 fun \u03b5 \u03b5pos => _ ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' M : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n N : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0) \u03b5 : \u211d \u03b5pos : \u03b5 > 0 \u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 n x) < \u03b5 ** filter_upwards [(tendsto_order.1 N).2 \u03b5 \u03b5pos] with n hn x hx ** case h \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' M : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n N : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0) \u03b5 : \u211d \u03b5pos : \u03b5 > 0 n : \u2115 hn : (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n < \u03b5 x : \u03b1 hx : x \u2208 s \\ u \u22a2 dist (OfNat.ofNat 0 x) (\u03c6 n x) < \u03b5 ** simp only [Pi.zero_apply, dist_zero_left, Real.norm_of_nonneg (hn\u03c6 n x hx.1)] ** case h \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' M : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n N : Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd 0) \u03b5 : \u211d \u03b5pos : \u03b5 > 0 n : \u2115 hn : (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n < \u03b5 x : \u03b1 hx : x \u2208 s \\ u \u22a2 (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n < \u03b5 ** exact (M n x hx).trans_lt hn ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u \u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t ** rcases eq_empty_or_nonempty (s \\ u) with (h | h) ** case inr \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u h : Set.Nonempty (s \\ u) \u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t ** obtain \u27e8x, hx, h'x\u27e9 : \u2203 x \u2208 s \\ u, \u2200 y \u2208 s \\ u, c y \u2264 c x :=\n  IsCompact.exists_isMaxOn (hs.diff u_open) h (hc.mono (diff_subset _ _)) ** case inr.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u h : Set.Nonempty (s \\ u) x : \u03b1 hx : x \u2208 s \\ u h'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x \u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t ** refine' \u27e8c x, hnc x hx.1, h'c x hx.1 _, h'x\u27e9 ** case inr.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u h : Set.Nonempty (s \\ u) x : \u03b1 hx : x \u2208 s \\ u h'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x \u22a2 x \u2260 x\u2080 ** rintro rfl ** case inr.intro.intro \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s c : \u03b1 \u2192 \u211d hc : ContinuousOn c s hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hmg : IntegrableOn g s \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u h : Set.Nonempty (s \\ u) x : \u03b1 hx : x \u2208 s \\ u h'x : \u2200 (y : \u03b1), y \u2208 s \\ u \u2192 c y \u2264 c x h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x \u2192 c y < c x hnc\u2080 : 0 < c x h\u2080 : x \u2208 s hcg : ContinuousWithinAt g s x x\u2080u : x \u2208 u \u22a2 False ** exact hx.2 x\u2080u ** case inl \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u h : s \\ u = \u2205 \u22a2 \u2203 t, 0 \u2264 t \u2227 t < c x\u2080 \u2227 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t ** exact\n  \u27e80, le_rfl, hnc\u2080, by simp only [h, mem_empty_iff_false, IsEmpty.forall_iff, imp_true_iff]\u27e9 ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u h : s \\ u = \u2205 \u22a2 \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 0 ** simp only [h, mem_empty_iff_false, IsEmpty.forall_iff, imp_true_iff] ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' \u22a2 \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n ** intro n x hx ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u B : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc \u22a2 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n ** simp_rw [\u2190 div_eq_inv_mul, div_pow, div_div] ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u B : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc \u22a2 c x ^ n / \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc \u2264 t ^ n / (t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s))) ** apply div_le_div (pow_nonneg t_pos n) _ _ B ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u \u22a2 t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) = \u222b (x : \u03b1) in v \u2229 s, t' ^ n \u2202\u03bc ** simp only [integral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,\n  Algebra.id.smul_eq_mul, mul_comm] ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u \u22a2 \u222b (x : \u03b1) in v \u2229 s, t' ^ n \u2202\u03bc \u2264 \u222b (y : \u03b1) in v \u2229 s, c y ^ n \u2202\u03bc ** apply set_integral_mono_on _ _ (v_open.measurableSet.inter hs.measurableSet) _ ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u \u22a2 IntegrableOn (fun a => t' ^ n) (v \u2229 s) ** apply integrableOn_const.2 (Or.inr _) ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u \u22a2 \u2191\u2191\u03bc (v \u2229 s) < \u22a4 ** exact lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) hs.measure_lt_top ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u \u22a2 IntegrableOn (fun a => c a ^ n) (v \u2229 s) ** exact (I n).mono (inter_subset_right _ _) le_rfl ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u \u22a2 \u2200 (x : \u03b1), x \u2208 v \u2229 s \u2192 t' ^ n \u2264 c x ^ n ** intro x hx ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x\u271d : \u03b1 hx\u271d : x\u271d \u2208 s \\ u x : \u03b1 hx : x \u2208 v \u2229 s \u22a2 t' ^ n \u2264 c x ^ n ** exact pow_le_pow_of_le_left t'_pos.le (le_of_lt (hv hx)) _ ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u B : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc \u22a2 c x ^ n \u2264 t ^ n ** exact pow_le_pow_of_le_left (hnc _ hx.1) (ht x hx) _ ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u B : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc \u22a2 0 < t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) ** apply mul_pos (pow_pos (t_pos.trans_lt tt') _) (ENNReal.toReal_pos (h\u03bc v v_open x\u2080_v).ne' _) ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u B : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc \u22a2 \u2191\u2191\u03bc (v \u2229 s) \u2260 \u22a4 ** have : \u03bc (v \u2229 s) \u2264 \u03bc s := measure_mono (inter_subset_right _ _) ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' n : \u2115 x : \u03b1 hx : x \u2208 s \\ u B : t' ^ n * ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)) \u2264 \u222b (y : \u03b1) in s, c y ^ n \u2202\u03bc this : \u2191\u2191\u03bc (v \u2229 s) \u2264 \u2191\u2191\u03bc s \u22a2 \u2191\u2191\u03bc (v \u2229 s) \u2260 \u22a4 ** exact ne_of_lt (lt_of_le_of_lt this hs.measure_lt_top) ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' M : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n \u22a2 Tendsto (fun n => (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n) atTop (\ud835\udcdd ((ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * 0)) ** apply Tendsto.mul tendsto_const_nhds _ ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' M : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n \u22a2 Tendsto (fun x => (t / t') ^ x) atTop (\ud835\udcdd 0) ** apply tendsto_pow_atTop_nhds_0_of_lt_1 (div_nonneg t_pos t'_pos.le) ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : BorelSpace \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6\u271d : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MetrizableSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc hs : IsCompact s h\u03bc : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 s hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u03c6 : \u2115 \u2192 \u03b1 \u2192 \u211d := fun n x => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 * c x ^ n hn\u03c6 : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 n x I : \u2200 (n : \u2115), IntegrableOn (fun x => c x ^ n) s J : \u2200 (n : \u2115), 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] fun x => c x ^ n P : \u2200 (n : \u2115), 0 < \u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc hi\u03c6 : \u2200 (n : \u2115), \u222b (x : \u03b1) in s, \u03c6 n x \u2202\u03bc = 1 u : Set \u03b1 u_open : IsOpen u x\u2080u : x\u2080 \u2208 u t : \u211d t_pos : 0 \u2264 t tx\u2080 : t < c x\u2080 ht : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 c x \u2264 t t' : \u211d tt' : t < t' t'x\u2080 : t' < c x\u2080 t'_pos : 0 < t' v : Set \u03b1 v_open : IsOpen v x\u2080_v : x\u2080 \u2208 v hv : v \u2229 s \u2286 c \u207b\u00b9' Ioi t' M : \u2200 (n : \u2115) (x : \u03b1), x \u2208 s \\ u \u2192 \u03c6 n x \u2264 (ENNReal.toReal (\u2191\u2191\u03bc (v \u2229 s)))\u207b\u00b9 * (t / t') ^ n \u22a2 t / t' < 1 ** exact (div_lt_one t'_pos).2 tt' ** Qed",
        "informal": ""
    },
    {
        "formal": "String.find_of_valid ** p : Char \u2192 Bool s : String \u22a2 find s p = { byteIdx := utf8Len (List.takeWhile (fun x => !p x) s.data) } ** simpa using findAux_of_valid p [] s.1 [] ** Qed",
        "informal": ""
    },
    {
        "formal": "IntervalIntegrable.div_const ** \u03b9 : Type u_1 \ud835\udd5c\u271d : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedRing A f\u271d g : \u211d \u2192 E a b : \u211d \u03bc : Measure \u211d \ud835\udd5c : Type u_6 f : \u211d \u2192 \ud835\udd5c inst\u271d : NormedField \ud835\udd5c h : IntervalIntegrable f \u03bc a b c : \ud835\udd5c \u22a2 IntervalIntegrable (fun x => f x / c) \u03bc a b ** simpa only [div_eq_mul_inv] using mul_const h c\u207b\u00b9 ** Qed",
        "informal": ""
    },
    {
        "formal": "WithBot.image_coe_Ici ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ici a = Ici \u2191a ** rw [\u2190 preimage_coe_Ici, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Ici_subset_Ioi.2 <| bot_lt_coe a)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.toJordanDecomposition_eq_of_eq_add_withDensity ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f \u22a2 IsFiniteMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) ** haveI := isFiniteMeasure_withDensity_ofReal hfi.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u22a2 IsFiniteMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) ** infer_instance ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f \u22a2 IsFiniteMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)) ** haveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x)) \u22a2 IsFiniteMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)) ** infer_instance ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f \u22a2 toJordanDecomposition s =     JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))       ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))       (_ :         ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098           (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)) ** haveI := isFiniteMeasure_withDensity_ofReal hfi.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u22a2 toJordanDecomposition s =     JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))       ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))       (_ :         ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098           (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)) ** haveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f this\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x)) \u22a2 toJordanDecomposition s =     JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))       ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))       (_ :         ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098           (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)) ** refine' toJordanDecomposition_eq _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f this\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x)) \u22a2 s =     JordanDecomposition.toSignedMeasure       (JordanDecomposition.mk ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x))         ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))         (_ :           ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u27c2\u2098             (toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x))) ** simp_rw [JordanDecomposition.toSignedMeasure, hadd] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f this\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x)) \u22a2 t + withDensity\u1d65 \u03bc f =     toSignedMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) -       toSignedMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)) ** ext i hi ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hf : Measurable f hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f this\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x)) i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(t + withDensity\u1d65 \u03bc f) i =     \u2191(toSignedMeasure ((toJordanDecomposition t).posPart + withDensity \u03bc fun x => ENNReal.ofReal (f x)) -           toSignedMeasure ((toJordanDecomposition t).negPart + withDensity \u03bc fun x => ENNReal.ofReal (-f x)))       i ** rw [VectorMeasure.sub_apply, toSignedMeasure_apply_measurable hi,\n    toSignedMeasure_apply_measurable hi, add_apply, add_apply, ENNReal.toReal_add,\n    ENNReal.toReal_add, add_sub_add_comm, \u2190 toSignedMeasure_apply_measurable hi,\n    \u2190 toSignedMeasure_apply_measurable hi, \u2190 VectorMeasure.sub_apply,\n    \u2190 JordanDecomposition.toSignedMeasure, toSignedMeasure_toJordanDecomposition,\n    VectorMeasure.add_apply, \u2190 toSignedMeasure_apply_measurable hi,\n    \u2190 toSignedMeasure_apply_measurable hi,\n    withDensity\u1d65_eq_withDensity_pos_part_sub_withDensity_neg_part hfi,\n    VectorMeasure.sub_apply] <;>\n  exact (measure_lt_top _ _).ne ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.ediv_add_emod' ** m k : Int \u22a2 m / k * k + m % k = m ** rw [Int.mul_comm] ** m k : Int \u22a2 k * (m / k) + m % k = m ** apply ediv_add_emod ** Qed",
        "informal": ""
    },
    {
        "formal": "ack_add_one_sq_lt_ack_add_four ** m n : \u2115 \u22a2 m \u2264 m + 2 ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.curry_inj ** c\u2081 c\u2082 : Code n\u2081 n\u2082 : \u2115 h : curry c\u2081 n\u2081 = curry c\u2082 n\u2082 \u22a2 c\u2081 = c\u2082 ** injection h ** c\u2081 c\u2082 : Code n\u2081 n\u2082 : \u2115 h : curry c\u2081 n\u2081 = curry c\u2082 n\u2082 \u22a2 n\u2081 = n\u2082 ** injection h with h\u2081 h\u2082 ** c\u2081 c\u2082 : Code n\u2081 n\u2082 : \u2115 h\u2081 : c\u2081 = c\u2082 h\u2082 : pair (Code.const n\u2081) Code.id = pair (Code.const n\u2082) Code.id \u22a2 n\u2081 = n\u2082 ** injection h\u2082 with h\u2083 h\u2084 ** c\u2081 c\u2082 : Code n\u2081 n\u2082 : \u2115 h\u2081 : c\u2081 = c\u2082 h\u2083 : Code.const n\u2081 = Code.const n\u2082 h\u2084 : Code.id = Code.id \u22a2 n\u2081 = n\u2082 ** exact const_inj h\u2083 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.restrict_iInf_restrict ** \u03b1 : Type u_1 \u03b9 : Sort u_2 s : Set \u03b1 m : \u03b9 \u2192 OuterMeasure \u03b1 \u22a2 \u2191(restrict s) (\u2a05 i, \u2191(restrict s) (m i)) = \u2191(restrict (range Subtype.val)) (\u2a05 i, \u2191(restrict s) (m i)) ** rw [Subtype.range_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.eq_rnDeriv ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t\u271d t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f \u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc ** set f' := hfi.1.mk f ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t\u271d t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f f' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) \u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc ** have hadd' : s = t + \u03bc.withDensity\u1d65 f' := by\n  convert hadd using 2\n  exact WithDensity\u1d65Eq.congr_ae hfi.1.ae_eq_mk.symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t\u271d t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f f' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) hadd' : s = t + withDensity\u1d65 \u03bc f' \u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc ** haveI := haveLebesgueDecomposition_mk \u03bc hfi.1.measurable_mk ht\u03bc hadd' ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t\u271d t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f f' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) hadd' : s = t + withDensity\u1d65 \u03bc f' this : HaveLebesgueDecomposition s \u03bc \u22a2 f =\u1da0[ae \u03bc] rnDeriv s \u03bc ** refine' (Integrable.ae_eq_of_withDensity\u1d65_eq (integrable_rnDeriv _ _) hfi _).symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t\u271d t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f f' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) hadd' : s = t + withDensity\u1d65 \u03bc f' this : HaveLebesgueDecomposition s \u03bc \u22a2 withDensity\u1d65 \u03bc (rnDeriv s \u03bc) = withDensity\u1d65 \u03bc f ** rw [\u2190 add_right_inj t, \u2190 hadd, eq_singularPart _ f ht\u03bc hadd,\n  singularPart_add_withDensity_rnDeriv_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t\u271d t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f f' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) \u22a2 s = t + withDensity\u1d65 \u03bc f' ** convert hadd using 2 ** case h.e'_3.h.e'_6 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s t\u271d t : SignedMeasure \u03b1 f : \u03b1 \u2192 \u211d hfi : Integrable f ht\u03bc : t \u27c2\u1d65 toENNRealVectorMeasure \u03bc hadd : s = t + withDensity\u1d65 \u03bc f f' : \u03b1 \u2192 \u211d := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) \u22a2 withDensity\u1d65 \u03bc f' = withDensity\u1d65 \u03bc f ** exact WithDensity\u1d65Eq.congr_ae hfi.1.ae_eq_mk.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_integral_pos_iff_support_of_nonneg_ae ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hfi : IntegrableOn f s \u22a2 0 < \u222b (x : \u03b1) in s, f x \u2202\u03bc \u2194 0 < \u2191\u2191\u03bc (support f \u2229 s) ** rw [integral_pos_iff_support_of_nonneg_ae hf hfi, Measure.restrict_apply\u2080] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hfi : IntegrableOn f s \u22a2 NullMeasurableSet (support f) ** rw [support_eq_preimage] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hfi : IntegrableOn f s \u22a2 NullMeasurableSet (f \u207b\u00b9' {0}\u1d9c) ** exact hfi.aestronglyMeasurable.aemeasurable.nullMeasurable (measurableSet_singleton 0).compl ** Qed",
        "informal": ""
    },
    {
        "formal": "String.contains_iff ** s : String c : Char \u22a2 contains s c = true \u2194 c \u2208 s.data ** simp [contains, any_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableEmbedding.map_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 \u22a2 \u2191\u2191(Measure.map f \u03bc) s = \u2191\u2191\u03bc (f \u207b\u00b9' s) ** refine' le_antisymm _ (le_map_apply hf.measurable.aemeasurable s) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 \u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s) ** set t := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 t : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c \u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s) ** have htm : MeasurableSet t :=\n  (hf.measurableSet_image.2 <| measurableSet_toMeasurable _ _).union\n    hf.measurableSet_range.compl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 t : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c htm : MeasurableSet t \u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s) ** have hst : s \u2286 t := by\n  rw [subset_union_compl_iff_inter_subset, \u2190 image_preimage_eq_inter_range]\n  exact image_subset _ (subset_toMeasurable _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 t : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c htm : MeasurableSet t hst : s \u2286 t \u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s) ** have hft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s) := by\n  rw [preimage_union, preimage_compl, preimage_range, compl_univ, union_empty,\n    hf.injective.preimage_image] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 t : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c htm : MeasurableSet t hst : s \u2286 t hft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s) \u22a2 \u2191\u2191(Measure.map f \u03bc) s \u2264 \u2191\u2191\u03bc (f \u207b\u00b9' s) ** calc\n  \u03bc.map f s \u2264 \u03bc.map f t := measure_mono hst\n  _ = \u03bc (f \u207b\u00b9' s) := by rw [map_apply hf.measurable htm, hft, measure_toMeasurable] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 t : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c htm : MeasurableSet t \u22a2 s \u2286 t ** rw [subset_union_compl_iff_inter_subset, \u2190 image_preimage_eq_inter_range] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 t : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c htm : MeasurableSet t \u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' toMeasurable \u03bc (f \u207b\u00b9' s) ** exact image_subset _ (subset_toMeasurable _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 t : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c htm : MeasurableSet t hst : s \u2286 t \u22a2 f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s) ** rw [preimage_union, preimage_compl, preimage_range, compl_univ, union_empty,\n  hf.injective.preimage_image] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 m1 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 hf : MeasurableEmbedding f \u03bc : Measure \u03b1 s : Set \u03b2 t : Set \u03b2 := f '' toMeasurable \u03bc (f \u207b\u00b9' s) \u222a (range f)\u1d9c htm : MeasurableSet t hst : s \u2286 t hft : f \u207b\u00b9' t = toMeasurable \u03bc (f \u207b\u00b9' s) \u22a2 \u2191\u2191(Measure.map f \u03bc) t = \u2191\u2191\u03bc (f \u207b\u00b9' s) ** rw [map_apply hf.measurable htm, hft, measure_toMeasurable] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.comap_iInf ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 \u22a2 \u2191(comap f) (\u2a05 i, m i) = \u2a05 i, \u2191(comap f) (m i) ** refine' ext_nonempty fun s hs => _ ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b1 hs : Set.Nonempty s \u22a2 \u2191(\u2191(comap f) (\u2a05 i, m i)) s = \u2191(\u2a05 i, \u2191(comap f) (m i)) s ** refine' ((comap_mono f).map_iInf_le s).antisymm _ ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b1 hs : Set.Nonempty s \u22a2 \u2191(\u2a05 i, \u2191(comap f) (m i)) s \u2264 \u2191(\u2191(comap f) (\u2a05 i, m i)) s ** simp only [comap_apply, iInf_apply' _ hs, iInf_apply' _ (hs.image _), le_iInf_iff,\n  Set.image_subset_iff, preimage_iUnion] ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b1 hs : Set.Nonempty s \u22a2 \u2200 (i : \u2115 \u2192 Set \u03b2),     s \u2286 \u22c3 i_1, f \u207b\u00b9' i i_1 \u2192       \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' t n) \u2264 \u2211' (n : \u2115), \u2a05 i_2, \u2191(m i_2) (i n) ** refine' fun t ht => iInf_le_of_le _ (iInf_le_of_le ht <| ENNReal.tsum_le_tsum fun k => _) ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b1 hs : Set.Nonempty s t : \u2115 \u2192 Set \u03b2 ht : s \u2286 \u22c3 i, f \u207b\u00b9' t i k : \u2115 \u22a2 \u2a05 i, \u2191(m i) (f '' (f \u207b\u00b9' t k)) \u2264 \u2a05 i, \u2191(m i) (t k) ** exact iInf_mono fun i => (m i).mono (image_preimage_subset _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec'.part_iff\u2081 ** f : \u2115 \u2192. \u2115 h : _root_.Partrec fun v => f (Vector.head v) v : \u2115 \u22a2 f (Vector.head (ofFn fun i => id v)) = f v ** simp only [id.def, head_ofFn] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.le_smul_caratheodory ** \u03b1 : Type u_1 a : \u211d\u22650\u221e m : OuterMeasure \u03b1 s : Set \u03b1 h : MeasurableSet s t : Set \u03b1 \u22a2 \u2191(a \u2022 m) t = \u2191(a \u2022 m) (t \u2229 s) + \u2191(a \u2022 m) (t \\ s) ** simp only [smul_apply, smul_eq_mul] ** \u03b1 : Type u_1 a : \u211d\u22650\u221e m : OuterMeasure \u03b1 s : Set \u03b1 h : MeasurableSet s t : Set \u03b1 \u22a2 a * \u2191m t = a * \u2191m (t \u2229 s) + a * \u2191m (t \\ s) ** rw [(isCaratheodory_iff m).mp h t] ** \u03b1 : Type u_1 a : \u211d\u22650\u221e m : OuterMeasure \u03b1 s : Set \u03b1 h : MeasurableSet s t : Set \u03b1 \u22a2 a * (\u2191m (t \u2229 s) + \u2191m (t \\ s)) = a * \u2191m (t \u2229 s) + a * \u2191m (t \\ s) ** simp [mul_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.map_le_restrict_range ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 R : Type u_3 R' : Type u_4 ms : Set (OuterMeasure \u03b1) m : OuterMeasure \u03b1 \u03b2 : Type u_5 ma : OuterMeasure \u03b1 mb : OuterMeasure \u03b2 f : \u03b1 \u2192 \u03b2 h : \u2191(map f) ma \u2264 mb s : Set \u03b2 \u22a2 \u2191(\u2191(map f) ma) s \u2264 \u2191(\u2191(restrict (range f)) mb) s ** simpa using h (s \u2229 range f) ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.exists_unique_equiv_nat ** m n a\u271d b\u271d c d a b : \u2124 hb : 0 < b z : \u2124 hz1 : 0 \u2264 z hz2 : z < b hz3 : z \u2261 a [ZMOD b] \u22a2 \u2191(natAbs z) < b \u2227 \u2191(natAbs z) \u2261 a [ZMOD b] ** constructor <;> rw [ofNat_natAbs_eq_of_nonneg z hz1] <;> assumption ** Qed",
        "informal": ""
    },
    {
        "formal": "Setoid.card_classes_ker_le ** \u03b1\u271d : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : Fintype \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Fintype \u2191(classes (ker f)) \u22a2 Fintype.card \u2191(classes (ker f)) \u2264 Fintype.card \u03b2 ** classical exact\n    le_trans (Set.card_le_of_subset (classes_ker_subset_fiber_set f)) (Fintype.card_range_le _) ** \u03b1\u271d : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : Fintype \u03b2 f : \u03b1 \u2192 \u03b2 inst\u271d : Fintype \u2191(classes (ker f)) \u22a2 Fintype.card \u2191(classes (ker f)) \u2264 Fintype.card \u03b2 ** exact\nle_trans (Set.card_le_of_subset (classes_ker_subset_fiber_set f)) (Fintype.card_range_le _) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsFundamentalDomain.aEStronglyMeasurable_on_iff ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2\u271d : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2\u271d inst\u271d\u2077 : MeasurableSpace \u03b2\u271d inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 \u03b2 : Type u_6 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 \u03b2 hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x \u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s) \u2194 AEStronglyMeasurable f (sum fun g => Measure.restrict \u03bc (g \u2022 t \u2229 s)) ** simp only [\u2190 ht.restrict_restrict,\n  ht.sum_restrict_of_ac restrict_le_self.absolutelyContinuous] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2\u271d : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2\u271d inst\u271d\u2077 : MeasurableSpace \u03b2\u271d inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 \u03b2 : Type u_6 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 \u03b2 hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x \u22a2 AEStronglyMeasurable f (sum fun g => Measure.restrict \u03bc (g \u2022 t \u2229 s)) \u2194     \u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 (g\u207b\u00b9 \u2022 s \u2229 t))) ** simp only [smul_set_inter, inter_comm, smul_inv_smul, aestronglyMeasurable_sum_measure_iff] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2\u271d : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2\u271d inst\u271d\u2077 : MeasurableSpace \u03b2\u271d inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 \u03b2 : Type u_6 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 \u03b2 hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x \u22a2 (\u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g\u207b\u00b9\u207b\u00b9 \u2022 s \u2229 t)))) \u2194     \u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t))) ** simp only [inv_inv] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2\u271d : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2\u271d inst\u271d\u2077 : MeasurableSpace \u03b2\u271d inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 \u03b2 : Type u_6 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 \u03b2 hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x \u22a2 (\u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t)))) \u2194     \u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t)) ** refine' forall_congr' fun g => _ ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2\u271d : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2\u271d inst\u271d\u2077 : MeasurableSpace \u03b2\u271d inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 \u03b2 : Type u_6 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 \u03b2 hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x g : G he : MeasurableEmbedding ((fun x x_1 => x \u2022 x_1) g\u207b\u00b9) \u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc (g\u207b\u00b9 \u2022 (g \u2022 s \u2229 t))) \u2194     AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t)) ** rw [\u2190 image_smul, \u2190 ((measurePreserving_smul g\u207b\u00b9 \u03bc).restrict_image_emb he\n  _).aestronglyMeasurable_comp_iff he] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2\u271d : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2\u271d inst\u271d\u2077 : MeasurableSpace \u03b2\u271d inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 \u03b2 : Type u_6 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t f : \u03b1 \u2192 \u03b2 hf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x \u22a2 (\u2200 (g : G), AEStronglyMeasurable f (Measure.restrict \u03bc (g \u2022 s \u2229 t))) \u2194 AEStronglyMeasurable f (Measure.restrict \u03bc t) ** simp only [\u2190 aestronglyMeasurable_sum_measure_iff, \u2190 hs.restrict_restrict,\n  hs.sum_restrict_of_ac restrict_le_self.absolutelyContinuous] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_ball ** E : Type u_1 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E inst\u271d\u2077 : MeasurableSpace E inst\u271d\u2076 : BorelSpace E inst\u271d\u2075 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u2074 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F s : Set E inst\u271d : Nontrivial E x : E r : \u211d hr : 0 \u2264 r \u22a2 \u2191\u2191\u03bc (ball x r) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 1) ** rw [\u2190 addHaar_ball_mul \u03bc x hr, mul_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.IdentDistrib.snorm_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e \u22a2 snorm f p \u03bc = snorm g p \u03bd ** by_cases h0 : p = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e h0 : \u00acp = 0 \u22a2 snorm f p \u03bc = snorm g p \u03bd ** by_cases h_top : p = \u221e ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e h0 : \u00acp = 0 h_top : \u00acp = \u22a4 \u22a2 snorm f p \u03bc = snorm g p \u03bd ** simp only [snorm_eq_snorm' h0 h_top, snorm', one_div] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e h0 : \u00acp = 0 h_top : \u00acp = \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9 =     (\u222b\u207b (a : \u03b2), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bd) ^ (ENNReal.toReal p)\u207b\u00b9 ** congr 1 ** case neg.e_a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e h0 : \u00acp = 0 h_top : \u00acp = \u22a4 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc = \u222b\u207b (a : \u03b2), \u2191\u2016g a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bd ** apply lintegral_eq ** case neg.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e h0 : \u00acp = 0 h_top : \u00acp = \u22a4 \u22a2 IdentDistrib (fun x => \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p) fun x => \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p ** exact h.comp (Measurable.pow_const (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)\n  p.toReal) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e h0 : p = 0 \u22a2 snorm f p \u03bc = snorm g p \u03bd ** simp [h0] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e h0 : \u00acp = 0 h_top : p = \u22a4 \u22a2 snorm f p \u03bc = snorm g p \u03bd ** simp only [h_top, snorm, snormEssSup, ENNReal.top_ne_zero, eq_self_iff_true, if_true, if_false] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e h0 : \u00acp = 0 h_top : p = \u22a4 \u22a2 essSup (fun x => \u2191\u2016f x\u2016\u208a) \u03bc = essSup (fun x => \u2191\u2016g x\u2016\u208a) \u03bd ** apply essSup_eq ** case pos.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bc : Measure \u03b1 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : OpensMeasurableSpace \u03b3 h : IdentDistrib f g p : \u211d\u22650\u221e h0 : \u00acp = 0 h_top : p = \u22a4 \u22a2 IdentDistrib (fun x => \u2191\u2016f x\u2016\u208a) fun x => \u2191\u2016g x\u2016\u208a ** exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.coeFn_compMeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : TopologicalSpace \u03b3 inst\u271d\u2077 : TopologicalSpace \u03b4 inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : PseudoMetrizableSpace \u03b2 inst\u271d\u2074 : BorelSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b3 inst\u271d\u00b9 : OpensMeasurableSpace \u03b3 inst\u271d : SecondCountableTopology \u03b3 g : \u03b2 \u2192 \u03b3 hg : Measurable g f : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 \u2191(compMeasurable g hg f) =\u1d50[\u03bc] g \u2218 \u2191f ** rw [compMeasurable_eq_mk] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : TopologicalSpace \u03b3 inst\u271d\u2077 : TopologicalSpace \u03b4 inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : PseudoMetrizableSpace \u03b2 inst\u271d\u2074 : BorelSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : PseudoMetrizableSpace \u03b3 inst\u271d\u00b9 : OpensMeasurableSpace \u03b3 inst\u271d : SecondCountableTopology \u03b3 g : \u03b2 \u2192 \u03b3 hg : Measurable g f : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 \u2191(mk (g \u2218 \u2191f) (_ : AEStronglyMeasurable (g \u2218 \u2191f) \u03bc)) =\u1d50[\u03bc] g \u2218 \u2191f ** apply coeFn_mk ** Qed",
        "informal": ""
    },
    {
        "formal": "blimsup_cthickening_ae_le_of_eventually_mul_le_aux ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i \u22a2 blimsup (fun i => cthickening (r\u2081 i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (r\u2082 i) (s i)) atTop p ** set Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) \u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup (fun i => cthickening (r\u2082 i) (s i)) atTop p ** set Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) \u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup Y\u2082 atTop p ** let Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 (j) (_ : p j \u2227 i \u2264 j), Y\u2082 j ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j \u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup Y\u2082 atTop p ** suffices \u2200 i, \u03bc (atTop.blimsup Y\u2081 p \\ Z i) = 0 by\n  rwa [ae_le_set, @blimsup_eq_iInf_biSup_of_nat _ _ _ Y\u2082, iInf_eq_iInter, diff_iInter,\n    measure_iUnion_null_iff] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j \u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bc (blimsup Y\u2081 atTop p \\ Z i) = 0 ** intros i ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 \u22a2 \u2191\u2191\u03bc (blimsup Y\u2081 atTop p \\ Z i) = 0 ** set W := atTop.blimsup Y\u2081 p \\ Z i ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i \u22a2 \u2191\u2191\u03bc W = 0 ** by_contra contra ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 \u22a2 False ** obtain \u27e8d, hd, hd'\u27e9 : \u2203 d, d \u2208 W \u2227 \u2200 {\u03b9 : Type _} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),\n    Tendsto \u03b4 l (\ud835\udcdd[>] 0) \u2192 (\u2200\u1da0 j in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192\n      Tendsto (fun j => \u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) :=\n  Measure.exists_mem_of_measure_ne_zero_of_ae contra\n    (IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div \u03bc W 2) ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd : d \u2208 W hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) \u22a2 False ** replace hd : d \u2208 blimsup Y\u2081 atTop p := ((mem_diff _).mp hd).1 ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p \u22a2 False ** obtain \u27e8f : \u2115 \u2192 \u2115, hf\u27e9 := exists_forall_mem_of_hasBasis_mem_blimsup' atTop_basis hd ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf : \u2200 (i : \u2115), d \u2208 Y\u2081 (f i) \u2227 p (f i) \u2227 f i \u2208 Ici i \u22a2 False ** simp only [forall_and] at hf ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf : (\u2200 (x : \u2115), d \u2208 cthickening (r\u2081 (f x)) (s (f x))) \u2227 (\u2200 (x : \u2115), p (f x)) \u2227 \u2200 (x : \u2115), f x \u2208 Ici x \u22a2 False ** obtain \u27e8hf\u2080 : \u2200 j, d \u2208 cthickening (r\u2081 (f j)) (s (f j)), hf\u2081, hf\u2082 : \u2200 j, j \u2264 f j\u27e9 := hf ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j \u22a2 False ** have hf\u2083 : Tendsto f atTop atTop :=\n  tendsto_atTop_atTop.mpr fun j => \u27e8f j, fun i hi => (hf\u2082 j).trans (hi.trans <| hf\u2082 i)\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop \u22a2 False ** replace hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[>] 0) := hr.comp hf\u2083 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) \u22a2 False ** replace hMr : \u2200\u1da0 j in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) := hf\u2083.eventually hMr ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) \u22a2 False ** replace hf\u2080 : \u2200 j, \u2203 w \u2208 s (f j), d \u2208 closedBall w (2 * r\u2081 (f j)) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) hf\u2080 : \u2200 (j : \u2115), \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j)) \u22a2 False ** choose w hw hw' using hf\u2080 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) \u22a2 False ** let C := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 \u22a2 False ** have hC : 0 < C :=\n  lt_of_lt_of_le zero_lt_one (IsUnifLocDoublingMeasure.one_le_scalingConstantOf \u03bc M\u207b\u00b9) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : 0 < C \u22a2 False ** suffices \u2203 \u03b7 < (1 : \u211d\u22650),\n    \u2200\u1da0 j in atTop, \u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u03b7 by\n  obtain \u27e8\u03b7, h\u03b7, h\u03b7'\u27e9 := this\n  replace h\u03b7' : 1 \u2264 \u03b7 := by\n    simpa only [ENNReal.one_le_coe_iff] using\n      le_of_tendsto (hd' w (fun j => r\u2081 (f j)) hr <| eventually_of_forall hw') h\u03b7'\n  exact (lt_self_iff_false _).mp (lt_of_lt_of_le h\u03b7 h\u03b7') ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : 0 < C \u22a2 \u2203 \u03b7, \u03b7 < 1 \u2227 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7 ** refine' \u27e81 - C\u207b\u00b9, tsub_lt_self zero_lt_one (inv_pos.mpr hC), _\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : 0 < C \u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9) ** replace hC : C \u2260 0 := ne_of_gt hC ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 \u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9) ** let b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) \u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9) ** let B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) \u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9) ** have h\u2081 : \u2200 j, b j \u2286 B j := fun j =>\n  closedBall_subset_closedBall (mul_le_of_le_one_left (hrp (f j)) hM'.le) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j \u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9) ** have h\u2082 : \u2200 j, W \u2229 B j \u2286 B j := fun j => inter_subset_right W (B j) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j \u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9) ** have h\u2083 : \u2200\u1da0 j in atTop, Disjoint (b j) (W \u2229 B j) := by\n  apply hMr.mp\n  rw [eventually_atTop]\n  refine'\n    \u27e8i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y\u2081 atTop p) _\u27e9\n  change Disjoint (b j) (Z i)\u1d9c\n  rw [disjoint_compl_right_iff_subset]\n  refine' (closedBall_subset_cthickening (hw j) (M * r\u2081 (f j))).trans\n    ((cthickening_mono hj' _).trans fun a ha => _)\n  simp only [mem_iUnion, exists_prop]\n  exact \u27e8f j, \u27e8hf\u2081 j, hj.le.trans (hf\u2082 j)\u27e9, ha\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) \u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9) ** have h\u2084 : \u2200\u1da0 j in atTop, \u03bc (B j) \u2264 C * \u03bc (b j) :=\n  (hr.eventually (IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul'\n    \u03bc M hM)).mono fun j hj => hj (w j) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) \u22a2 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9) ** refine' (h\u2083.and h\u2084).mono fun j hj\u2080 => _ ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) \u22a2 \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191(1 - C\u207b\u00b9) ** change \u03bc (W \u2229 B j) / \u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) \u22a2 \u2191\u2191\u03bc (W \u2229 B j) / \u2191\u2191\u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9) ** rcases eq_or_ne (\u03bc (B j)) \u221e with (hB | hB) ** case intro.intro.intro.intro.intro.inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) hB : \u2191\u2191\u03bc (B j) \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc (W \u2229 B j) / \u2191\u2191\u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9) ** apply ENNReal.div_le_of_le_mul ** case intro.intro.intro.intro.intro.inr.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) hB : \u2191\u2191\u03bc (B j) \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191(1 - C\u207b\u00b9) * \u2191\u2191\u03bc (B j) ** rw [ENNReal.coe_sub, ENNReal.coe_one, ENNReal.sub_mul fun _ _ => hB, one_mul] ** case intro.intro.intro.intro.intro.inr.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) hB : \u2191\u2191\u03bc (B j) \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) ** replace hB : \u2191C\u207b\u00b9 * \u03bc (B j) \u2260 \u221e ** case intro.intro.intro.intro.intro.inr.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) ** obtain \u27e8hj\u2081 : Disjoint (b j) (W \u2229 B j), hj\u2082 : \u03bc (B j) \u2264 C * \u03bc (b j)\u27e9 := hj\u2080 ** case intro.intro.intro.intro.intro.inr.h.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 hj\u2081 : Disjoint (b j) (W \u2229 B j) hj\u2082 : \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) \u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) ** replace hj\u2082 : \u2191C\u207b\u00b9 * \u03bc (B j) \u2264 \u03bc (b j) ** case intro.intro.intro.intro.intro.inr.h.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 hj\u2081 : Disjoint (b j) (W \u2229 B j) hj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j) \u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) ** have hj\u2083 : \u2191C\u207b\u00b9 * \u03bc (B j) + \u03bc (W \u2229 B j) \u2264 \u03bc (B j) := by\n  refine' le_trans (add_le_add_right hj\u2082 _) _\n  rw [\u2190 measure_union' hj\u2081 measurableSet_closedBall]\n  exact measure_mono (union_subset (h\u2081 j) (h\u2082 j)) ** case intro.intro.intro.intro.intro.inr.h.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 hj\u2081 : Disjoint (b j) (W \u2229 B j) hj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j) hj\u2083 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) + \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) \u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) ** replace hj\u2083 := tsub_le_tsub_right hj\u2083 (\u2191C\u207b\u00b9 * \u03bc (B j)) ** case intro.intro.intro.intro.intro.inr.h.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 hj\u2081 : Disjoint (b j) (W \u2229 B j) hj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j) hj\u2083 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) + \u2191\u2191\u03bc (W \u2229 B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u22a2 \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) - \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) ** rwa [ENNReal.add_sub_cancel_left hB] at hj\u2083 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hr : Tendsto r\u2081 atTop (\ud835\udcdd[Ioi 0] 0) hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 hMr : \u2200\u1da0 (i : \u2115) in atTop, M * r\u2081 i \u2264 r\u2082 i Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j this : \u2200 (i : \u2115), \u2191\u2191\u03bc (blimsup Y\u2081 atTop p \\ Z i) = 0 \u22a2 blimsup Y\u2081 atTop p \u2264\u1d50[\u03bc] blimsup Y\u2082 atTop p ** rwa [ae_le_set, @blimsup_eq_iInf_biSup_of_nat _ _ _ Y\u2082, iInf_eq_iInter, diff_iInter,\n  measure_iUnion_null_iff] ** case hf\u2080 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) \u22a2 \u2200 (j : \u2115), \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j)) ** intro j ** case hf\u2080 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) j : \u2115 \u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j)) ** specialize hrp (f j) ** case hf\u2080 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) j : \u2115 hrp : OfNat.ofNat 0 (f j) \u2264 r\u2081 (f j) \u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j)) ** rw [Pi.zero_apply] at hrp ** case hf\u2080 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) j : \u2115 hrp : 0 \u2264 r\u2081 (f j) \u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j)) ** rcases eq_or_lt_of_le hrp with (hr0 | hrp') ** case hf\u2080.inl \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) j : \u2115 hrp : 0 \u2264 r\u2081 (f j) hr0 : 0 = r\u2081 (f j) \u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j)) ** specialize hf\u2080 j ** case hf\u2080.inl \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) j : \u2115 hrp : 0 \u2264 r\u2081 (f j) hr0 : 0 = r\u2081 (f j) hf\u2080 : d \u2208 cthickening (r\u2081 (f j)) (s (f j)) \u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j)) ** rw [\u2190 hr0, cthickening_zero, (hs (f j)).closure_eq] at hf\u2080 ** case hf\u2080.inl \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) j : \u2115 hrp : 0 \u2264 r\u2081 (f j) hr0 : 0 = r\u2081 (f j) hf\u2080 : d \u2208 s (f j) \u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j)) ** exact \u27e8d, hf\u2080, by simp [\u2190 hr0]\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) j : \u2115 hrp : 0 \u2264 r\u2081 (f j) hr0 : 0 = r\u2081 (f j) hf\u2080 : d \u2208 s (f j) \u22a2 d \u2208 closedBall d (2 * r\u2081 (f j)) ** simp [\u2190 hr0] ** case hf\u2080.inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) j : \u2115 hrp : 0 \u2264 r\u2081 (f j) hrp' : 0 < r\u2081 (f j) \u22a2 \u2203 w, w \u2208 s (f j) \u2227 d \u2208 closedBall w (2 * r\u2081 (f j)) ** simpa using mem_iUnion\u2082.mp (cthickening_subset_iUnion_closedBall_of_lt (s (f j))\n  (by positivity) (lt_two_mul_self hrp') (hf\u2080 j)) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2080 : \u2200 (j : \u2115), d \u2208 cthickening (r\u2081 (f j)) (s (f j)) hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) j : \u2115 hrp : 0 \u2264 r\u2081 (f j) hrp' : 0 < r\u2081 (f j) \u22a2 0 < 2 * r\u2081 (f j) ** positivity ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : 0 < C this : \u2203 \u03b7, \u03b7 < 1 \u2227 \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7 \u22a2 False ** obtain \u27e8\u03b7, h\u03b7, h\u03b7'\u27e9 := this ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : 0 < C \u03b7 : \u211d\u22650 h\u03b7 : \u03b7 < 1 h\u03b7' : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7 \u22a2 False ** replace h\u03b7' : 1 \u2264 \u03b7 := by\n  simpa only [ENNReal.one_le_coe_iff] using\n    le_of_tendsto (hd' w (fun j => r\u2081 (f j)) hr <| eventually_of_forall hw') h\u03b7' ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : 0 < C \u03b7 : \u211d\u22650 h\u03b7 : \u03b7 < 1 h\u03b7' : 1 \u2264 \u03b7 \u22a2 False ** exact (lt_self_iff_false _).mp (lt_of_lt_of_le h\u03b7 h\u03b7') ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type ?u.5949} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : 0 < C \u03b7 : \u211d\u22650 h\u03b7 : \u03b7 < 1 h\u03b7' : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (W \u2229 closedBall (w j) (r\u2081 (f j))) / \u2191\u2191\u03bc (closedBall (w j) (r\u2081 (f j))) \u2264 \u2191\u03b7 \u22a2 1 \u2264 \u03b7 ** simpa only [ENNReal.one_le_coe_iff] using\n  le_of_tendsto (hd' w (fun j => r\u2081 (f j)) hr <| eventually_of_forall hw') h\u03b7' ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j \u22a2 \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) ** apply hMr.mp ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j \u22a2 \u2200\u1da0 (x : \u2115) in atTop, M * r\u2081 (f x) \u2264 r\u2082 (f x) \u2192 Disjoint (b x) (W \u2229 B x) ** rw [eventually_atTop] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j \u22a2 \u2203 a, \u2200 (b_1 : \u2115), b_1 \u2265 a \u2192 M * r\u2081 (f b_1) \u2264 r\u2082 (f b_1) \u2192 Disjoint (b b_1) (W \u2229 B b_1) ** refine'\n  \u27e8i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y\u2081 atTop p) _\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j j : \u2115 hj : j \u2265 i hj' : M * r\u2081 (f j) \u2264 r\u2082 (f j) \u22a2 Disjoint (b j) fun a => a \u2208 Z i \u2192 False ** change Disjoint (b j) (Z i)\u1d9c ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j j : \u2115 hj : j \u2265 i hj' : M * r\u2081 (f j) \u2264 r\u2082 (f j) \u22a2 Disjoint (b j) (Z i)\u1d9c ** rw [disjoint_compl_right_iff_subset] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j j : \u2115 hj : j \u2265 i hj' : M * r\u2081 (f j) \u2264 r\u2082 (f j) \u22a2 b j \u2286 Z i ** refine' (closedBall_subset_cthickening (hw j) (M * r\u2081 (f j))).trans\n  ((cthickening_mono hj' _).trans fun a ha => _) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j j : \u2115 hj : j \u2265 i hj' : M * r\u2081 (f j) \u2264 r\u2082 (f j) a : \u03b1 ha : a \u2208 cthickening (r\u2082 (f j)) (s (f j)) \u22a2 a \u2208 Z i ** simp only [mem_iUnion, exists_prop] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j j : \u2115 hj : j \u2265 i hj' : M * r\u2081 (f j) \u2264 r\u2082 (f j) a : \u03b1 ha : a \u2208 cthickening (r\u2082 (f j)) (s (f j)) \u22a2 \u2203 i_1, (p i_1 \u2227 i \u2264 i_1) \u2227 a \u2208 cthickening (r\u2082 i_1) (s i_1) ** exact \u27e8f j, \u27e8hf\u2081 j, hj.le.trans (hf\u2082 j)\u27e9, ha\u27e9 ** case intro.intro.intro.intro.intro.inl \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) hB : \u2191\u2191\u03bc (B j) = \u22a4 \u22a2 \u2191\u2191\u03bc (W \u2229 B j) / \u2191\u2191\u03bc (B j) \u2264 \u2191(1 - C\u207b\u00b9) ** simp [hB] ** case hB \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) hB : \u2191\u2191\u03bc (B j) \u2260 \u22a4 \u22a2 \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 ** refine' ENNReal.mul_ne_top _ hB ** case hB \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hj\u2080 : Disjoint (b j) (W \u2229 B j) \u2227 \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) hB : \u2191\u2191\u03bc (B j) \u2260 \u22a4 \u22a2 \u2191C\u207b\u00b9 \u2260 \u22a4 ** rwa [ENNReal.coe_inv hC, Ne.def, ENNReal.inv_eq_top, ENNReal.coe_eq_zero] ** case hj\u2082 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 hj\u2081 : Disjoint (b j) (W \u2229 B j) hj\u2082 : \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) \u22a2 \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j) ** rw [ENNReal.coe_inv hC, \u2190 ENNReal.div_eq_inv_mul] ** case hj\u2082 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 hj\u2081 : Disjoint (b j) (W \u2229 B j) hj\u2082 : \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) \u22a2 \u2191\u2191\u03bc (B j) / \u2191C \u2264 \u2191\u2191\u03bc (b j) ** exact ENNReal.div_le_of_le_mul' hj\u2082 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 hj\u2081 : Disjoint (b j) (W \u2229 B j) hj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j) \u22a2 \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) + \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) ** refine' le_trans (add_le_add_right hj\u2082 _) _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 hj\u2081 : Disjoint (b j) (W \u2229 B j) hj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j) \u22a2 \u2191\u2191\u03bc (b j) + \u2191\u2191\u03bc (W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) ** rw [\u2190 measure_union' hj\u2081 measurableSet_closedBall] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : SecondCountableTopology \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsUnifLocDoublingMeasure \u03bc p : \u2115 \u2192 Prop s : \u2115 \u2192 Set \u03b1 hs : \u2200 (i : \u2115), IsClosed (s i) r\u2081 r\u2082 : \u2115 \u2192 \u211d hrp : 0 \u2264 r\u2081 M : \u211d hM : 0 < M hM' : M < 1 Y\u2081 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2081 i) (s i) Y\u2082 : \u2115 \u2192 Set \u03b1 := fun i => cthickening (r\u2082 i) (s i) Z : \u2115 \u2192 Set \u03b1 := fun i => \u22c3 j, \u22c3 (_ : p j \u2227 i \u2264 j), Y\u2082 j i : \u2115 W : Set \u03b1 := blimsup Y\u2081 atTop p \\ Z i contra : \u00ac\u2191\u2191\u03bc W = 0 d : \u03b1 hd' :   \u2200 {\u03b9 : Type} {l : Filter \u03b9} (w : \u03b9 \u2192 \u03b1) (\u03b4 : \u03b9 \u2192 \u211d),     Tendsto \u03b4 l (\ud835\udcdd[Ioi 0] 0) \u2192       (\u2200\u1da0 (j : \u03b9) in l, d \u2208 closedBall (w j) (2 * \u03b4 j)) \u2192         Tendsto (fun j => \u2191\u2191\u03bc (W \u2229 closedBall (w j) (\u03b4 j)) / \u2191\u2191\u03bc (closedBall (w j) (\u03b4 j))) l (\ud835\udcdd 1) hd : d \u2208 blimsup Y\u2081 atTop p f : \u2115 \u2192 \u2115 hf\u2081 : \u2200 (x : \u2115), p (f x) hf\u2082 : \u2200 (j : \u2115), j \u2264 f j hf\u2083 : Tendsto f atTop atTop hr : Tendsto (r\u2081 \u2218 f) atTop (\ud835\udcdd[Ioi 0] 0) hMr : \u2200\u1da0 (j : \u2115) in atTop, M * r\u2081 (f j) \u2264 r\u2082 (f j) w : \u2115 \u2192 \u03b1 hw : \u2200 (j : \u2115), w j \u2208 s (f j) hw' : \u2200 (j : \u2115), d \u2208 closedBall (w j) (2 * r\u2081 (f j)) C : \u211d\u22650 := IsUnifLocDoublingMeasure.scalingConstantOf \u03bc M\u207b\u00b9 hC : C \u2260 0 b : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (M * r\u2081 (f j)) B : \u2115 \u2192 Set \u03b1 := fun j => closedBall (w j) (r\u2081 (f j)) h\u2081 : \u2200 (j : \u2115), b j \u2286 B j h\u2082 : \u2200 (j : \u2115), W \u2229 B j \u2286 B j h\u2083 : \u2200\u1da0 (j : \u2115) in atTop, Disjoint (b j) (W \u2229 B j) h\u2084 : \u2200\u1da0 (j : \u2115) in atTop, \u2191\u2191\u03bc (B j) \u2264 \u2191C * \u2191\u2191\u03bc (b j) j : \u2115 hB : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2260 \u22a4 hj\u2081 : Disjoint (b j) (W \u2229 B j) hj\u2082 : \u2191C\u207b\u00b9 * \u2191\u2191\u03bc (B j) \u2264 \u2191\u2191\u03bc (b j) \u22a2 \u2191\u2191\u03bc (b j \u222a W \u2229 B j) \u2264 \u2191\u2191\u03bc (B j) ** exact measure_mono (union_subset (h\u2081 j) (h\u2082 j)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.smulInvariantMeasure_tfae ** G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 \u22a2 List.TFAE     [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x] ** tfae_have 1 \u2194 2 ** G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s \u22a2 List.TFAE     [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x] ** tfae_have 1 \u2192 6 ** G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc \u22a2 List.TFAE     [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x] ** tfae_have 6 \u2192 7 ** G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x \u22a2 List.TFAE     [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x] ** tfae_have 7 \u2192 4 ** G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s \u22a2 List.TFAE     [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x] ** tfae_have 4 \u2192 5 ** G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s \u22a2 List.TFAE     [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x] ** tfae_have 5 \u2192 3 ** G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s tfae_5_to_3 :   (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s \u22a2 List.TFAE     [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x] ** tfae_have 3 \u2192 2 ** G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s tfae_5_to_3 :   (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s tfae_3_to_2 :   (\u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192     \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s \u22a2 List.TFAE     [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,       \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x] ** tfae_finish ** case tfae_1_iff_2 G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 \u22a2 SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s ** exact \u27e8fun h => h.1, fun h => \u27e8h\u27e9\u27e9 ** case tfae_1_to_6 G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s \u22a2 SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc ** intro h c ** case tfae_1_to_6 G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c\u271d : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s h : SMulInvariantMeasure G \u03b1 \u03bc c : G \u22a2 map (fun x => c \u2022 x) \u03bc = \u03bc ** exact (measurePreserving_smul c \u03bc).map_eq ** case tfae_6_to_7 G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc \u22a2 (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x ** exact fun H c => \u27e8measurable_const_smul c, H c\u27e9 ** case tfae_7_to_4 G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x \u22a2 (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s ** exact fun H c => (H c).measure_preimage_emb (measurableEmbedding_const_smul c) ** case tfae_4_to_5 G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s \u22a2 (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s ** exact fun H c s => by\n  rw [\u2190 preimage_smul_inv]\n  apply H ** G : Type u M : Type v \u03b1 : Type w s\u271d : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c\u271d : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s H : \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s c : G s : Set \u03b1 \u22a2 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s ** rw [\u2190 preimage_smul_inv] ** G : Type u M : Type v \u03b1 : Type w s\u271d : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c\u271d : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s H : \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s c : G s : Set \u03b1 \u22a2 \u2191\u2191\u03bc ((fun x => c\u207b\u00b9 \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s ** apply H ** case tfae_5_to_3 G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s \u22a2 (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s ** exact fun H c s _ => H c s ** case tfae_3_to_2 G : Type u M : Type v \u03b1 : Type w s : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s tfae_5_to_3 :   (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s \u22a2 (\u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192     \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s ** intro H c s hs ** case tfae_3_to_2 G : Type u M : Type v \u03b1 : Type w s\u271d : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c\u271d : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s tfae_5_to_3 :   (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s H : \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s c : G s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s ** rw [preimage_smul] ** case tfae_3_to_2 G : Type u M : Type v \u03b1 : Type w s\u271d : Set \u03b1 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSpace G inst\u271d : MeasurableSMul G \u03b1 c\u271d : G \u03bc : Measure \u03b1 tfae_1_iff_2 :   SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc tfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x tfae_7_to_4 :   (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s tfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s tfae_5_to_3 :   (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s H : \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s c : G s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191\u03bc (c\u207b\u00b9 \u2022 s) = \u2191\u2191\u03bc s ** exact H c\u207b\u00b9 s hs ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.gcd_mul_right_right ** m n : Nat \u22a2 gcd n (n * m) = n ** rw [gcd_comm, gcd_mul_right_left] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb \u22a2 (fun t =>       \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -         (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)) =o[lt]     fun t => \u2016\u222b (x : \u211d) in ua t..va t, 1 \u2202\u03bc\u2016 + \u2016\u222b (x : \u211d) in ub t..vb t, 1 \u2202\u03bc\u2016 ** haveI := FTCFilter.meas_gen la ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this : IsMeasurablyGenerated la' \u22a2 (fun t =>       \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -         (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)) =o[lt]     fun t => \u2016\u222b (x : \u211d) in ua t..va t, 1 \u2202\u03bc\u2016 + \u2016\u222b (x : \u211d) in ub t..vb t, 1 \u2202\u03bc\u2016 ** haveI := FTCFilter.meas_gen lb ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' \u22a2 (fun t =>       \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -         (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc)) =o[lt]     fun t => \u2016\u222b (x : \u211d) in ua t..va t, 1 \u2202\u03bc\u2016 + \u2016\u222b (x : \u211d) in ub t..vb t, 1 \u2202\u03bc\u2016 ** refine'\n  ((measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_a ha_lim hua hva).neg_left.add_add\n        (measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_b hb_lim hub hvb)).congr'\n    _ EventuallyEq.rfl ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' \u22a2 (fun x =>       -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +         (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]     fun t =>     \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -       (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc) ** have A : \u2200\u1da0 t in lt, IntervalIntegrable f \u03bc (ua t) (va t) :=\n  ha_lim.eventually_intervalIntegrable_ae hmeas_a (FTCFilter.finiteAt_inner la) hua hva ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' A : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t) \u22a2 (fun x =>       -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +         (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]     fun t =>     \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -       (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc) ** have A' : \u2200\u1da0 t in lt, IntervalIntegrable f \u03bc a (ua t) :=\n  ha_lim.eventually_intervalIntegrable_ae hmeas_a (FTCFilter.finiteAt_inner la)\n    (tendsto_const_pure.mono_right FTCFilter.pure_le) hua ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' A : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t) A' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t) \u22a2 (fun x =>       -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +         (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]     fun t =>     \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -       (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc) ** have B : \u2200\u1da0 t in lt, IntervalIntegrable f \u03bc (ub t) (vb t) :=\n  hb_lim.eventually_intervalIntegrable_ae hmeas_b (FTCFilter.finiteAt_inner lb) hub hvb ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' A : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t) A' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t) B : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t) \u22a2 (fun x =>       -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +         (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]     fun t =>     \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -       (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc) ** have B' : \u2200\u1da0 t in lt, IntervalIntegrable f \u03bc b (ub t) :=\n  hb_lim.eventually_intervalIntegrable_ae hmeas_b (FTCFilter.finiteAt_inner lb)\n    (tendsto_const_pure.mono_right FTCFilter.pure_le) hub ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' A : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t) A' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t) B : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t) B' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t) \u22a2 (fun x =>       -(\u222b (x : \u211d) in ua x..va x, f x \u2202\u03bc - \u222b (x : \u211d) in ua x..va x, ca \u2202\u03bc) +         (\u222b (x : \u211d) in ub x..vb x, f x \u2202\u03bc - \u222b (x : \u211d) in ub x..vb x, cb \u2202\u03bc)) =\u1da0[lt]     fun t =>     \u222b (x : \u211d) in va t..vb t, f x \u2202\u03bc - \u222b (x : \u211d) in ua t..ub t, f x \u2202\u03bc -       (\u222b (x : \u211d) in ub t..vb t, cb \u2202\u03bc - \u222b (x : \u211d) in ua t..va t, ca \u2202\u03bc) ** filter_upwards [A, A', B, B'] with _ ua_va a_ua ub_vb b_ub ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' A : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t) A' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t) B : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t) B' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t) a\u271d : \u03b9 ua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d) a_ua : IntervalIntegrable f \u03bc a (ua a\u271d) ub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d) b_ub : IntervalIntegrable f \u03bc b (ub a\u271d) \u22a2 -(\u222b (x : \u211d) in ua a\u271d..va a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc) +       (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc) =     \u222b (x : \u211d) in va a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..ub a\u271d, f x \u2202\u03bc -       (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc) ** rw [\u2190 integral_interval_sub_interval_comm'] ** case h.hab \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' A : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t) A' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t) B : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t) B' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t) a\u271d : \u03b9 ua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d) a_ua : IntervalIntegrable f \u03bc a (ua a\u271d) ub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d) b_ub : IntervalIntegrable f \u03bc b (ub a\u271d) \u22a2 IntervalIntegrable (fun x => f x) \u03bc (ub a\u271d) (vb a\u271d)  case h.hcd \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' A : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t) A' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t) B : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t) B' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t) a\u271d : \u03b9 ua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d) a_ua : IntervalIntegrable f \u03bc a (ua a\u271d) ub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d) b_ub : IntervalIntegrable f \u03bc b (ub a\u271d) \u22a2 IntervalIntegrable (fun x => f x) \u03bc (ua a\u271d) (va a\u271d)  case h.hac \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' A : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t) A' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t) B : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t) B' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t) a\u271d : \u03b9 ua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d) a_ua : IntervalIntegrable f \u03bc a (ua a\u271d) ub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d) b_ub : IntervalIntegrable f \u03bc b (ub a\u271d) \u22a2 IntervalIntegrable (fun x => f x) \u03bc (ub a\u271d) (ua a\u271d) ** exacts [ub_vb, ua_va, b_ub.symm.trans <| hab.symm.trans a_ua] ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedSpace \u211d E f : \u211d \u2192 E a b : \u211d c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 \u03bc : Measure \u211d u v ua va ub vb : \u03b9 \u2192 \u211d inst\u271d\u00b2 : FTCFilter a la la' inst\u271d\u00b9 : FTCFilter b lb lb' inst\u271d : IsLocallyFiniteMeasure \u03bc hab : IntervalIntegrable f \u03bc a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' \u2293 Measure.ae \u03bc) (\ud835\udcdd ca) hb_lim : Tendsto f (lb' \u2293 Measure.ae \u03bc) (\ud835\udcdd cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb this\u271d : IsMeasurablyGenerated la' this : IsMeasurablyGenerated lb' A : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ua t) (va t) A' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc a (ua t) B : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc (ub t) (vb t) B' : \u2200\u1da0 (t : \u03b9) in lt, IntervalIntegrable f \u03bc b (ub t) a\u271d : \u03b9 ua_va : IntervalIntegrable f \u03bc (ua a\u271d) (va a\u271d) a_ua : IntervalIntegrable f \u03bc a (ua a\u271d) ub_vb : IntervalIntegrable f \u03bc (ub a\u271d) (vb a\u271d) b_ub : IntervalIntegrable f \u03bc b (ub a\u271d) \u22a2 -(\u222b (x : \u211d) in ua a\u271d..va a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc) +       (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc) =     \u222b (x : \u211d) in ub a\u271d..vb a\u271d, f x \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, f x \u2202\u03bc -       (\u222b (x : \u211d) in ub a\u271d..vb a\u271d, cb \u2202\u03bc - \u222b (x : \u211d) in ua a\u271d..va a\u271d, ca \u2202\u03bc) ** abel ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_sum_measure ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 \u22a2 \u222b\u207b (a : \u03b1), f a \u2202Measure.sum \u03bc = \u2211' (i : \u03b9), \u222b\u207b (a : \u03b1), f a \u2202\u03bc i ** simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_sum, ENNReal.tsum_eq_iSup_sum] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 \u22a2 \u2a06 x, \u2a06 s, \u2211 i in s, SimpleFunc.lintegral (\u2191x) (\u03bc i) = \u2a06 s, \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) ** rw [iSup_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 \u22a2 \u2a06 j, \u2a06 i, \u2211 i_1 in j, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2a06 s, \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) ** congr ** case e_s \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 \u22a2 (fun j => \u2a06 i, \u2211 i_1 in j, SimpleFunc.lintegral (\u2191i) (\u03bc i_1)) = fun s =>     \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) ** funext s ** case e_s.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 s : Finset \u03b9 \u22a2 \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) ** induction' s using Finset.induction_on with i s hi hs ** case e_s.h.insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 i : \u03b9 s : Finset \u03b9 hi : \u00aci \u2208 s hs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) \u22a2 \u2a06 i_1, \u2211 i in insert i s, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) = \u2211 a in insert i s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) ** simp only [Finset.sum_insert hi, \u2190 hs] ** case e_s.h.insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 i : \u03b9 s : Finset \u03b9 hi : \u00aci \u2208 s hs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) \u22a2 \u2a06 i_1, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191i_1) (\u03bc i) =     (\u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc i)) + \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) ** refine' (ENNReal.iSup_add_iSup _).symm ** case e_s.h.insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 i : \u03b9 s : Finset \u03b9 hi : \u00aci \u2208 s hs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) \u22a2 \u2200 (i_1 j : { i // \u2191i \u2264 fun a => f a }),     \u2203 k,       SimpleFunc.lintegral (\u2191i_1) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191j) (\u03bc i) \u2264         SimpleFunc.lintegral (\u2191k) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191k) (\u03bc i) ** intro \u03c6 \u03c8 ** case e_s.h.insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 i : \u03b9 s : Finset \u03b9 hi : \u00aci \u2208 s hs : \u2a06 i, \u2211 i_1 in s, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in s, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) \u03c6 \u03c8 : { i // \u2191i \u2264 fun a => f a } \u22a2 \u2203 k,     SimpleFunc.lintegral (\u2191\u03c6) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191\u03c8) (\u03bc i) \u2264       SimpleFunc.lintegral (\u2191k) (\u03bc i) + \u2211 i in s, SimpleFunc.lintegral (\u2191k) (\u03bc i) ** exact\n  \u27e8\u27e8\u03c6 \u2294 \u03c8, fun x => sup_le (\u03c6.2 x) (\u03c8.2 x)\u27e9,\n    add_le_add (SimpleFunc.lintegral_mono le_sup_left le_rfl)\n      (Finset.sum_le_sum fun j _ => SimpleFunc.lintegral_mono le_sup_right le_rfl)\u27e9 ** case e_s.h.empty \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 \u22a2 \u2a06 i, \u2211 i_1 in \u2205, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) = \u2211 a in \u2205, \u2a06 x, SimpleFunc.lintegral (\u2191x) (\u03bc a) ** apply bot_unique ** case e_s.h.empty.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 m : MeasurableSpace \u03b1 \u03b9 : Type u_5 f : \u03b1 \u2192 \u211d\u22650\u221e \u03bc : \u03b9 \u2192 Measure \u03b1 \u22a2 \u2a06 i, \u2211 i_1 in \u2205, SimpleFunc.lintegral (\u2191i) (\u03bc i_1) \u2264 \u22a5 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.normalize_eq_zero ** d : Nat n : Int d0 : d \u2260 0 \u22a2 normalize n d = 0 \u2194 n = 0 ** have' := normalize_eq_iff d0 Nat.one_ne_zero ** case refine'_3 d : Nat n : Int d0 : d \u2260 0 this : normalize ?refine'_1 d = normalize ?refine'_2 1 \u2194 ?refine'_1 * \u21911 = ?refine'_2 * \u2191d \u22a2 normalize n d = 0 \u2194 n = 0  case refine'_1 d : Nat n : Int d0 : d \u2260 0 \u22a2 Int  case refine'_2 d : Nat n : Int d0 : d \u2260 0 \u22a2 Int ** rw [normalize_zero (d := 1)] at this ** case refine'_3 d : Nat n : Int d0 : d \u2260 0 this : normalize ?refine'_1 d = 0 \u2194 ?refine'_1 * \u21911 = 0 * \u2191d \u22a2 normalize n d = 0 \u2194 n = 0  case refine'_1 d : Nat n : Int d0 : d \u2260 0 \u22a2 Int  case refine'_1 d : Nat n : Int d0 : d \u2260 0 \u22a2 Int ** rw [this] ** case refine'_3 d : Nat n : Int d0 : d \u2260 0 this : normalize n d = 0 \u2194 n * \u21911 = 0 * \u2191d \u22a2 n * \u21911 = 0 * \u2191d \u2194 n = 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBSet.toStream_toList ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t : RBSet \u03b1 cmp \u22a2 RBNode.Stream.toList (toStream t) = toList t ** simp [toStream_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_div_right_eq_self ** G : Type u_1 inst\u271d\u00b3 : MeasurableSpace G \u03bc : Measure G g\u271d : G inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MeasurableMul G inst\u271d : IsMulRightInvariant \u03bc f : G \u2192 \u211d\u22650\u221e g : G \u22a2 \u222b\u207b (x : G), f (x / g) \u2202\u03bc = \u222b\u207b (x : G), f x \u2202\u03bc ** simp_rw [div_eq_mul_inv, lintegral_mul_right_eq_self f g\u207b\u00b9] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toMeasure_pure ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a a' : \u03b1 s\u271d : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(toMeasure (pure a)) s = \u2191\u2191(Measure.dirac a) s ** rw [toMeasure_pure_apply a s hs, Measure.dirac_apply' a hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a a' : \u03b1 s\u271d : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 (if a \u2208 s then 1 else 0) = Set.indicator s 1 a ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Holor.mul_assoc ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Semigroup \u03b1 x : Holor \u03b1 ds\u2081 y : Holor \u03b1 ds\u2082 z : Holor \u03b1 ds\u2083 \u22a2 HEq (x \u2297 y \u2297 z) (x \u2297 (y \u2297 z)) ** simp [cast_heq, mul_assoc0, assocLeft] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.foldl_eq ** \u03b1 : Type u_1 f : \u03b1 \u2192 Char \u2192 \u03b1 s : String a : \u03b1 \u22a2 foldl f a s = List.foldl f a s.data ** simpa using foldlAux_of_valid f [] s.1 [] a ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.compProd_eq_sum_compProd_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7 ** by_cases h : IsSFiniteKernel \u03b7 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } h : IsSFiniteKernel \u03b7 \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } h : \u00acIsSFiniteKernel \u03b7 \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7 ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } h : IsSFiniteKernel \u03b7 \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7 ** rw [compProd_eq_sum_compProd] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } h : IsSFiniteKernel \u03b7 \u22a2 (kernel.sum fun n => kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7 ** congr with n a s hs ** case pos.e_\u03ba.h.h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } h : IsSFiniteKernel \u03b7 n : \u2115 a : \u03b1 s : Set (\u03b2 \u00d7 \u03b3) hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(kernel.sum fun m => seq \u03ba n \u2297\u2096 seq \u03b7 m) a) s = \u2191\u2191(\u2191(seq \u03ba n \u2297\u2096 \u03b7) a) s ** simp_rw [kernel.sum_apply' _ _ hs, compProd_apply_eq_compProdFun _ _ _ hs,\n  compProdFun_tsum_right _ \u03b7 a hs] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } h : \u00acIsSFiniteKernel \u03b7 \u22a2 \u03ba \u2297\u2096 \u03b7 = kernel.sum fun n => seq \u03ba n \u2297\u2096 \u03b7 ** simp_rw [compProd_of_not_isSFiniteKernel_right _ _ h] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } h : \u00acIsSFiniteKernel \u03b7 \u22a2 0 = kernel.sum fun n => 0 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_add_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f g : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc ** refine' le_antisymm _ (le_lintegral_add _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f g : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc ** rcases exists_measurable_le_lintegral_eq \u03bc fun a => f a + g a with \u27e8\u03c6, h\u03c6m, h\u03c6_le, h\u03c6_eq\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f g \u03c6 : \u03b1 \u2192 \u211d\u22650\u221e h\u03c6m : Measurable \u03c6 h\u03c6_le : \u03c6 \u2264 fun a => f a + g a h\u03c6_eq : \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u03c6 a \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc ** calc\n  \u222b\u207b a, f a + g a \u2202\u03bc = \u222b\u207b a, \u03c6 a \u2202\u03bc := h\u03c6_eq\n  _ \u2264 \u222b\u207b a, f a + (\u03c6 a - f a) \u2202\u03bc := (lintegral_mono fun a => le_add_tsub)\n  _ = \u222b\u207b a, f a \u2202\u03bc + \u222b\u207b a, \u03c6 a - f a \u2202\u03bc := (lintegral_add_aux hf (h\u03c6m.sub hf))\n  _ \u2264 \u222b\u207b a, f a \u2202\u03bc + \u222b\u207b a, g a \u2202\u03bc :=\n    add_le_add_left (lintegral_mono fun a => tsub_le_iff_left.2 <| h\u03c6_le a) _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.cast_bit1 ** \u03b1 : Type u_1 inst\u271d : Semiring \u03b1 n : Num \u22a2 \u2191(Num.bit1 n) = _root_.bit1 \u2191n ** rw [\u2190 bit1_of_bit1, _root_.bit1, bit0_of_bit0, cast_add, cast_bit0] ** \u03b1 : Type u_1 inst\u271d : Semiring \u03b1 n : Num \u22a2 _root_.bit0 \u2191n + \u21911 = _root_.bit1 \u2191n ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_pos ** \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 0 < ncard s \u2194 Set.Nonempty s ** rw [pos_iff_ne_zero, Ne.def, ncard_eq_zero hs, nonempty_iff_ne_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_preimage_mul_right ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d h : a \u2260 0 s : Set \u211d \u22a2 \u2191\u2191(Measure.map (fun x => x * a) volume) s = ofReal |a\u207b\u00b9| * \u2191\u2191volume s ** rw [map_volume_mul_right h] ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 a : \u211d h : a \u2260 0 s : Set \u211d \u22a2 \u2191\u2191(ofReal |a\u207b\u00b9| \u2022 volume) s = ofReal |a\u207b\u00b9| * \u2191\u2191volume s ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.hG ** \u22a2 Primrec Nat.Partrec.Code.G ** have a := (Primrec.ofNat (\u2115 \u00d7 Code)).comp (Primrec.list_length (\u03b1 := List (Option \u2115))) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) \u22a2 Primrec Nat.Partrec.Code.G ** have k := Primrec.fst.comp a ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).1 \u22a2 Primrec Nat.Partrec.Code.G ** refine' Primrec.option_some.comp (Primrec.list_map (Primrec.list_range.comp k) (_ : Primrec _)) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).1 \u22a2 Primrec fun p =>     (fun a n =>         Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' =>           Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1)             (some (unpair n).2)             (fun cf cg x x => do               let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n               let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n               some (Nat.pair x y))             (fun cf cg x x => do               let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n               Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x)             (fun cf cg x x =>               let z := (unpair n).1;               Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do                 let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y)                 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i)))             fun cf x =>             let z := (unpair n).1;             let m := (unpair n).2;             do             let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m)             Nat.casesOn x (some m) fun x =>                 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1)))       p.1 p.2 ** replace k := k.comp (Primrec.fst (\u03b2 := \u2115)) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 \u22a2 Primrec fun p =>     (fun a n =>         Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' =>           Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1)             (some (unpair n).2)             (fun cf cg x x => do               let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n               let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n               some (Nat.pair x y))             (fun cf cg x x => do               let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n               Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x)             (fun cf cg x x =>               let z := (unpair n).1;               Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do                 let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y)                 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i)))             fun cf x =>             let z := (unpair n).1;             let m := (unpair n).2;             do             let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m)             Nat.casesOn x (some m) fun x =>                 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1)))       p.1 p.2 ** have n := Primrec.snd (\u03b1 := List (List (Option \u2115))) (\u03b2 := \u2115) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n : Primrec Prod.snd \u22a2 Primrec fun p =>     (fun a n =>         Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' =>           Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1)             (some (unpair n).2)             (fun cf cg x x => do               let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n               let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n               some (Nat.pair x y))             (fun cf cg x x => do               let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n               Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x)             (fun cf cg x x =>               let z := (unpair n).1;               Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do                 let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y)                 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i)))             fun cf x =>             let z := (unpair n).1;             let m := (unpair n).2;             do             let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m)             Nat.casesOn x (some m) fun x =>                 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1)))       p.1 p.2 ** refine' Primrec.nat_casesOn k (_root_.Primrec.const Option.none) (_ : Primrec _) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n : Primrec Prod.snd \u22a2 Primrec fun p =>     (fun p n =>         (fun k' =>             Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)               (some (unpair p.2).2)               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2                 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 some (Nat.pair x y))               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)               (fun cf cg x x =>                 let z := (unpair p.2).1;                 Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)                   fun y => do                   let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)                   Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))               fun cf x =>               let z := (unpair p.2).1;               let m := (unpair p.2).2;               do               let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)               Nat.casesOn x (some m) fun x =>                   Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))           n)       p.1 p.2 ** have k := k.comp (Primrec.fst (\u03b2 := \u2115)) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 \u22a2 Primrec fun p =>     (fun p n =>         (fun k' =>             Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)               (some (unpair p.2).2)               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2                 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 some (Nat.pair x y))               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)               (fun cf cg x x =>                 let z := (unpair p.2).1;                 Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)                   fun y => do                   let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)                   Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))               fun cf x =>               let z := (unpair p.2).1;               let m := (unpair p.2).2;               do               let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)               Nat.casesOn x (some m) fun x =>                   Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))           n)       p.1 p.2 ** have n := n.comp (Primrec.fst (\u03b2 := \u2115)) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 \u22a2 Primrec fun p =>     (fun p n =>         (fun k' =>             Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)               (some (unpair p.2).2)               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2                 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 some (Nat.pair x y))               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)               (fun cf cg x x =>                 let z := (unpair p.2).1;                 Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)                   fun y => do                   let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)                   Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))               fun cf x =>               let z := (unpair p.2).1;               let m := (unpair p.2).2;               do               let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)               Nat.casesOn x (some m) fun x =>                   Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))           n)       p.1 p.2 ** have k' := Primrec.snd (\u03b1 := List (List (Option \u2115)) \u00d7 \u2115) (\u03b2 := \u2115) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd \u22a2 Primrec fun p =>     (fun p n =>         (fun k' =>             Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)               (some (unpair p.2).2)               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2                 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 some (Nat.pair x y))               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)               (fun cf cg x x =>                 let z := (unpair p.2).1;                 Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)                   fun y => do                   let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)                   Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))               fun cf x =>               let z := (unpair p.2).1;               let m := (unpair p.2).2;               do               let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)               Nat.casesOn x (some m) fun x =>                   Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))           n)       p.1 p.2 ** have c := Primrec.snd.comp (a.comp <| (Primrec.fst (\u03b2 := \u2115)).comp (Primrec.fst (\u03b2 := \u2115))) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun p =>     (fun p n =>         (fun k' =>             Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1)               (some (unpair p.2).2)               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2                 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 some (Nat.pair x y))               (fun cf cg x x => do                 let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2                 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x)               (fun cf cg x x =>                 let z := (unpair p.2).1;                 Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z)                   fun y => do                   let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y)                   Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i)))               fun cf x =>               let z := (unpair p.2).1;               let m := (unpair p.2).2;               do               let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m)               Nat.casesOn x (some m) fun x =>                   Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1)))           n)       p.1 p.2 ** apply\n  Nat.Partrec.Code.rec_prim c\n    (_root_.Primrec.const (some 0))\n    (Primrec.option_some.comp (_root_.Primrec.succ.comp n))\n    (Primrec.option_some.comp (Primrec.fst.comp <| Primrec.unpair.comp n))\n    (Primrec.option_some.comp (Primrec.snd.comp <| Primrec.unpair.comp n)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2     let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     some (Nat.pair x y) ** have L := (Primrec.fst.comp Primrec.fst).comp\n  (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2     let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     some (Nat.pair x y) ** have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2     let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     some (Nat.pair x y) ** have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2     let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     some (Nat.pair x y) ** have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2     let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     some (Nat.pair x y) ** have cg := (Primrec.fst.comp Primrec.snd).comp\n  (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2     let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     some (Nat.pair x y) ** refine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cf).pair n) ?_ ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec\u2082 fun a x => do     let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     some (Nat.pair x y) ** unfold Primrec\u2082 ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun p =>     Option.map (fun y => Nat.pair p.2 y)       (Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1) p.1.1.1.2) ** refine Primrec.option_map ((hlup.comp <| L.pair <| (k.pair cg).pair n).comp Primrec.fst) ?_ ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec\u2082 fun p y => Nat.pair p.2 y ** unfold Primrec\u2082 ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun p => (fun p y => Nat.pair p.2 y) p.1 p.2 ** exact Primrec\u2082.natPair.comp (Primrec.snd.comp Primrec.fst) Primrec.snd ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have L := (Primrec.fst.comp Primrec.fst).comp\n  (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have cg := (Primrec.fst.comp Primrec.snd).comp\n  (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun a => do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2     Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** refine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cg).pair n) ?_ ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec\u2082 fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** unfold Primrec\u2082 ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun p =>     (fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x) p.1 p.2 ** have h :=\n  hlup.comp ((L.comp Primrec.fst).pair <| ((k.pair cf).comp Primrec.fst).pair Primrec.snd) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 h :   Primrec fun a =>     Nat.Partrec.Code.lup (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).1       (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).2.1       (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).2.2 \u22a2 Primrec fun p =>     (fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x) p.1 p.2 ** exact h ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)       fun y => do       let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)       Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have L := (Primrec.fst.comp Primrec.fst).comp\n  (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)       fun y => do       let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)       Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)       fun y => do       let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)       Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)       fun y => do       let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)       Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)       fun y => do       let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)       Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have cg := (Primrec.fst.comp Primrec.snd).comp\n  (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)       fun y => do       let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)       Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have z := Primrec.fst.comp (Primrec.unpair.comp n) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z : Primrec fun a => (unpair a.1.1.2).1 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z)       fun y => do       let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y)       Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** refine'\n  Primrec.nat_casesOn (Primrec.snd.comp (Primrec.unpair.comp n))\n    (hlup.comp <| L.pair <| (k.pair cf).pair z)\n    (_ : Primrec _) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z : Primrec fun a => (unpair a.1.1.2).1 \u22a2 Primrec fun p =>     (fun a n =>         (fun y => do             let i \u2190               Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)                   (Nat.pair (unpair a.1.1.2).1 y)             Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)                 (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))           n)       p.1 p.2 ** have L := L.comp (Primrec.fst (\u03b2 := \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 \u22a2 Primrec fun p =>     (fun a n =>         (fun y => do             let i \u2190               Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)                   (Nat.pair (unpair a.1.1.2).1 y)             Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)                 (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))           n)       p.1 p.2 ** have z := z.comp (Primrec.fst (\u03b2 := \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z : Primrec fun a => (unpair a.1.1.1.2).1 \u22a2 Primrec fun p =>     (fun a n =>         (fun y => do             let i \u2190               Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)                   (Nat.pair (unpair a.1.1.2).1 y)             Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)                 (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))           n)       p.1 p.2 ** have y := Primrec.snd\n  (\u03b1 := ((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) (\u03b2 := \u2115) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z : Primrec fun a => (unpair a.1.1.1.2).1 y : Primrec Prod.snd \u22a2 Primrec fun p =>     (fun a n =>         (fun y => do             let i \u2190               Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)                   (Nat.pair (unpair a.1.1.2).1 y)             Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)                 (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))           n)       p.1 p.2 ** have h\u2081 := hlup.comp <| L.pair <| (((k'.pair c).comp Primrec.fst).comp Primrec.fst).pair\n  (Primrec\u2082.natPair.comp z y) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z : Primrec fun a => (unpair a.1.1.1.2).1 y : Primrec Prod.snd h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 \u22a2 Primrec fun p =>     (fun a n =>         (fun y => do             let i \u2190               Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)                   (Nat.pair (unpair a.1.1.2).1 y)             Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1)                 (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i)))           n)       p.1 p.2 ** refine' Primrec.option_bind h\u2081 (_ : Primrec _) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z : Primrec fun a => (unpair a.1.1.1.2).1 y : Primrec Prod.snd h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 \u22a2 Primrec fun p =>     (fun p i =>         Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)           (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))       p.1 p.2 ** have z := z.comp (Primrec.fst (\u03b2 := \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z\u271d : Primrec fun a => (unpair a.1.1.1.2).1 y : Primrec Prod.snd h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 z : Primrec fun a => (unpair a.1.1.1.1.2).1 \u22a2 Primrec fun p =>     (fun p i =>         Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)           (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))       p.1 p.2 ** have y := y.comp (Primrec.fst (\u03b2 := \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z\u271d : Primrec fun a => (unpair a.1.1.1.2).1 y\u271d : Primrec Prod.snd h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 z : Primrec fun a => (unpair a.1.1.1.1.2).1 y : Primrec fun a => a.1.2 \u22a2 Primrec fun p =>     (fun p i =>         Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)           (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))       p.1 p.2 ** have i := Primrec.snd\n  (\u03b1 := (((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) \u00d7 \u2115)\n  (\u03b2 := \u2115) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z\u271d : Primrec fun a => (unpair a.1.1.1.2).1 y\u271d : Primrec Prod.snd h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 z : Primrec fun a => (unpair a.1.1.1.1.2).1 y : Primrec fun a => a.1.2 i : Primrec Prod.snd \u22a2 Primrec fun p =>     (fun p i =>         Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)           (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))       p.1 p.2 ** have h\u2082 := hlup.comp ((L.comp Primrec.fst).pair <|\n  ((k.pair cg).comp <| Primrec.fst.comp Primrec.fst).pair <|\n    Primrec\u2082.natPair.comp z <| Primrec\u2082.natPair.comp y i) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z\u271d : Primrec fun a => (unpair a.1.1.1.2).1 y\u271d : Primrec Prod.snd h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1       (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 z : Primrec fun a => (unpair a.1.1.1.1.2).1 y : Primrec fun a => a.1.2 i : Primrec Prod.snd h\u2082 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1),           Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).1       (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1),             Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).2.1       (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1),             Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).2.2 \u22a2 Primrec fun p =>     (fun p i =>         Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1)           (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i)))       p.1 p.2 ** exact h\u2082 ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     let m := (unpair a.1.1.2).2;     do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)     Nat.casesOn x (some m) fun x =>         Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have L := (Primrec.fst.comp Primrec.fst).comp\n  (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Option \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     let m := (unpair a.1.1.2).2;     do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)     Nat.casesOn x (some m) fun x =>         Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Option \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     let m := (unpair a.1.1.2).2;     do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)     Nat.casesOn x (some m) fun x =>         Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Option \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     let m := (unpair a.1.1.2).2;     do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)     Nat.casesOn x (some m) fun x =>         Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n    (\u03b2 := Code \u00d7 Option \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     let m := (unpair a.1.1.2).2;     do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)     Nat.casesOn x (some m) fun x =>         Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have z := Primrec.fst.comp (Primrec.unpair.comp n) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     let m := (unpair a.1.1.2).2;     do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)     Nat.casesOn x (some m) fun x =>         Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have m := Primrec.snd.comp (Primrec.unpair.comp n) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m : Primrec fun a => (unpair a.1.1.2).2 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     let m := (unpair a.1.1.2).2;     do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)     Nat.casesOn x (some m) fun x =>         Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have h\u2081 := hlup.comp <| L.pair <| (k.pair cf).pair (Primrec\u2082.natPair.comp z m) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m : Primrec fun a => (unpair a.1.1.2).2 h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 \u22a2 Primrec fun a =>     let z := (unpair a.1.1.2).1;     let m := (unpair a.1.1.2).2;     do     let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m)     Nat.casesOn x (some m) fun x =>         Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** refine' Primrec.option_bind h\u2081 (_ : Primrec _) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m : Primrec fun a => (unpair a.1.1.2).2 h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 \u22a2 Primrec fun p =>     (fun a x =>         Nat.casesOn x (some (unpair a.1.1.2).2) fun x =>           Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)             (Nat.pair (unpair a.1.1.2).1 ((unpair a.1.1.2).2 + 1)))       p.1 p.2 ** have m := m.comp (Primrec.fst (\u03b2 := \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m\u271d : Primrec fun a => (unpair a.1.1.2).2 h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 m : Primrec fun a => (unpair a.1.1.1.2).2 \u22a2 Primrec fun p =>     (fun a x =>         Nat.casesOn x (some (unpair a.1.1.2).2) fun x =>           Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2)             (Nat.pair (unpair a.1.1.2).1 ((unpair a.1.1.2).2 + 1)))       p.1 p.2 ** refine Primrec.nat_casesOn Primrec.snd (Primrec.option_some.comp m) ?_ ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m\u271d : Primrec fun a => (unpair a.1.1.2).2 h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 m : Primrec fun a => (unpair a.1.1.1.2).2 \u22a2 Primrec\u2082 fun p n =>     (fun x =>         Nat.Partrec.Code.lup p.1.1.1.1 (p.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).2)           (Nat.pair (unpair p.1.1.1.2).1 ((unpair p.1.1.1.2).2 + 1)))       n ** unfold Primrec\u2082 ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m\u271d : Primrec fun a => (unpair a.1.1.2).2 h\u2081 :   Primrec fun a =>     Nat.Partrec.Code.lup       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1       (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 m : Primrec fun a => (unpair a.1.1.1.2).2 \u22a2 Primrec fun p =>     (fun p n =>         (fun x =>             Nat.Partrec.Code.lup p.1.1.1.1 (p.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).2)               (Nat.pair (unpair p.1.1.1.2).1 ((unpair p.1.1.1.2).2 + 1)))           n)       p.1 p.2 ** exact (hlup.comp ((L.comp Primrec.fst).pair <|\n  ((k'.pair c).comp <| Primrec.fst.comp Primrec.fst).pair\n    (Primrec\u2082.natPair.comp (z.comp Primrec.fst) (_root_.Primrec.succ.comp m)))).comp\n  Primrec.fst ** Qed",
        "informal": ""
    },
    {
        "formal": "ContinuousLinearMap.integral_comp_comm' ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F K : \u211d\u22650 hL : AntilipschitzWith K \u2191L \u03c6 : \u03b1 \u2192 E \u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc) ** by_cases h : Integrable \u03c6 \u03bc ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F K : \u211d\u22650 hL : AntilipschitzWith K \u2191L \u03c6 : \u03b1 \u2192 E h : \u00acIntegrable \u03c6 \u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc) ** have : \u00acIntegrable (fun a => L (\u03c6 a)) \u03bc := by\n  erw [LipschitzWith.integrable_comp_iff_of_antilipschitz L.lipschitz hL L.map_zero]\n  assumption ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F K : \u211d\u22650 hL : AntilipschitzWith K \u2191L \u03c6 : \u03b1 \u2192 E h : \u00acIntegrable \u03c6 this : \u00acIntegrable fun a => \u2191L (\u03c6 a) \u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc) ** simp [integral_undef, h, this] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F K : \u211d\u22650 hL : AntilipschitzWith K \u2191L \u03c6 : \u03b1 \u2192 E h : Integrable \u03c6 \u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc) ** exact integral_comp_comm L h ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F K : \u211d\u22650 hL : AntilipschitzWith K \u2191L \u03c6 : \u03b1 \u2192 E h : \u00acIntegrable \u03c6 \u22a2 \u00acIntegrable fun a => \u2191L (\u03c6 a) ** erw [LipschitzWith.integrable_comp_iff_of_antilipschitz L.lipschitz hL L.map_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F K : \u211d\u22650 hL : AntilipschitzWith K \u2191L \u03c6 : \u03b1 \u2192 E h : \u00acIntegrable \u03c6 \u22a2 \u00acIntegrable fun a => \u03c6 a ** assumption ** Qed",
        "informal": ""
    },
    {
        "formal": "map_finset_inf ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u2074 : SemilatticeInf \u03b1 inst\u271d\u00b3 : OrderTop \u03b1 s\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b2 f\u271d g\u271d : \u03b2 \u2192 \u03b1 a : \u03b1 inst\u271d\u00b2 : SemilatticeInf \u03b2 inst\u271d\u00b9 : OrderTop \u03b2 inst\u271d : InfTopHomClass F \u03b1 \u03b2 f : F s\u271d : Finset \u03b9 g : \u03b9 \u2192 \u03b1 i : \u03b9 s : Finset \u03b9 x\u271d : \u00aci \u2208 s h : \u2191f (inf s g) = inf s (\u2191f \u2218 g) \u22a2 \u2191f (inf (cons i s x\u271d) g) = inf (cons i s x\u271d) (\u2191f \u2218 g) ** rw [inf_cons, inf_cons, map_inf, h, Function.comp_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.progMeasurable_min_stopping_time ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 \u22a2 ProgMeasurable f fun i \u03c9 => min i (\u03c4 \u03c9) ** intro i ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 \u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2 ** let m_prod : MeasurableSpace (Set.Iic i \u00d7 \u03a9) := Subtype.instMeasurableSpace.prod (f i) ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) \u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2 ** let m_set : \u2200 t : Set (Set.Iic i \u00d7 \u03a9), MeasurableSpace t := fun _ =>\n  @Subtype.instMeasurableSpace (Set.Iic i \u00d7 \u03a9) _ m_prod ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace \u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2 ** let s := {p : Set.Iic i \u00d7 \u03a9 | \u03c4 p.2 \u2264 i} ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} \u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2 ** have hs : MeasurableSet[m_prod] s := @measurable_snd (Set.Iic i) \u03a9 _ (f i) _ (h\u03c4 i) ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s \u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2 ** have h_meas_fst : \u2200 t : Set (Set.Iic i \u00d7 \u03a9),\n    Measurable[m_set t] fun x : t => ((x : Set.Iic i \u00d7 \u03a9).fst : \u03b9) :=\n  fun t => (@measurable_subtype_coe (Set.Iic i \u00d7 \u03a9) m_prod _).fst.subtype_val ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 \u22a2 StronglyMeasurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2 ** apply Measurable.stronglyMeasurable ** case hf \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 \u22a2 Measurable fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2 ** refine' measurable_of_restrict_of_restrict_compl hs _ _ ** case hf.refine'_1 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 \u22a2 Measurable (Set.restrict s fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2) ** refine @Measurable.min _ _ _ _ _ (m_set s) _ _ _ _ _ (h_meas_fst s) ?_ ** case hf.refine'_1 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 \u22a2 Measurable fun a => \u03c4 (\u2191a).2 ** refine' @measurable_of_Iic \u03b9 s _ _ _ (m_set s) _ _ _ _ fun j => _ ** case hf.refine'_1 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 j : \u03b9 \u22a2 MeasurableSet ((fun a => \u03c4 (\u2191a).2) \u207b\u00b9' Set.Iic j) ** have h_set_eq : (fun x : s => \u03c4 (x : Set.Iic i \u00d7 \u03a9).snd) \u207b\u00b9' Set.Iic j =\n    (fun x : s => (x : Set.Iic i \u00d7 \u03a9).snd) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j} := by\n  ext1 \u03c9\n  simp only [Set.mem_preimage, Set.mem_Iic, iff_and_self, le_min_iff, Set.mem_setOf_eq]\n  exact fun _ => \u03c9.prop ** case hf.refine'_1 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 j : \u03b9 h_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j} \u22a2 MeasurableSet ((fun a => \u03c4 (\u2191a).2) \u207b\u00b9' Set.Iic j) ** rw [h_set_eq] ** case hf.refine'_1 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 j : \u03b9 h_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j} \u22a2 MeasurableSet ((fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j}) ** suffices h_meas : @Measurable _ _ (m_set s) (f i) fun x : s => (x : Set.Iic i \u00d7 \u03a9).snd ** case hf.refine'_1 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 j : \u03b9 h_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j} h_meas : Measurable fun x => (\u2191x).2 \u22a2 MeasurableSet ((fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j})  case h_meas \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 j : \u03b9 h_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j} \u22a2 Measurable fun x => (\u2191x).2 ** exact h_meas (f.mono (min_le_left _ _) _ (h\u03c4.measurableSet_le (min i j))) ** case h_meas \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 j : \u03b9 h_set_eq : (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j} \u22a2 Measurable fun x => (\u2191x).2 ** exact measurable_snd.comp (@measurable_subtype_coe _ m_prod _) ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 j : \u03b9 \u22a2 (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j = (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j} ** ext1 \u03c9 ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 j : \u03b9 \u03c9 : \u2191s \u22a2 \u03c9 \u2208 (fun x => \u03c4 (\u2191x).2) \u207b\u00b9' Set.Iic j \u2194 \u03c9 \u2208 (fun x => (\u2191x).2) \u207b\u00b9' {\u03c9 | \u03c4 \u03c9 \u2264 min i j} ** simp only [Set.mem_preimage, Set.mem_Iic, iff_and_self, le_min_iff, Set.mem_setOf_eq] ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 j : \u03b9 \u03c9 : \u2191s \u22a2 \u03c4 (\u2191\u03c9).2 \u2264 j \u2192 \u03c4 (\u2191\u03c9).2 \u2264 i ** exact fun _ => \u03c9.prop ** case hf.refine'_2 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 \u22a2 Measurable (Set.restrict s\u1d9c fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2) ** letI sc := s\u1d9c ** case hf.refine'_2 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 sc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c \u22a2 Measurable (Set.restrict s\u1d9c fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2) ** suffices h_min_eq_left :\n  (fun x : sc => min (\u2191(x : Set.Iic i \u00d7 \u03a9).fst) (\u03c4 (x : Set.Iic i \u00d7 \u03a9).snd)) = fun x : sc =>\n    \u2191(x : Set.Iic i \u00d7 \u03a9).fst ** case h_min_eq_left \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 sc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c \u22a2 (fun x => min (\u2191(\u2191x).1) (\u03c4 (\u2191x).2)) = fun x => \u2191(\u2191x).1 ** ext1 \u03c9 ** case h_min_eq_left.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 sc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c \u03c9 : \u2191sc \u22a2 min (\u2191(\u2191\u03c9).1) (\u03c4 (\u2191\u03c9).2) = \u2191(\u2191\u03c9).1 ** rw [min_eq_left] ** case h_min_eq_left.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 sc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c \u03c9 : \u2191sc \u22a2 \u2191(\u2191\u03c9).1 \u2264 \u03c4 (\u2191\u03c9).2 ** have hx_fst_le : \u2191(\u03c9 : Set.Iic i \u00d7 \u03a9).fst \u2264 i := (\u03c9 : Set.Iic i \u00d7 \u03a9).fst.prop ** case h_min_eq_left.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 sc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c \u03c9 : \u2191sc hx_fst_le : \u2191(\u2191\u03c9).1 \u2264 i \u22a2 \u2191(\u2191\u03c9).1 \u2264 \u03c4 (\u2191\u03c9).2 ** refine' hx_fst_le.trans (le_of_lt _) ** case h_min_eq_left.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 sc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c \u03c9 : \u2191sc hx_fst_le : \u2191(\u2191\u03c9).1 \u2264 i \u22a2 i < \u03c4 (\u2191\u03c9).2 ** convert \u03c9.prop ** case a \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 sc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c \u03c9 : \u2191sc hx_fst_le : \u2191(\u2191\u03c9).1 \u2264 i \u22a2 i < \u03c4 (\u2191\u03c9).2 \u2194 \u2191\u03c9 \u2208 sc ** simp only [not_le, Set.mem_compl_iff, Set.mem_setOf_eq] ** case hf.refine'_2 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 sc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c h_min_eq_left : (fun x => min (\u2191(\u2191x).1) (\u03c4 (\u2191x).2)) = fun x => \u2191(\u2191x).1 \u22a2 Measurable (Set.restrict s\u1d9c fun p => (fun i \u03c9 => min i (\u03c4 \u03c9)) (\u2191p.1) p.2) ** simp_rw [Set.restrict, h_min_eq_left] ** case hf.refine'_2 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2077 : LinearOrder \u03b9 inst\u271d\u2076 : MeasurableSpace \u03b9 inst\u271d\u2075 : TopologicalSpace \u03b9 inst\u271d\u2074 : OrderTopology \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : BorelSpace \u03b9 inst\u271d\u00b9 : TopologicalSpace \u03b2 u : \u03b9 \u2192 \u03a9 \u2192 \u03b2 \u03c4 : \u03a9 \u2192 \u03b9 f : Filtration \u03b9 m inst\u271d : MetrizableSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 i : \u03b9 m_prod : MeasurableSpace (\u2191(Set.Iic i) \u00d7 \u03a9) := MeasurableSpace.prod Subtype.instMeasurableSpace (\u2191f i) m_set : (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)) \u2192 MeasurableSpace \u2191t := fun x => Subtype.instMeasurableSpace s : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := {p | \u03c4 p.2 \u2264 i} hs : MeasurableSet s h_meas_fst : \u2200 (t : Set (\u2191(Set.Iic i) \u00d7 \u03a9)), Measurable fun x => \u2191(\u2191x).1 sc : Set (\u2191(Set.Iic i) \u00d7 \u03a9) := s\u1d9c h_min_eq_left : (fun x => min (\u2191(\u2191x).1) (\u03c4 (\u2191x).2)) = fun x => \u2191(\u2191x).1 \u22a2 Measurable fun x => \u2191(\u2191x).1 ** exact h_meas_fst _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.HeapNode.rankTR_eq ** \u22a2 @Std.BinomialHeap.Imp.HeapNode.rankTR = @rank ** funext \u03b1 s ** case h.h \u03b1 : Type u_1 s : HeapNode \u03b1 \u22a2 Std.BinomialHeap.Imp.HeapNode.rankTR s = rank s ** exact go s 0 ** \u03b1 : Type ?u.3227 a\u271d : \u03b1 child\u271d sibling\u271d : HeapNode \u03b1 x\u271d : Nat \u22a2 Std.BinomialHeap.Imp.HeapNode.rankTR.go (node a\u271d child\u271d sibling\u271d) x\u271d = rank (node a\u271d child\u271d sibling\u271d) + x\u271d ** simp_arith only [rankTR.go, go, rank] ** Qed",
        "informal": ""
    },
    {
        "formal": "tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : MetrizableSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsOpenPosMeasure \u03bc hs : IsCompact s c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 closure (interior s) hmg : ContinuousOn g s \u22a2 x\u2080 \u2208 s ** rw [\u2190 hs.isClosed.closure_eq] ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : MetrizableSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsOpenPosMeasure \u03bc hs : IsCompact s c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 closure (interior s) hmg : ContinuousOn g s \u22a2 x\u2080 \u2208 closure s ** exact closure_mono interior_subset h\u2080 ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.isPiSystem_Ioo_rat ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 \u22a2 IsPiSystem (\u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b}) ** convert isPiSystem_Ioo ((\u2191) : \u211a \u2192 \u211d) ((\u2191) : \u211a \u2192 \u211d) ** case h.e'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 \u22a2 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b} = {S | \u2203 l u, \u2191l < \u2191u \u2227 Ioo \u2191l \u2191u = S} ** ext x ** case h.e'_2.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 x : Set \u211d \u22a2 x \u2208 \u22c3 a, \u22c3 b, \u22c3 (_ : a < b), {Ioo \u2191a \u2191b} \u2194 x \u2208 {S | \u2203 l u, \u2191l < \u2191u \u2227 Ioo \u2191l \u2191u = S} ** simp [eq_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.boundedContinuousFunction_dense ** \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc \u22a2 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc) = \u22a4 ** rw [AddSubgroup.eq_top_iff'] ** \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc \u22a2 \u2200 (x : { x // x \u2208 Lp E p }), x \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc) ** intro f ** \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u22a2 f \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc) ** refine' mem_closure_iff_frequently.mpr _ ** \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u22a2 \u2203\u1da0 (x : { x // x \u2208 Lp E p }) in \ud835\udcdd f, x \u2208 \u2191(boundedContinuousFunction E p \u03bc) ** rw [Metric.nhds_basis_closedBall.frequently_iff] ** \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u22a2 \u2200 (i : \u211d), 0 < i \u2192 \u2203 x, x \u2208 Metric.closedBall f i \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc) ** intro \u03b5 h\u03b5 ** \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc) ** have A : ENNReal.ofReal \u03b5 \u2260 0 := by simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, h\u03b5] ** \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 A : ENNReal.ofReal \u03b5 \u2260 0 \u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc) ** obtain \u27e8g, hg, g_mem\u27e9 :\n    \u2203 g : \u03b1 \u2192\u1d47 E, snorm ((f : \u03b1 \u2192 E) - (g : \u03b1 \u2192 E)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u2227 Mem\u2112p g p \u03bc :=\n  (Lp.mem\u2112p f).exists_boundedContinuous_snorm_sub_le hp A ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 A : ENNReal.ofReal \u03b5 \u2260 0 g : \u03b1 \u2192\u1d47 E hg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc) ** refine' \u27e8g_mem.toLp _, _, \u27e8g, rfl\u27e9\u27e9 ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 A : ENNReal.ofReal \u03b5 \u2260 0 g : \u03b1 \u2192\u1d47 E hg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 Mem\u2112p.toLp (\u2191g) g_mem \u2208 Metric.closedBall f \u03b5 ** simp only [dist_eq_norm, Metric.mem_closedBall'] ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 A : ENNReal.ofReal \u03b5 \u2260 0 g : \u03b1 \u2192\u1d47 E hg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 \u2016f - Mem\u2112p.toLp (\u2191g) g_mem\u2016 \u2264 \u03b5 ** rw [Lp.norm_def] ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 A : ENNReal.ofReal \u03b5 \u2260 0 g : \u03b1 \u2192\u1d47 E hg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5 ** have key : snorm ((f : \u03b1 \u2192 E) - (g : \u03b1 \u2192 E)) p \u03bc = snorm (f - Mem\u2112p.toLp (\u2191g) g_mem) p \u03bc := by\n  apply snorm_congr_ae\n  filter_upwards [coeFn_sub f (g_mem.toLp g), g_mem.coeFn_toLp] with x hx h'x\n  simp only [hx, Pi.sub_apply, sub_right_inj, h'x] ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 A : ENNReal.ofReal \u03b5 \u2260 0 g : \u03b1 \u2192\u1d47 E hg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5 g_mem : Mem\u2112p (\u2191g) p key : snorm (\u2191\u2191f - \u2191g) p \u03bc = snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc \u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5 ** simpa only [key] using ENNReal.toReal_le_of_le_ofReal h\u03b5.le hg ** \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 ENNReal.ofReal \u03b5 \u2260 0 ** simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, h\u03b5] ** \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 A : ENNReal.ofReal \u03b5 \u2260 0 g : \u03b1 \u2192\u1d47 E hg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 snorm (\u2191\u2191f - \u2191g) p \u03bc = snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc ** apply snorm_congr_ae ** case hfg \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 A : ENNReal.ofReal \u03b5 \u2260 0 g : \u03b1 \u2192\u1d47 E hg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 \u2191\u2191f - \u2191g =\u1d50[\u03bc] \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) ** filter_upwards [coeFn_sub f (g_mem.toLp g), g_mem.coeFn_toLp] with x hx h'x ** case h \u03b1 : Type u_1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 inst\u271d\u2075 : T4Space \u03b1 inst\u271d\u2074 : BorelSpace \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E _i : Fact (1 \u2264 p) hp : p \u2260 \u22a4 inst\u271d : Measure.WeaklyRegular \u03bc f : { x // x \u2208 Lp E p } \u03b5 : \u211d h\u03b5 : 0 < \u03b5 A : ENNReal.ofReal \u03b5 \u2260 0 g : \u03b1 \u2192\u1d47 E hg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5 g_mem : Mem\u2112p (\u2191g) p x : \u03b1 hx : \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x = (\u2191\u2191f - \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem)) x h'x : \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem) x = \u2191g x \u22a2 (\u2191\u2191f - \u2191g) x = \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x ** simp only [hx, Pi.sub_apply, sub_right_inj, h'x] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.erase_erase ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a a' : \u03b1 s\u271d : Finmap \u03b2 s : AList \u03b2 \u22a2 (erase a (erase a' \u27e6s\u27e7)).entries = (erase a' (erase a \u27e6s\u27e7)).entries ** simp only [AList.erase_erase, erase_toFinmap] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 \u22a2 \u2191\u2191\u03bc (f '' s) = 0 ** suffices H : \u2200 R, \u03bc (f '' (s \u2229 closedBall 0 R)) = 0 ** case H E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 \u22a2 \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 ** intro R ** case H E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 R : \u211d \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 ** have A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u03b5 * \u03bc (closedBall 0 R) :=\n  fun \u03b5 \u03b5pos =>\n  addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux \u03bc\n    (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _)) R (inter_subset_right _ _) \u03b5 \u03b5pos\n    fun x hx => h'f' x hx.1 ** case H E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 R : \u211d A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 ** have B : Tendsto (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (closedBall 0 R)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) := by\n  have :\n    Tendsto (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (closedBall 0 R)) (\ud835\udcdd 0)\n      (\ud835\udcdd (((0 : \u211d\u22650) : \u211d\u22650\u221e) * \u03bc (closedBall 0 R))) :=\n    ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 tendsto_id)\n      (Or.inr measure_closedBall_lt_top.ne)\n  simp only [zero_mul, ENNReal.coe_zero] at this\n  exact Tendsto.mono_left this nhdsWithin_le_nhds ** case H E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 R : \u211d A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) B : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 ** apply le_antisymm _ (zero_le _) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 R : \u211d A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) B : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 0 ** apply ge_of_tendsto B ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 R : \u211d A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) B : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191c * \u2191\u2191\u03bc (closedBall 0 R) ** filter_upwards [self_mem_nhdsWithin] ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 R : \u211d A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) B : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200 (a : \u211d\u22650), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191a * \u2191\u2191\u03bc (closedBall 0 R) ** exact A ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 H : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 \u22a2 \u2191\u2191\u03bc (f '' s) = 0 ** apply le_antisymm _ (zero_le _) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 H : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 0 ** rw [\u2190 iUnion_inter_closedBall_nat s 0] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 H : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 \u22a2 \u2191\u2191\u03bc (f '' \u22c3 n, s \u2229 closedBall 0 \u2191n) \u2264 0 ** calc\n  \u03bc (f '' \u22c3 n : \u2115, s \u2229 closedBall 0 n) \u2264 \u2211' n : \u2115, \u03bc (f '' (s \u2229 closedBall 0 n)) := by\n    rw [image_iUnion]; exact measure_iUnion_le _\n  _ \u2264 0 := by simp only [H, tsum_zero, nonpos_iff_eq_zero] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 H : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 \u22a2 \u2191\u2191\u03bc (f '' \u22c3 n, s \u2229 closedBall 0 \u2191n) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n)) ** rw [image_iUnion] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 H : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 \u22a2 \u2191\u2191\u03bc (\u22c3 i, f '' (s \u2229 closedBall 0 \u2191i)) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n)) ** exact measure_iUnion_le _ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 H : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0 \u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n)) \u2264 0 ** simp only [H, tsum_zero, nonpos_iff_eq_zero] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 R : \u211d A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) \u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** have :\n  Tendsto (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (closedBall 0 R)) (\ud835\udcdd 0)\n    (\ud835\udcdd (((0 : \u211d\u22650) : \u211d\u22650\u221e) * \u03bc (closedBall 0 R))) :=\n  ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 tendsto_id)\n    (Or.inr measure_closedBall_lt_top.ne) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 R : \u211d A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) this : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd (\u21910 * \u2191\u2191\u03bc (closedBall 0 R))) \u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** simp only [zero_mul, ENNReal.coe_zero] at this ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 R : \u211d A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) this : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd 0) \u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** exact Tendsto.mono_left this nhdsWithin_le_nhds ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_le_coe_iff_finite_ncard_le ** \u03b1 : Type u_1 s t : Set \u03b1 k : \u2115 \u22a2 encard s \u2264 \u2191k \u2194 Set.Finite s \u2227 ncard s \u2264 k ** rw [encard_le_coe_iff, and_congr_right_iff] ** \u03b1 : Type u_1 s t : Set \u03b1 k : \u2115 \u22a2 Set.Finite s \u2192 ((\u2203 n\u2080, encard s = \u2191n\u2080 \u2227 n\u2080 \u2264 k) \u2194 ncard s \u2264 k) ** exact fun hfin \u21a6 \u27e8fun \u27e8n\u2080, hn\u2080, hle\u27e9 \u21a6 by rwa [ncard_def, hn\u2080, ENat.toNat_coe],\n  fun h \u21a6 \u27e8s.ncard, by rw [hfin.cast_ncard_eq], h\u27e9\u27e9 ** \u03b1 : Type u_1 s t : Set \u03b1 k : \u2115 hfin : Set.Finite s x\u271d : \u2203 n\u2080, encard s = \u2191n\u2080 \u2227 n\u2080 \u2264 k n\u2080 : \u2115 hn\u2080 : encard s = \u2191n\u2080 hle : n\u2080 \u2264 k \u22a2 ncard s \u2264 k ** rwa [ncard_def, hn\u2080, ENat.toNat_coe] ** \u03b1 : Type u_1 s t : Set \u03b1 k : \u2115 hfin : Set.Finite s h : ncard s \u2264 k \u22a2 encard s = \u2191(ncard s) ** rw [hfin.cast_ncard_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "sign_sum ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 \u03b9 : Type u_2 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 hs : Finset.Nonempty s t : SignType h : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = t \u22a2 \u2191sign (\u2211 i in s, f i) = t ** cases t ** case zero \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 \u03b9 : Type u_2 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 hs : Finset.Nonempty s h : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = zero \u22a2 \u2191sign (\u2211 i in s, f i) = zero ** simp_rw [zero_eq_zero, sign_eq_zero_iff] at h \u22a2 ** case zero \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 \u03b9 : Type u_2 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 hs : Finset.Nonempty s h : \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 0 \u22a2 \u2211 i in s, f i = 0 ** exact Finset.sum_eq_zero h ** case neg \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 \u03b9 : Type u_2 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 hs : Finset.Nonempty s h : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = neg \u22a2 \u2191sign (\u2211 i in s, f i) = neg ** simp_rw [neg_eq_neg_one, sign_eq_neg_one_iff] at h \u22a2 ** case neg \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 \u03b9 : Type u_2 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 hs : Finset.Nonempty s h : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < 0 \u22a2 \u2211 i in s, f i < 0 ** exact Finset.sum_neg h hs ** case pos \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 \u03b9 : Type u_2 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 hs : Finset.Nonempty s h : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191sign (f i) = pos \u22a2 \u2191sign (\u2211 i in s, f i) = pos ** simp_rw [pos_eq_one, sign_eq_one_iff] at h \u22a2 ** case pos \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 \u03b9 : Type u_2 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 hs : Finset.Nonempty s h : \u2200 (i : \u03b9), i \u2208 s \u2192 0 < f i \u22a2 0 < \u2211 i in s, f i ** exact Finset.sum_pos h hs ** Qed",
        "informal": ""
    },
    {
        "formal": "List.range_eq_range' ** n : Nat \u22a2 range' 0 (0 + n) = range' 0 n ** rw [Nat.zero_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.not_empty_toList ** n : \u2115 \u03b1 : Type u_1 v : Vector \u03b1 (n + 1) \u22a2 \u00acList.isEmpty (toList v) = true ** simp only [empty_toList_eq_ff, Bool.coe_sort_false, not_false_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "lowerClosure_infs ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeInf \u03b1 s t : Set \u03b1 \u22a2 lowerClosure (s \u22bc t) = lowerClosure s \u2293 lowerClosure t ** ext a ** case a.h F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeInf \u03b1 s t : Set \u03b1 a : \u03b1 \u22a2 a \u2208 \u2191(lowerClosure (s \u22bc t)) \u2194 a \u2208 \u2191(lowerClosure s \u2293 lowerClosure t) ** simp only [SetLike.mem_coe, mem_lowerClosure, Set.mem_infs, exists_and_left, exists_prop,\n  LowerSet.coe_sup, Set.mem_inter_iff] ** case a.h F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeInf \u03b1 s t : Set \u03b1 a : \u03b1 \u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1) \u2194 a \u2208 lowerClosure s \u2293 lowerClosure t ** constructor ** case a.h.mp F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeInf \u03b1 s t : Set \u03b1 a : \u03b1 \u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1) \u2192 a \u2208 lowerClosure s \u2293 lowerClosure t ** rintro \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, ha\u27e9 ** case a.h.mp.intro.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeInf \u03b1 s t : Set \u03b1 a b : \u03b1 hb : b \u2208 s c : \u03b1 hc : c \u2208 t ha : a \u2264 b \u2293 c \u22a2 a \u2208 lowerClosure s \u2293 lowerClosure t ** exact \u27e8\u27e8b, hb, ha.trans inf_le_left\u27e9, c, hc, ha.trans inf_le_right\u27e9 ** case a.h.mpr F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeInf \u03b1 s t : Set \u03b1 a : \u03b1 \u22a2 a \u2208 lowerClosure s \u2293 lowerClosure t \u2192 \u2203 a_2, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_2) \u2227 a \u2264 a_2 ** rintro \u27e8\u27e8b, hb, hab\u27e9, c, hc, hac\u27e9 ** case a.h.mpr.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeInf \u03b1 s t : Set \u03b1 a b : \u03b1 hb : b \u2208 s hab : a \u2264 b c : \u03b1 hc : c \u2208 t hac : a \u2264 c \u22a2 \u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2293 b = a_1) \u2227 a \u2264 a_1 ** exact \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, le_inf hab hac\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) ** rcases eq_empty_or_nonempty s with (rfl | h's) ** case inr E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) ** have :\n    \u2200 A : E \u2192L[\u211d] E, \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227\n      \u2200 (t : Set E), ApproximatesLinearOn f A t \u03b4 \u2192\n        \u03bc (f '' t) \u2264 (Real.toNNReal |A.det| + \u03b5 : \u211d\u22650) * \u03bc t := by\n  intro A\n  let m : \u211d\u22650 := Real.toNNReal |A.det| + \u03b5\n  have I : ENNReal.ofReal |A.det| < m := by\n    simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]\n  rcases ((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, h'\u27e9\n  exact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9 ** case inr E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s this :   \u2200 (A : E \u2192L[\u211d] E),     \u2203 \u03b4,       0 < \u03b4 \u2227         \u2200 (t : Set E),           ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) ** choose \u03b4 h\u03b4 using this ** case inr E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) ** obtain \u27e8t, A, t_disj, t_meas, t_cover, ht, Af'\u27e9 :\n  \u2203 (t : \u2115 \u2192 Set E) (A : \u2115 \u2192 E \u2192L[\u211d] E),\n    Pairwise (Disjoint on t) \u2227\n      (\u2200 n : \u2115, MeasurableSet (t n)) \u2227\n        (s \u2286 \u22c3 n : \u2115, t n) \u2227\n          (\u2200 n : \u2115, ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))) \u2227\n            (s.Nonempty \u2192 \u2200 n, \u2203 y \u2208 s, A n = f' y) :=\n  exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' \u03b4 fun A => (h\u03b4 A).1.ne' ** case inl E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc R : \u211d \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 hf' : \u2200 (x : E), x \u2208 \u2205 \u2192 HasFDerivWithinAt f (f' x) \u2205 x hs : \u2205 \u2286 closedBall 0 R h'f' : \u2200 (x : E), x \u2208 \u2205 \u2192 ContinuousLinearMap.det (f' x) = 0 \u22a2 \u2191\u2191\u03bc (f '' \u2205) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) ** simp only [measure_empty, zero_le, image_empty] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u22a2 \u2200 (A : E \u2192L[\u211d] E),     \u2203 \u03b4,       0 < \u03b4 \u2227         \u2200 (t : Set E),           ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t ** intro A ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s A : E \u2192L[\u211d] E \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t ** let m : \u211d\u22650 := Real.toNNReal |A.det| + \u03b5 ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s A : E \u2192L[\u211d] E m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5 \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t ** have I : ENNReal.ofReal |A.det| < m := by\n  simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s A : E \u2192L[\u211d] E m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5 I : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t ** rcases ((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, h'\u27e9 ** case intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s A : E \u2192L[\u211d] E m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5 I : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m \u03b4 : \u211d\u22650 h : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s h' : 0 < \u03b4 \u22a2 \u2203 \u03b4,     0 < \u03b4 \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t ** exact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9 ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s A : E \u2192L[\u211d] E m : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5 \u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m ** simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 n, f '' (s \u2229 t n)) ** apply measure_mono ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 f '' s \u2286 \u22c3 n, f '' (s \u2229 t n) ** rw [\u2190 image_iUnion, \u2190 inter_iUnion] ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 i, t i) ** exact image_subset f (subset_inter Subset.rfl t_cover) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) ** apply ENNReal.tsum_le_tsum fun n => ?_ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y n : \u2115 \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) ** apply (h\u03b4 (A n)).2 ** case a E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y n : \u2115 \u22a2 ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) ** exact ht n ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2211' (n : \u2115), \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) ** congr with n ** case e_f.h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y n : \u2115 \u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) ** rcases Af' h's n with \u27e8y, ys, hy\u27e9 ** case e_f.h.intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y n : \u2115 y : E ys : y \u2208 s hy : A n = f' y \u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) ** simp only [hy, h'f' y ys, Real.toNNReal_zero, abs_zero, zero_add] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2211' (n : \u2115), \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) \u2264 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n) ** rw [ENNReal.tsum_mul_left] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i) \u2264 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n) ** refine' mul_le_mul_left' (ENNReal.tsum_le_tsum fun n => measure_mono _) _ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y n : \u2115 \u22a2 s \u2229 t n \u2286 closedBall 0 R \u2229 t n ** exact inter_subset_inter_left _ hs ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, closedBall 0 R \u2229 t n) ** rw [measure_iUnion] ** case hn E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 Pairwise (Disjoint on fun n => closedBall 0 R \u2229 t n) ** exact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _ ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2200 (i : \u2115), MeasurableSet (closedBall 0 R \u2229 t i) ** intro n ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y n : \u2115 \u22a2 MeasurableSet (closedBall 0 R \u2229 t n) ** exact measurableSet_closedBall.inter (t_meas n) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 n, closedBall 0 R \u2229 t n) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) ** rw [\u2190 inter_iUnion] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x R : \u211d hs : s \u2286 closedBall 0 R \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0 h's : Set.Nonempty s \u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650 h\u03b4 :   \u2200 (A : E \u2192L[\u211d] E),     0 < \u03b4 A \u2227       \u2200 (t : Set E),         ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t t : \u2115 \u2192 Set E A : \u2115 \u2192 E \u2192L[\u211d] E t_disj : Pairwise (Disjoint on t) t_meas : \u2200 (n : \u2115), MeasurableSet (t n) t_cover : s \u2286 \u22c3 n, t n ht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n)) Af' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y \u22a2 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R \u2229 \u22c3 i, t i) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R) ** exact mul_le_mul_left' (measure_mono (inter_subset_left _ _)) _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.orderEmbOfFin_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : LinearOrder \u03b1 s : Finset \u03b1 k : \u2115 h : card s = k hz : 0 < k \u22a2 \u2191(orderEmbOfFin s h) { val := 0, isLt := hz } = min' s (_ : Finset.Nonempty s) ** simp only [orderEmbOfFin_apply, Fin.val_mk, sorted_zero_eq_min'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.natAbs_eq_iff_sq_eq ** a\u271d b\u271d : \u2124 n : \u2115 a b : \u2124 \u22a2 natAbs a = natAbs b \u2194 a ^ 2 = b ^ 2 ** rw [sq, sq] ** a\u271d b\u271d : \u2124 n : \u2115 a b : \u2124 \u22a2 natAbs a = natAbs b \u2194 a * a = b * b ** exact natAbs_eq_iff_mul_self_eq ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.bind_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p\u271d : PMF \u03b1 f : \u03b1 \u2192 PMF \u03b2 g : \u03b2 \u2192 PMF \u03b3 p : PMF \u03b1 q : PMF \u03b2 x : \u03b2 \u22a2 \u2191(bind p fun x => q) x = \u2191q x ** rw [bind_apply, ENNReal.tsum_mul_right, tsum_coe, one_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.borel_eq_generateFrom_Iio_rat ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 \u22a2 borel \u211d = MeasurableSpace.generateFrom (\u22c3 a, {Iio \u2191a}) ** rw [borel_eq_generateFrom_Iio] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 \u22a2 MeasurableSpace.generateFrom (range Iio) = MeasurableSpace.generateFrom (\u22c3 a, {Iio \u2191a}) ** refine le_antisymm\n  (generateFrom_le ?_)\n  (generateFrom_mono <| iUnion_subset fun q \u21a6 singleton_subset_iff.mpr <| mem_range_self _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 \u22a2 \u2200 (t : Set \u211d), t \u2208 range Iio \u2192 MeasurableSet t ** rintro _ \u27e8a, rfl\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 a : \u211d \u22a2 MeasurableSet (Iio a) ** have : IsLUB (range ((\u2191) : \u211a \u2192 \u211d) \u2229 Iio a) a := by\n  simp [isLUB_iff_le_iff, mem_upperBounds, \u2190 le_iff_forall_rat_lt_imp_le] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 a : \u211d this : IsLUB (range Rat.cast \u2229 Iio a) a \u22a2 MeasurableSet (Iio a) ** rw [\u2190 this.biUnion_Iio_eq, \u2190 image_univ, \u2190 image_inter_preimage, univ_inter, biUnion_image] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 a : \u211d this : IsLUB (range Rat.cast \u2229 Iio a) a \u22a2 MeasurableSet (\u22c3 y \u2208 Rat.cast \u207b\u00b9' Iio a, Iio \u2191y) ** exact MeasurableSet.biUnion (to_countable _)\n  fun b _ => GenerateMeasurable.basic (Iio (b : \u211d)) (by simp) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 a : \u211d \u22a2 IsLUB (range Rat.cast \u2229 Iio a) a ** simp [isLUB_iff_le_iff, mem_upperBounds, \u2190 le_iff_forall_rat_lt_imp_le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 a : \u211d this : IsLUB (range Rat.cast \u2229 Iio a) a b : \u211a x\u271d : b \u2208 Rat.cast \u207b\u00b9' Iio a \u22a2 Iio \u2191b \u2208 \u22c3 a, {Iio \u2191a} ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.JordanDecomposition.eq_of_posPart_eq_posPart ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 \u22a2 j\u2081 = j\u2082 ** ext1 ** case posPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 \u22a2 j\u2081.posPart = j\u2082.posPart ** exact hj ** case negPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 \u22a2 j\u2081.negPart = j\u2082.negPart ** rw [\u2190 toSignedMeasure_eq_toSignedMeasure_iff] ** case negPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 \u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart ** unfold toSignedMeasure at hj' ** case negPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' :   Measure.toSignedMeasure j\u2081.posPart - Measure.toSignedMeasure j\u2081.negPart =     Measure.toSignedMeasure j\u2082.posPart - Measure.toSignedMeasure j\u2082.negPart \u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart ** simp_rw [hj, sub_right_inj] at hj' ** case negPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart \u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart ** exact hj' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** have A : Tendsto (fun r : \u211d => \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc (closedBall x r)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) := by\n  apply\n    tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds h\n      (eventually_of_forall fun b => zero_le _)\n  filter_upwards [self_mem_nhdsWithin]\n  rintro r (rpos : 0 < r)\n  apply mul_le_mul_right' (measure_mono (inter_subset_inter_right _ _)) _\n  intro y hy\n  have : y - x \u2208 r \u2022 closedBall (0 : E) 1 := by\n    apply smul_set_mono t_bound\n    simpa [neg_add_eq_sub] using hy\n  simpa only [smul_closedBall _ _ zero_le_one, Real.norm_of_nonneg rpos.le,\n    mem_closedBall_iff_norm, mul_one, sub_zero, smul_zero] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** have B :\n  Tendsto (fun r : \u211d => \u03bc (closedBall x r) / \u03bc ({x} + r \u2022 u)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (closedBall x 1) / \u03bc ({x} + u))) := by\n  apply tendsto_const_nhds.congr' _\n  filter_upwards [self_mem_nhdsWithin]\n  rintro r (rpos : 0 < r)\n  have : closedBall x r = {x} + r \u2022 closedBall (0 : E) 1 := by\n    simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero,\n      mul_one, singleton_add_closedBall, smul_zero]\n  simp only [this, addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne']\n  simp only [addHaar_closedBall_center, image_add_left, measure_preimage_add, singleton_add] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** have C : Tendsto (fun r : \u211d =>\n      \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc (closedBall x r) * (\u03bc (closedBall x r) / \u03bc ({x} + r \u2022 u)))\n    (\ud835\udcdd[>] 0) (\ud835\udcdd (0 * (\u03bc (closedBall x 1) / \u03bc ({x} + u)))) := by\n  apply ENNReal.Tendsto.mul A _ B (Or.inr ENNReal.zero_ne_top)\n  simp only [ne_eq, not_true, singleton_add, image_add_left, measure_preimage_add, false_or,\n    ENNReal.div_eq_top, h'u, false_or_iff, not_and, and_false_iff]\n  intro aux\n  exact (measure_closedBall_lt_top.ne aux).elim ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) C :   Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** simp only [zero_mul] at C ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) C :   Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** apply C.congr' _ ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) C :   Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u))) =\u1da0[\ud835\udcdd[Ioi 0] 0]     fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** filter_upwards [self_mem_nhdsWithin] ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) C :   Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200 (a : \u211d),     a \u2208 Ioi 0 \u2192       \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc (closedBall x a) * (\u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc ({x} + a \u2022 u)) =         \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 u) ** rintro r (rpos : 0 < r) ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) C :   Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =     \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** calc\n  \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc (closedBall x r) * (\u03bc (closedBall x r) / \u03bc ({x} + r \u2022 u)) =\n      \u03bc (closedBall x r) * (\u03bc (closedBall x r))\u207b\u00b9 * (\u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 u)) :=\n    by simp only [div_eq_mul_inv]; ring\n  _ = \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 u) := by\n    rw [ENNReal.mul_inv_cancel (measure_closedBall_pos \u03bc x rpos).ne'\n        measure_closedBall_lt_top.ne,\n      one_mul] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** apply\n  tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds h\n    (eventually_of_forall fun b => zero_le _) ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0,     \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc (closedBall x b) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) ** filter_upwards [self_mem_nhdsWithin] ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 \u22a2 \u2200 (a : \u211d),     a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc (closedBall x a) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a) ** rintro r (rpos : 0 < r) ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 r : \u211d rpos : 0 < r \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) ** apply mul_le_mul_right' (measure_mono (inter_subset_inter_right _ _)) _ ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 r : \u211d rpos : 0 < r \u22a2 {x} + r \u2022 t \u2286 closedBall x r ** intro y hy ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 r : \u211d rpos : 0 < r y : E hy : y \u2208 {x} + r \u2022 t \u22a2 y \u2208 closedBall x r ** have : y - x \u2208 r \u2022 closedBall (0 : E) 1 := by\n  apply smul_set_mono t_bound\n  simpa [neg_add_eq_sub] using hy ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 r : \u211d rpos : 0 < r y : E hy : y \u2208 {x} + r \u2022 t this : y - x \u2208 r \u2022 closedBall 0 1 \u22a2 y \u2208 closedBall x r ** simpa only [smul_closedBall _ _ zero_le_one, Real.norm_of_nonneg rpos.le,\n  mem_closedBall_iff_norm, mul_one, sub_zero, smul_zero] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 r : \u211d rpos : 0 < r y : E hy : y \u2208 {x} + r \u2022 t \u22a2 y - x \u2208 r \u2022 closedBall 0 1 ** apply smul_set_mono t_bound ** case a E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 r : \u211d rpos : 0 < r y : E hy : y \u2208 {x} + r \u2022 t \u22a2 y - x \u2208 r \u2022 t ** simpa [neg_add_eq_sub] using hy ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) ** apply tendsto_const_nhds.congr' _ ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 (fun x_1 => \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** filter_upwards [self_mem_nhdsWithin] ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc ({x} + a \u2022 u) ** rintro r (rpos : 0 < r) ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r \u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** have : closedBall x r = {x} + r \u2022 closedBall (0 : E) 1 := by\n  simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero,\n    mul_one, singleton_add_closedBall, smul_zero] ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r this : closedBall x r = {x} + r \u2022 closedBall 0 1 \u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** simp only [this, addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne'] ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r this : closedBall x r = {x} + r \u2022 closedBall 0 1 \u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall 0 1) / \u2191\u2191\u03bc u ** simp only [addHaar_closedBall_center, image_add_left, measure_preimage_add, singleton_add] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r \u22a2 closedBall x r = {x} + r \u2022 closedBall 0 1 ** simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero,\n  mul_one, singleton_add_closedBall, smul_zero] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))) ** apply ENNReal.Tendsto.mul A _ B (Or.inr ENNReal.zero_ne_top) ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) \u22a2 0 \u2260 0 \u2228 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) \u2260 \u22a4 ** simp only [ne_eq, not_true, singleton_add, image_add_left, measure_preimage_add, false_or,\n  ENNReal.div_eq_top, h'u, false_or_iff, not_and, and_false_iff] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) \u22a2 \u2191\u2191\u03bc (closedBall x 1) = \u22a4 \u2192 \u00ac\u00ac\u2191\u2191\u03bc u = \u22a4 ** intro aux ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) aux : \u2191\u2191\u03bc (closedBall x 1) = \u22a4 \u22a2 \u00ac\u00ac\u2191\u2191\u03bc u = \u22a4 ** exact (measure_closedBall_lt_top.ne aux).elim ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) C :   Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =     \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) ** simp only [div_eq_mul_inv] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) C :   Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc ({x} + r \u2022 u))\u207b\u00b9) =     \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) * (\u2191\u2191\u03bc ({x} + r \u2022 u))\u207b\u00b9) ** ring ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t u : Set E h'u : \u2191\u2191\u03bc u \u2260 0 t_bound : t \u2286 closedBall 0 1 A : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) B : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))) C :   Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))     (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d rpos : 0 < r \u22a2 \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =     \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u) ** rw [ENNReal.mul_inv_cancel (measure_closedBall_pos \u03bc x rpos).ne'\n    measure_closedBall_lt_top.ne,\n  one_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.quasiMeasurePreserving_div_of_right_invariant ** G : Type u_1 inst\u271d\u2076 : MeasurableSpace G inst\u271d\u2075 : Group G inst\u271d\u2074 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b3 : SigmaFinite \u03bd inst\u271d\u00b2 : SigmaFinite \u03bc s : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulRightInvariant \u03bc \u22a2 QuasiMeasurePreserving fun p => p.1 / p.2 ** refine' QuasiMeasurePreserving.prod_of_left measurable_div (eventually_of_forall fun y => _) ** G : Type u_1 inst\u271d\u2076 : MeasurableSpace G inst\u271d\u2075 : Group G inst\u271d\u2074 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b3 : SigmaFinite \u03bd inst\u271d\u00b2 : SigmaFinite \u03bc s : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulRightInvariant \u03bc y : G \u22a2 QuasiMeasurePreserving fun x => (x, y).1 / (x, y).2 ** exact (measurePreserving_div_right \u03bc y).quasiMeasurePreserving ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.ae_tendsto_of_cauchy_snorm' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** have h_summable : \u2200\u1d50 x \u2202\u03bc, Summable fun i : \u2115 => f (i + 1) x - f i x := by\n  have h1 :\n    \u2200 n, snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n    snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau\n  have h2 :\n    \u2200 n,\n      (\u222b\u207b a, (\u2211 i in Finset.range (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n        (\u2211' i, B i) ^ p :=\n    fun n => lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hp1 n (h1 n)\n  have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n    lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2\n  have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n    tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3\n  exact\n    h4.mono fun x hx =>\n      summable_of_summable_nnnorm\n        (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx)) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** have h :\n  \u2200\u1d50 x \u2202\u03bc, \u2203 l : E,\n    atTop.Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) (\ud835\udcdd l) := by\n  refine' h_summable.mono fun x hx => _\n  let hx_sum := hx.hasSum.tendsto_sum_nat\n  exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** refine' h.mono fun x hx => _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 hx : \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** cases' hx with l hx ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** have h_rw_sum :\n    (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x := by\n  ext1 n\n  change\n    (\u2211 i : \u2115 in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x\n  rw [Finset.sum_range_sub (fun m => f m x)] ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) h_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** rw [h_rw_sum] at hx ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** have hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x := by\n  ext1 n\n  abel ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x \u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l) ** rw [hf_rw] ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x \u22a2 \u2203 l, Tendsto (fun n => f n x - f 0 x + f 0 x) atTop (\ud835\udcdd l) ** exact \u27e8l + f 0 x, Tendsto.add_const _ hx\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** have h1 :\n  \u2200 n, snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n  snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** have h2 :\n  \u2200 n,\n    (\u222b\u207b a, (\u2211 i in Finset.range (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n      (\u2211' i, B i) ^ p :=\n  fun n => lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hp1 n (h1 n) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i h2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n  lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i h2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p h3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n  tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i h2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p h3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i h4 : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x ** exact\n  h4.mono fun x hx =>\n    summable_of_summable_nnnorm\n      (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx)) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) ** refine' h_summable.mono fun x hx => _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x x : \u03b1 hx : Summable fun i => f (i + 1) x - f i x \u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) ** let hx_sum := hx.hasSum.tendsto_sum_nat ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x x : \u03b1 hx : Summable fun i => f (i + 1) x - f i x hx_sum : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop   (\ud835\udcdd (\u2211' (b : \u2115), (f (b + 1) x - f b x))) :=   HasSum.tendsto_sum_nat (Summable.hasSum hx) \u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) ** exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) \u22a2 (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x ** ext1 n ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) n : \u2115 \u22a2 \u2211 i in Finset.range n, (f (i + 1) x - f i x) = f n x - f 0 x ** change\n  (\u2211 i : \u2115 in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) n : \u2115 \u22a2 \u2211 i in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i) = f n x - f 0 x ** rw [Finset.sum_range_sub (fun m => f m x)] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x \u22a2 (fun n => f n x) = fun n => f n x - f 0 x + f 0 x ** ext1 n ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03b1 \u2192 E p : \u211d hf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc hp1 : 1 \u2264 p B : \u2115 \u2192 \u211d\u22650\u221e hB : \u2211' (i : \u2115), B i \u2260 \u22a4 h_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N h_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l) x : \u03b1 l : E hx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l) h_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x n : \u2115 \u22a2 f n x = f n x - f 0 x + f 0 x ** abel ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_const_sub_uIcc ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 (fun x => a - x) \u207b\u00b9' [[b, c]] = [[a - b, a - c]] ** simp_rw [\u2190 Icc_min_max, preimage_const_sub_Icc] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 a b c d : \u03b1 \u22a2 Icc (a - max b c) (a - min b c) = Icc (min (a - b) (a - c)) (max (a - b) (a - c)) ** simp only [sub_eq_add_neg, min_add_add_left, max_add_add_left, min_neg_neg, max_neg_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.takeD_eq_takeDTR ** \u22a2 @takeD = @takeDTR ** funext \u03b1 f n l ** case h.h.h.h \u03b1 : Type u_1 f : Nat n : List \u03b1 l : \u03b1 \u22a2 takeD f n l = takeDTR f n l ** simp [takeDTR, takeDTR_go_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpL2_const_inner ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E \u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]     fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) ** have h_mem_Lp : Mem\u2112p (fun a => \u27eac, (condexpL2 E \ud835\udd5c hm f : \u03b1 \u2192 E) a\u27eb) 2 \u03bc := by\n  refine' Mem\u2112p.const_inner _ _; rw [lpMeas_coe]; exact Lp.mem\u2112p _ ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 \u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]     fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) ** have h_eq : h_mem_Lp.toLp _ =\u1d50[\u03bc] fun a => \u27eac, (condexpL2 E \ud835\udd5c hm f : \u03b1 \u2192 E) a\u27eb :=\n  h_mem_Lp.coeFn_toLp ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) \u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]     fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) ** refine' EventuallyEq.trans _ h_eq ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) \u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]     \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) ** refine' Lp.ae_eq_of_forall_set_integral_eq' \ud835\udd5c hm _ _ two_ne_zero ENNReal.coe_ne_top\n  (fun s _ h\u03bcs => integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs.ne _) _ _ _ _ ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E \u22a2 Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 ** refine' Mem\u2112p.const_inner _ _ ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E \u22a2 Mem\u2112p (fun a => \u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) 2 ** exact Lp.mem\u2112p _ ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) \u22a2 \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) s ** intro s _ h\u03bcs ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) s : Set \u03b1 a\u271d : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 IntegrableOn (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) s ** rw [IntegrableOn, integrable_congr (ae_restrict_of_ae h_eq)] ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) s : Set \u03b1 a\u271d : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 Integrable fun x => (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) x ** exact (integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs.ne _).const_inner _ ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) \u22a2 \u2200 (s : Set \u03b1),     MeasurableSet s \u2192       \u2191\u2191\u03bc s < \u22a4 \u2192         \u222b (x : \u03b1) in s,             \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)))               x \u2202\u03bc =           \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) x \u2202\u03bc ** intro s hs h\u03bcs ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s,       \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) x \u2202\u03bc =     \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) x \u2202\u03bc ** rw [\u2190 lpMeas_coe, integral_condexpL2_eq_of_fin_meas_real _ hs h\u03bcs.ne,\n  integral_congr_ae (ae_restrict_of_ae h_eq), lpMeas_coe, \u2190\n  L2.inner_indicatorConstLp_eq_set_integral_inner \ud835\udd5c (\u2191(condexpL2 E \ud835\udd5c hm f)) (hm s hs) c h\u03bcs.ne,\n  \u2190 inner_condexpL2_left_eq_right, condexpL2_indicator_of_measurable _ hs,\n  L2.inner_indicatorConstLp_eq_set_integral_inner \ud835\udd5c f (hm s hs) c h\u03bcs.ne,\n  set_integral_congr_ae (hm s hs)\n    ((Mem\u2112p.coeFn_toLp ((Lp.mem\u2112p f).const_inner c)).mono fun x hx _ => hx)] ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) \u22a2 AEStronglyMeasurable' m     (\u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)))) \u03bc ** exact lpMeas.aeStronglyMeasurable' _ ** case refine'_4 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) \u22a2 AEStronglyMeasurable' m (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) \u03bc ** refine' AEStronglyMeasurable'.congr _ h_eq.symm ** case refine'_4 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } c : E h_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2 h_eq :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>     inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) \u22a2 AEStronglyMeasurable' m (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) \u03bc ** exact (lpMeas.aeStronglyMeasurable' _).const_inner _ ** Qed",
        "informal": ""
    },
    {
        "formal": "essSup_indicator_eq_essSup_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 \u22a2 essSup (indicator s f) \u03bc = essSup f (Measure.restrict \u03bc s) ** refine'\n  le_antisymm _\n    (limsSup_le_limsSup_of_le (map_restrict_ae_le_map_indicator_ae hs)\n      (by isBoundedDefault) (by isBoundedDefault) ) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 \u22a2 essSup (indicator s f) \u03bc \u2264 essSup f (Measure.restrict \u03bc s) ** refine' limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_restrict_le => _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1da0 (n : \u03b2) in map f (Measure.ae (Measure.restrict \u03bc s)), n \u2264 c \u22a2 \u2200\u1da0 (n : \u03b2) in map (indicator s f) (Measure.ae \u03bc), n \u2264 c ** rw [eventually_map] at h_restrict_le \u22a2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (a : \u03b1) \u2202Measure.restrict \u03bc s, f a \u2264 c \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, indicator s f a \u2264 c ** rw [ae_restrict_iff' hs] at h_restrict_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hc : 0 \u2264 c \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, indicator s f a \u2264 c ** refine' h_restrict_le.mono fun x hxc => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hc : 0 \u2264 c x : \u03b1 hxc : x \u2208 s \u2192 f x \u2264 c \u22a2 indicator s f x \u2264 c ** by_cases hxs : x \u2208 s ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 \u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map f (Measure.ae (Measure.restrict \u03bc s))) ** isBoundedDefault ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 \u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map (indicator s f) (Measure.ae \u03bc)) ** isBoundedDefault ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 \u22a2 IsCobounded (fun x x_1 => x \u2264 x_1) (map (indicator s f) (Measure.ae \u03bc)) ** isBoundedDefault ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 \u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map f (Measure.ae (Measure.restrict \u03bc s))) ** isBoundedDefault ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c \u22a2 0 \u2264 c ** rsuffices \u27e8x, hx\u27e9 : \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c x : \u03b1 hx : 0 \u2264 f x \u2227 f x \u2264 c \u22a2 0 \u2264 c  \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c \u22a2 \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c ** exact hx.1.trans hx.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c \u22a2 \u2203 x, 0 \u2264 f x \u2227 f x \u2264 c ** refine' Frequently.exists _ ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c \u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c ** rw [EventuallyLE, ae_restrict_iff' hs] at hf ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c \u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c ** have hs' : \u2203\u1d50 x \u2202\u03bc, x \u2208 s := by\n  contrapose! hs_not_null\n  rw [not_frequently, ae_iff] at hs_not_null\n  suffices { a : \u03b1 | \u00aca \u2209 s } = s by rwa [\u2190 this]\n  simp ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 f x \u2227 f x \u2264 c ** refine' hs'.mp (hf.mp (h_restrict_le.mono fun x hxs_imp_c hxf_nonneg hxs => _)) ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s x : \u03b1 hxs_imp_c : x \u2208 s \u2192 f x \u2264 c hxf_nonneg : x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hxs : x \u2208 s \u22a2 0 \u2264 f x \u2227 f x \u2264 c ** rw [Pi.zero_apply] at hxf_nonneg ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hs' : \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s x : \u03b1 hxs_imp_c : x \u2208 s \u2192 f x \u2264 c hxf_nonneg : x \u2208 s \u2192 0 \u2264 f x hxs : x \u2208 s \u22a2 0 \u2264 f x \u2227 f x \u2264 c ** exact \u27e8hxf_nonneg hxs, hxs_imp_c hxs\u27e9 ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c \u22a2 Filter \u03b1 ** exact \u03bc.ae ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c \u22a2 \u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s ** contrapose! hs_not_null ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hs : MeasurableSet s c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hs_not_null : \u00ac\u2203\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u22a2 \u2191\u2191\u03bc s = 0 ** rw [not_frequently, ae_iff] at hs_not_null ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hs : MeasurableSet s c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0 \u22a2 \u2191\u2191\u03bc s = 0 ** suffices { a : \u03b1 | \u00aca \u2209 s } = s by rwa [\u2190 this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hs : MeasurableSet s c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0 \u22a2 {a | \u00ac\u00aca \u2208 s} = s ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 OfNat.ofNat 0 x \u2264 f x hs : MeasurableSet s c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hs_not_null : \u2191\u2191\u03bc {a | \u00ac\u00aca \u2208 s} = 0 this : {a | \u00ac\u00aca \u2208 s} = s \u22a2 \u2191\u2191\u03bc s = 0 ** rwa [\u2190 this] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hc : 0 \u2264 c x : \u03b1 hxc : x \u2208 s \u2192 f x \u2264 c hxs : x \u2208 s \u22a2 indicator s f x \u2264 c ** simpa [hxs] using hxc hxs ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : CompleteLinearOrder \u03b2 inst\u271d : Zero \u03b2 s : Set \u03b1 f : \u03b1 \u2192 \u03b2 hf : 0 \u2264\u1d50[Measure.restrict \u03bc s] f hs : MeasurableSet s hs_not_null : \u2191\u2191\u03bc s \u2260 0 c : \u03b2 h_restrict_le : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 c hc : 0 \u2264 c x : \u03b1 hxc : x \u2208 s \u2192 f x \u2264 c hxs : \u00acx \u2208 s \u22a2 indicator s f x \u2264 c ** simpa [hxs] using hc ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.size_swap! ** \u03b1 : Type u_1 a : Array \u03b1 i j : Nat hi : i < size a hj : j < size a \u22a2 size (swap! a i j) = size a ** simp [swap!, hi, hj] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.isCaratheodory_inter ** \u03b1 : Type u m : OuterMeasure \u03b1 s s\u2081 s\u2082 : Set \u03b1 h\u2081 : IsCaratheodory m s\u2081 h\u2082 : IsCaratheodory m s\u2082 \u22a2 IsCaratheodory m (s\u2081 \u2229 s\u2082) ** rw [\u2190 isCaratheodory_compl_iff, Set.compl_inter] ** \u03b1 : Type u m : OuterMeasure \u03b1 s s\u2081 s\u2082 : Set \u03b1 h\u2081 : IsCaratheodory m s\u2081 h\u2082 : IsCaratheodory m s\u2082 \u22a2 IsCaratheodory m (s\u2081\u1d9c \u222a s\u2082\u1d9c) ** exact isCaratheodory_union _ (isCaratheodory_compl _ h\u2081) (isCaratheodory_compl _ h\u2082) ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.insert_toList_zoom ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t' : RBNode \u03b1 p : Path \u03b1 v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n e : zoom (cmp v) t Path.root = (t', p) \u22a2 toList (insert cmp t v) = Path.withList p (toList (setRoot v t')) ** rw [\u2190 setBlack_toList, \u2190 Path.zoom_insert ht e, setBlack_toList, Path.insert_toList] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurePreserving_funUnique ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b2 : Type u m : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 \u03b1 : Type v inst\u271d : Unique \u03b1 \u22a2 MeasurePreserving \u2191(MeasurableEquiv.funUnique \u03b1 \u03b2) ** set e := MeasurableEquiv.funUnique \u03b1 \u03b2 ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b2 : Type u m : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 \u03b1 : Type v inst\u271d : Unique \u03b1 e : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2 \u22a2 MeasurePreserving \u2191e ** have : (piPremeasure fun _ : \u03b1 => \u03bc.toOuterMeasure) = Measure.map e.symm \u03bc := by\n  ext1 s\n  rw [piPremeasure, Fintype.prod_unique, e.symm.map_apply]\n  congr 1; exact e.toEquiv.image_eq_preimage s ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b2 : Type u m : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 \u03b1 : Type v inst\u271d : Unique \u03b1 e : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2 this : (piPremeasure fun x => \u2191\u03bc) = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc) \u22a2 MeasurePreserving \u2191e ** simp only [Measure.pi, OuterMeasure.pi, this, boundedBy_measure, toOuterMeasure_toMeasure] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b2 : Type u m : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 \u03b1 : Type v inst\u271d : Unique \u03b1 e : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2 this : (piPremeasure fun x => \u2191\u03bc) = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc) \u22a2 MeasurePreserving \u2191(MeasurableEquiv.funUnique \u03b1 \u03b2) ** exact (e.symm.measurable.measurePreserving _).symm e.symm ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b2 : Type u m : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 \u03b1 : Type v inst\u271d : Unique \u03b1 e : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2 \u22a2 (piPremeasure fun x => \u2191\u03bc) = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc) ** ext1 s ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b2 : Type u m : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 \u03b1 : Type v inst\u271d : Unique \u03b1 e : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2 s : Set (\u03b1 \u2192 \u03b2) \u22a2 piPremeasure (fun x => \u2191\u03bc) s = \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.symm e)) \u03bc) s ** rw [piPremeasure, Fintype.prod_unique, e.symm.map_apply] ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b2 : Type u m : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 \u03b1 : Type v inst\u271d : Unique \u03b1 e : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2 s : Set (\u03b1 \u2192 \u03b2) \u22a2 \u2191\u2191\u03bc (eval default '' s) = \u2191\u2191\u03bc (\u2191(MeasurableEquiv.symm e) \u207b\u00b9' s) ** congr 1 ** case h.e_a \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b2 : Type u m : MeasurableSpace \u03b2 \u03bc : Measure \u03b2 \u03b1 : Type v inst\u271d : Unique \u03b1 e : (\u03b1 \u2192 \u03b2) \u2243\u1d50 \u03b2 := MeasurableEquiv.funUnique \u03b1 \u03b2 s : Set (\u03b1 \u2192 \u03b2) \u22a2 eval default '' s = \u2191(MeasurableEquiv.symm e) \u207b\u00b9' s ** exact e.toEquiv.image_eq_preimage s ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.setToL1S_smul_real ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T c : \u211d f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 setToL1S T (c \u2022 f) = c \u2022 setToL1S T f ** simp_rw [setToL1S] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T c : \u211d f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (c \u2022 f)) = c \u2022 SimpleFunc.setToSimpleFunc T (toSimpleFunc f) ** rw [\u2190 SimpleFunc.setToSimpleFunc_smul_real T h_add c (SimpleFunc.integrable f)] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T c : \u211d f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 SimpleFunc.setToSimpleFunc T (toSimpleFunc (c \u2022 f)) = SimpleFunc.setToSimpleFunc T (c \u2022 toSimpleFunc f) ** refine' SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E T : Set \u03b1 \u2192 E \u2192L[\u211d] F h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T c : \u211d f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(toSimpleFunc (c \u2022 f)) =\u1d50[\u03bc] \u2191(c \u2022 toSimpleFunc f) ** exact smul_toSimpleFunc c f ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.mapAccumr_bisim_tail ** \u03b1 : Type u_2 n : \u2115 xs : Vector \u03b1 n \u03c3\u2081 : Type \u03b2 : Type u_1 \u03c3\u2082 : Type f\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2 f\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 h : \u2203 R, R s\u2081 s\u2082 \u2227 \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).1 (f\u2082 a q).1 \u2227 (f\u2081 a s).2 = (f\u2082 a q).2 \u22a2 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2 ** rcases h with \u27e8R, h\u2080, hR\u27e9 ** case intro.intro \u03b1 : Type u_2 n : \u2115 xs : Vector \u03b1 n \u03c3\u2081 : Type \u03b2 : Type u_1 \u03c3\u2082 : Type f\u2081 : \u03b1 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b2 f\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b2 s\u2081 : \u03c3\u2081 s\u2082 : \u03c3\u2082 R : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop h\u2080 : R s\u2081 s\u2082 hR : \u2200 {s : \u03c3\u2081} {q : \u03c3\u2082} (a : \u03b1), R s q \u2192 R (f\u2081 a s).1 (f\u2082 a q).1 \u2227 (f\u2081 a s).2 = (f\u2082 a q).2 \u22a2 (mapAccumr f\u2081 xs s\u2081).2 = (mapAccumr f\u2082 xs s\u2082).2 ** exact (mapAccumr_bisim R h\u2080 hR).2 ** Qed",
        "informal": ""
    },
    {
        "formal": "BoundedContinuousFunction.toLp_inj ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E inst\u271d\u2074 : IsFiniteMeasure \u03bc \ud835\udd5c : Type u_5 inst\u271d\u00b3 : Fact (1 \u2264 p) f g : \u03b1 \u2192\u1d47 E inst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E \u22a2 \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g \u2194 f = g ** refine' \u27e8fun h => _, by tauto\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E inst\u271d\u2074 : IsFiniteMeasure \u03bc \ud835\udd5c : Type u_5 inst\u271d\u00b3 : Fact (1 \u2264 p) f g : \u03b1 \u2192\u1d47 E inst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E h : \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g \u22a2 f = g ** rw [\u2190 FunLike.coe_fn_eq, \u2190 (map_continuous f).ae_eq_iff_eq \u03bc (map_continuous g)] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E inst\u271d\u2074 : IsFiniteMeasure \u03bc \ud835\udd5c : Type u_5 inst\u271d\u00b3 : Fact (1 \u2264 p) f g : \u03b1 \u2192\u1d47 E inst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E h : \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g \u22a2 \u2191f =\u1d50[\u03bc] \u2191g ** refine' (coeFn_toLp p \u03bc \ud835\udd5c f).symm.trans (EventuallyEq.trans _ <| coeFn_toLp p \u03bc \ud835\udd5c g) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E inst\u271d\u2074 : IsFiniteMeasure \u03bc \ud835\udd5c : Type u_5 inst\u271d\u00b3 : Fact (1 \u2264 p) f g : \u03b1 \u2192\u1d47 E inst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E h : \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g \u22a2 \u2191\u2191(\u2191(toLp p \u03bc \ud835\udd5c) f) =\u1d50[\u03bc] \u2191\u2191(\u2191(toLp p \u03bc \ud835\udd5c) g) ** rw [h] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : SecondCountableTopologyEither \u03b1 E inst\u271d\u2074 : IsFiniteMeasure \u03bc \ud835\udd5c : Type u_5 inst\u271d\u00b3 : Fact (1 \u2264 p) f g : \u03b1 \u2192\u1d47 E inst\u271d\u00b2 : Measure.IsOpenPosMeasure \u03bc inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedSpace \ud835\udd5c E \u22a2 f = g \u2192 \u2191(toLp p \u03bc \ud835\udd5c) f = \u2191(toLp p \u03bc \ud835\udd5c) g ** tauto ** Qed",
        "informal": ""
    },
    {
        "formal": "tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080)) ** let h := g - fun _ => g x\u2080 ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 A : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (0 + 1 \u2022 g x\u2080)) \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080)) ** simp only [one_smul, zero_add] at A ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 A : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080)) \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc) l (\ud835\udcdd (g x\u2080)) ** refine' Tendsto.congr' _ A ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 A : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080)) \u22a2 (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) =\u1da0[l] fun i =>     \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc ** filter_upwards [integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt hs hl\u03c6 hi\u03c6 hmg hcg,\n  hi\u03c6] with i hi h'i ** case h \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 A : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080)) i : \u03b9 hi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i \u22a2 \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080 = \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc ** simp only [Pi.sub_apply, smul_sub] ** case h \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 A : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080)) i : \u03b9 hi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i \u22a2 \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x - \u03c6 i x \u2022 g x\u2080 \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080 = \u222b (x : \u03b1) in s, \u03c6 i x \u2022 g x \u2202\u03bc ** rw [integral_sub hi, integral_smul_const, sub_add_cancel] ** case h \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 A : Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 (g - fun x => g x\u2080) x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (g x\u2080)) i : \u03b9 hi : IntegrableOn (fun x => \u03c6 i x \u2022 g x) s h'i : \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc = OfNat.ofNat 1 i \u22a2 Integrable fun x => \u03c6 i x \u2022 g x\u2080 ** exact Integrable.smul_const (integrable_of_integral_eq_one h'i) _ ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc + (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) \u2022 g x\u2080) l (\ud835\udcdd (0 + 1 \u2022 g x\u2080)) ** refine' Tendsto.add _ (Tendsto.smul (tendsto_const_nhds.congr' hi\u03c6.symm) tendsto_const_nhds) ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2022 h x \u2202\u03bc) l (\ud835\udcdd 0) ** apply tendsto_set_integral_peak_smul_of_integrableOn_of_continuousWithinAt_aux hs hn\u03c6 hl\u03c6 hi\u03c6 ** case hmg \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 \u22a2 IntegrableOn (fun x => h x) s ** apply Integrable.sub hmg ** case hmg \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 \u22a2 Integrable fun x => g x\u2080 ** apply integrableOn_const.2 ** case hmg \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 \u22a2 g x\u2080 = 0 \u2228 \u2191\u2191\u03bc s < \u22a4 ** simp only [h's.lt_top, or_true_iff] ** case h'g \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 \u22a2 h x\u2080 = 0 ** simp only [Pi.sub_apply, sub_self] ** case hcg \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d : CompleteSpace E hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 \u03c6 i x hl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u) hi\u03c6 : (fun i => \u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) =\u1da0[l] 1 hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 h : \u03b1 \u2192 E := g - fun x => g x\u2080 \u22a2 ContinuousWithinAt (fun x => h x) s x\u2080 ** exact hcg.sub continuousWithinAt_const ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.degreeOf_mul_X_ne ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R i j : \u03c3 f : MvPolynomial \u03c3 R h : i \u2260 j \u22a2 degreeOf i (f * X j) = degreeOf i f ** classical\nrepeat' rw [degreeOf_eq_sup (R := R) i]\nrw [support_mul_X]\nsimp only [Finset.sup_map]\ncongr\next\nsimp only [Finsupp.single, Nat.one_ne_zero, add_right_eq_self, addRightEmbedding_apply, coe_mk,\n  Pi.add_apply, comp_apply, ite_eq_right_iff, Finsupp.coe_add, Pi.single_eq_of_ne h] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R i j : \u03c3 f : MvPolynomial \u03c3 R h : i \u2260 j \u22a2 degreeOf i (f * X j) = degreeOf i f ** repeat' rw [degreeOf_eq_sup (R := R) i] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R i j : \u03c3 f : MvPolynomial \u03c3 R h : i \u2260 j \u22a2 (Finset.sup (support (f * X j)) fun m => \u2191m i) = Finset.sup (support f) fun m => \u2191m i ** rw [support_mul_X] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R i j : \u03c3 f : MvPolynomial \u03c3 R h : i \u2260 j \u22a2 (Finset.sup (Finset.map (addRightEmbedding fun\u2080 | j => 1) (support f)) fun m => \u2191m i) =     Finset.sup (support f) fun m => \u2191m i ** simp only [Finset.sup_map] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R i j : \u03c3 f : MvPolynomial \u03c3 R h : i \u2260 j \u22a2 Finset.sup (support f) ((fun m => \u2191m i) \u2218 \u2191(addRightEmbedding fun\u2080 | j => 1)) = Finset.sup (support f) fun m => \u2191m i ** congr ** case e_f R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R i j : \u03c3 f : MvPolynomial \u03c3 R h : i \u2260 j \u22a2 (fun m => \u2191m i) \u2218 \u2191(addRightEmbedding fun\u2080 | j => 1) = fun m => \u2191m i ** ext ** case e_f.h R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R i j : \u03c3 f : MvPolynomial \u03c3 R h : i \u2260 j x\u271d : \u03c3 \u2192\u2080 \u2115 \u22a2 ((fun m => \u2191m i) \u2218 \u2191(addRightEmbedding fun\u2080 | j => 1)) x\u271d = \u2191x\u271d i ** simp only [Finsupp.single, Nat.one_ne_zero, add_right_eq_self, addRightEmbedding_apply, coe_mk,\n  Pi.add_apply, comp_apply, ite_eq_right_iff, Finsupp.coe_add, Pi.single_eq_of_ne h] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R i j : \u03c3 f : MvPolynomial \u03c3 R h : i \u2260 j \u22a2 (Finset.sup (support (f * X j)) fun m => \u2191m i) = degreeOf i f ** rw [degreeOf_eq_sup (R := R) i] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_zero ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u22a2 snorm 0 p \u03bc = 0 ** by_cases h0 : p = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G h0 : \u00acp = 0 \u22a2 snorm 0 p \u03bc = 0 ** by_cases h_top : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G h0 : \u00acp = 0 h_top : \u00acp = \u22a4 \u22a2 snorm 0 p \u03bc = 0 ** rw [\u2190 Ne.def] at h0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G h0 : p \u2260 0 h_top : \u00acp = \u22a4 \u22a2 snorm 0 p \u03bc = 0 ** simp [snorm_eq_snorm' h0 h_top, ENNReal.toReal_pos h0 h_top] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G h0 : p = 0 \u22a2 snorm 0 p \u03bc = 0 ** simp [h0] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G h0 : \u00acp = 0 h_top : p = \u22a4 \u22a2 snorm 0 p \u03bc = 0 ** simp only [h_top, snorm_exponent_top, snormEssSup_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_integral_map ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g\u271d : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E \u03b2 : Type u_5 inst\u271d : MeasurableSpace \u03b2 g : \u03b1 \u2192 \u03b2 f : \u03b2 \u2192 E s : Set \u03b2 hs : MeasurableSet s hf : AEStronglyMeasurable f (Measure.map g \u03bc) hg : AEMeasurable g \u22a2 \u222b (y : \u03b2) in s, f y \u2202Measure.map g \u03bc = \u222b (x : \u03b1) in g \u207b\u00b9' s, f (g x) \u2202\u03bc ** rw [Measure.restrict_map_of_aemeasurable hg hs,\n  integral_map (hg.mono_measure Measure.restrict_le_self) (hf.mono_measure _)] ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g\u271d : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E \u03b2 : Type u_5 inst\u271d : MeasurableSpace \u03b2 g : \u03b1 \u2192 \u03b2 f : \u03b2 \u2192 E s : Set \u03b2 hs : MeasurableSet s hf : AEStronglyMeasurable f (Measure.map g \u03bc) hg : AEMeasurable g \u22a2 Measure.map g (Measure.restrict \u03bc (g \u207b\u00b9' s)) \u2264 Measure.map g \u03bc ** exact Measure.map_mono_of_aemeasurable Measure.restrict_le_self hg ** Qed",
        "informal": ""
    },
    {
        "formal": "String.data_dropWhile ** p : Char \u2192 Bool s : String \u22a2 (dropWhile s p).data = List.dropWhile p s.data ** rw [dropWhile_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "ContextFreeRule.rewrites_iff ** T : Type uT N : Type uN r : ContextFreeRule T N u v : List (Symbol T N) \u22a2 (\u2203 p q, u = p ++ [Symbol.nonterminal r.input] ++ q \u2227 v = p ++ r.output ++ q) \u2192 Rewrites r u v ** rintro \u27e8p, q, rfl, rfl\u27e9 ** case intro.intro.intro T : Type uT N : Type uN r : ContextFreeRule T N p q : List (Symbol T N) \u22a2 Rewrites r (p ++ [Symbol.nonterminal r.input] ++ q) (p ++ r.output ++ q) ** apply rewrites_of_exists_parts ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedSpace \u211d E f : \u211d \u2192 E c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 a b z : \u211d u v ua ub va vb : \u03b9 \u2192 \u211d inst\u271d\u00b9 : FTCFilter a la la' inst\u271d : FTCFilter b lb lb' hab : IntervalIntegrable f volume a b hmeas : StronglyMeasurableAtFilter f lb' hf : Tendsto f (lb' \u2293 Measure.ae volume) (\ud835\udcdd c) hu : Tendsto u lt lb hv : Tendsto v lt lb \u22a2 (fun t => ((\u222b (x : \u211d) in a..v t, f x) - \u222b (x : \u211d) in a..u t, f x) - (v t - u t) \u2022 c) =o[lt] (v - u) ** simpa only [integral_const, smul_eq_mul, mul_one] using\n  measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right hab hmeas hf hu hv ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.toList_drop ** \u03b1 : Type u \u03b2 : Type v \u03c6 : Type w n\u271d n m : \u2115 v : Vector \u03b1 m \u22a2 toList (drop n v) = List.drop n (toList v) ** cases v ** case mk \u03b1 : Type u \u03b2 : Type v \u03c6 : Type w n\u271d n m : \u2115 val\u271d : List \u03b1 property\u271d : List.length val\u271d = m \u22a2 toList (drop n { val := val\u271d, property := property\u271d }) = List.drop n (toList { val := val\u271d, property := property\u271d }) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.isSFiniteKernel_sum ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n) \u22a2 IsSFiniteKernel (kernel.sum \u03bas) ** cases fintypeOrInfinite \u03b9 ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n) val\u271d : Infinite \u03b9 \u22a2 IsSFiniteKernel (kernel.sum \u03bas) ** cases nonempty_denumerable \u03b9 ** case inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n) val\u271d\u00b9 : Infinite \u03b9 val\u271d : Denumerable \u03b9 \u22a2 IsSFiniteKernel (kernel.sum \u03bas) ** exact isSFiniteKernel_sum_of_denumerable h\u03bas ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n) val\u271d : Fintype \u03b9 \u22a2 IsSFiniteKernel (kernel.sum \u03bas) ** rw [sum_fintype] ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Countable \u03b9 \u03bas : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03bas : \u2200 (n : \u03b9), IsSFiniteKernel (\u03bas n) val\u271d : Fintype \u03b9 \u22a2 IsSFiniteKernel (\u2211 i : \u03b9, \u03bas i) ** exact IsSFiniteKernel.finset_sum Finset.univ fun i _ => h\u03bas i ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ComplexMeasure.singularPart_add_withDensity_rnDeriv_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc \u22a2 singularPart c \u03bc + Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc) = c ** conv_rhs => rw [\u2190 c.toComplexMeasure_to_signedMeasure] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc \u22a2 singularPart c \u03bc + Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc) = SignedMeasure.toComplexMeasure (\u2191re c) (\u2191im c) ** ext i hi : 1 ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(singularPart c \u03bc + Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i = \u2191(SignedMeasure.toComplexMeasure (\u2191re c) (\u2191im c)) i ** rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(singularPart c \u03bc) i + \u2191(Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i = { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i } ** ext ** case h.a \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 (\u2191(singularPart c \u03bc) i + \u2191(Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i).re = { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.re ** rw [Complex.add_re, withDensity\u1d65_apply (c.integrable_rnDeriv \u03bc) hi, \u2190 IsROrC.re_eq_complex_re,\n  \u2190 integral_re (c.integrable_rnDeriv \u03bc).integrableOn, IsROrC.re_eq_complex_re,\n  \u2190 withDensity\u1d65_apply _ hi] ** case h.a \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 (\u2191(singularPart c \u03bc) i).re + \u2191(Measure.withDensity\u1d65 \u03bc fun a => (rnDeriv c \u03bc a).re) i =     { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.re ** change (c.re.singularPart \u03bc + \u03bc.withDensity\u1d65 (c.re.rnDeriv \u03bc)) i = _ ** case h.a \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(SignedMeasure.singularPart (\u2191re c) \u03bc + Measure.withDensity\u1d65 \u03bc (SignedMeasure.rnDeriv (\u2191re c) \u03bc)) i =     { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.re ** rw [c.re.singularPart_add_withDensity_rnDeriv_eq \u03bc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 Integrable fun a => (rnDeriv c \u03bc a).re ** exact SignedMeasure.integrable_rnDeriv _ _ ** case h.a \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 (\u2191(singularPart c \u03bc) i + \u2191(Measure.withDensity\u1d65 \u03bc (rnDeriv c \u03bc)) i).im = { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.im ** rw [Complex.add_im, withDensity\u1d65_apply (c.integrable_rnDeriv \u03bc) hi, \u2190 IsROrC.im_eq_complex_im,\n  \u2190 integral_im (c.integrable_rnDeriv \u03bc).integrableOn, IsROrC.im_eq_complex_im,\n  \u2190 withDensity\u1d65_apply _ hi] ** case h.a \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 (\u2191(singularPart c \u03bc) i).im + \u2191(Measure.withDensity\u1d65 \u03bc fun a => (rnDeriv c \u03bc a).im) i =     { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.im ** change (c.im.singularPart \u03bc + \u03bc.withDensity\u1d65 (c.im.rnDeriv \u03bc)) i = _ ** case h.a \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(SignedMeasure.singularPart (\u2191im c) \u03bc + Measure.withDensity\u1d65 \u03bc (SignedMeasure.rnDeriv (\u2191im c) \u03bc)) i =     { re := \u2191(\u2191re c) i, im := \u2191(\u2191im c) i }.im ** rw [c.im.singularPart_add_withDensity_rnDeriv_eq \u03bc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : ComplexMeasure \u03b1 inst\u271d : HaveLebesgueDecomposition c \u03bc i : Set \u03b1 hi : MeasurableSet i \u22a2 Integrable fun a => (rnDeriv c \u03bc a).im ** exact SignedMeasure.integrable_rnDeriv _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.comp_eq_snd_compProd ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 f : \u03b2 \u2192 \u03b3 g : \u03b3 \u2192 \u03b1 \u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 } inst\u271d\u00b9 : IsSFiniteKernel \u03b7 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba \u22a2 \u03b7 \u2218\u2096 \u03ba = snd (\u03ba \u2297\u2096 prodMkLeft \u03b1 \u03b7) ** ext a s hs ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 f : \u03b2 \u2192 \u03b3 g : \u03b3 \u2192 \u03b1 \u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 } inst\u271d\u00b9 : IsSFiniteKernel \u03b7 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba a : \u03b1 s : Set \u03b3 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(\u03b7 \u2218\u2096 \u03ba) a) s = \u2191\u2191(\u2191(snd (\u03ba \u2297\u2096 prodMkLeft \u03b1 \u03b7)) a) s ** rw [comp_apply' _ _ _ hs, snd_apply' _ _ hs, compProd_apply] ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 f : \u03b2 \u2192 \u03b3 g : \u03b3 \u2192 \u03b1 \u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 } inst\u271d\u00b9 : IsSFiniteKernel \u03b7 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba a : \u03b1 s : Set \u03b3 hs : MeasurableSet s \u22a2 \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 b) s \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191(prodMkLeft \u03b1 \u03b7) (a, b)) {c | (b, c) \u2208 {p | p.2 \u2208 s}} \u2202\u2191\u03ba a  case h.h.hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 f : \u03b2 \u2192 \u03b3 g : \u03b3 \u2192 \u03b1 \u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 } inst\u271d\u00b9 : IsSFiniteKernel \u03b7 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba a : \u03b1 s : Set \u03b3 hs : MeasurableSet s \u22a2 MeasurableSet {p | p.2 \u2208 s} ** swap ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 f : \u03b2 \u2192 \u03b3 g : \u03b3 \u2192 \u03b1 \u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 } inst\u271d\u00b9 : IsSFiniteKernel \u03b7 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba a : \u03b1 s : Set \u03b3 hs : MeasurableSet s \u22a2 \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 b) s \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191(prodMkLeft \u03b1 \u03b7) (a, b)) {c | (b, c) \u2208 {p | p.2 \u2208 s}} \u2202\u2191\u03ba a ** simp only [Set.mem_setOf_eq, Set.setOf_mem_eq, prodMkLeft_apply' _ _ s] ** case h.h.hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 f : \u03b2 \u2192 \u03b3 g : \u03b3 \u2192 \u03b1 \u03b7 : { x // x \u2208 kernel \u03b2 \u03b3 } inst\u271d\u00b9 : IsSFiniteKernel \u03b7 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba a : \u03b1 s : Set \u03b3 hs : MeasurableSet s \u22a2 MeasurableSet {p | p.2 \u2208 s} ** exact measurable_snd hs ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.mem_sym_iff ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n : \u2115 m : Sym \u03b1 n \u22a2 m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s ** induction' n with n ih ** case succ \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s m : Sym \u03b1 (Nat.succ n) \u22a2 m \u2208 Finset.sym s (Nat.succ n) \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s ** refine' mem_sup.trans \u27e8_, fun h \u21a6 _\u27e9 ** case zero \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n : \u2115 m\u271d : Sym \u03b1 n m : Sym \u03b1 Nat.zero \u22a2 m \u2208 Finset.sym s Nat.zero \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s ** refine' mem_singleton.trans \u27e8_, fun _ \u21a6 Sym.eq_nil_of_card_zero _\u27e9 ** case zero \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n : \u2115 m\u271d : Sym \u03b1 n m : Sym \u03b1 Nat.zero \u22a2 m = \u2205 \u2192 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s ** rintro rfl ** case zero \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n : \u2115 m : Sym \u03b1 n \u22a2 \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 a \u2208 s ** exact fun a ha \u21a6 (Finset.not_mem_empty _ ha).elim ** case succ.refine'_1 \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s m : Sym \u03b1 (Nat.succ n) \u22a2 (\u2203 v, v \u2208 s \u2227 m \u2208 image (Sym.cons v) (Finset.sym s n)) \u2192 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s ** rintro \u27e8a, ha, he\u27e9 b hb ** case succ.refine'_1.intro.intro \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a\u271d b\u271d : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s m : Sym \u03b1 (Nat.succ n) a : \u03b1 ha : a \u2208 s he : m \u2208 image (Sym.cons a) (Finset.sym s n) b : \u03b1 hb : b \u2208 m \u22a2 b \u2208 s ** rw [mem_image] at he ** case succ.refine'_1.intro.intro \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a\u271d b\u271d : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s m : Sym \u03b1 (Nat.succ n) a : \u03b1 ha : a \u2208 s he : \u2203 a_1, a_1 \u2208 Finset.sym s n \u2227 a ::\u209b a_1 = m b : \u03b1 hb : b \u2208 m \u22a2 b \u2208 s ** obtain \u27e8m, he, rfl\u27e9 := he ** case succ.refine'_1.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a\u271d b\u271d : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s a : \u03b1 ha : a \u2208 s b : \u03b1 m : Sym \u03b1 n he : m \u2208 Finset.sym s n hb : b \u2208 a ::\u209b m \u22a2 b \u2208 s ** rw [Sym.mem_cons] at hb ** case succ.refine'_1.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a\u271d b\u271d : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s a : \u03b1 ha : a \u2208 s b : \u03b1 m : Sym \u03b1 n he : m \u2208 Finset.sym s n hb : b = a \u2228 b \u2208 m \u22a2 b \u2208 s ** obtain rfl | hb := hb ** case succ.refine'_1.intro.intro.intro.intro.inl \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b\u271d : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s b : \u03b1 m : Sym \u03b1 n he : m \u2208 Finset.sym s n ha : b \u2208 s \u22a2 b \u2208 s ** exact ha ** case succ.refine'_1.intro.intro.intro.intro.inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a\u271d b\u271d : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s a : \u03b1 ha : a \u2208 s b : \u03b1 m : Sym \u03b1 n he : m \u2208 Finset.sym s n hb : b \u2208 m \u22a2 b \u2208 s ** exact ih.1 he _ hb ** case succ.refine'_2 \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s m : Sym \u03b1 (Nat.succ n) h : \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s \u22a2 \u2203 v, v \u2208 s \u2227 m \u2208 image (Sym.cons v) (Finset.sym s n) ** obtain \u27e8a, m, rfl\u27e9 := m.exists_eq_cons_of_succ ** case succ.refine'_2.intro.intro \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a\u271d b : \u03b1 n\u271d : \u2115 m\u271d : Sym \u03b1 n\u271d n : \u2115 ih : \u2200 {m : Sym \u03b1 n}, m \u2208 Finset.sym s n \u2194 \u2200 (a : \u03b1), a \u2208 m \u2192 a \u2208 s a : \u03b1 m : Sym \u03b1 n h : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u209b m \u2192 a_1 \u2208 s \u22a2 \u2203 v, v \u2208 s \u2227 a ::\u209b m \u2208 image (Sym.cons v) (Finset.sym s n) ** exact\n  \u27e8a, h _ <| Sym.mem_cons_self _ _,\n    mem_image_of_mem _ <| ih.2 fun b hb \u21a6 h _ <| Sym.mem_cons_of_mem hb\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "LazyList.append_bind ** \u03b1 : Type u_1 \u03b2 : Type u_2 xs : LazyList \u03b1 ys : Thunk (LazyList \u03b1) f : \u03b1 \u2192 LazyList \u03b2 \u22a2 LazyList.bind (append xs ys) f = append (LazyList.bind xs f) { fn := fun x => LazyList.bind (Thunk.get ys) f } ** match xs with\n| LazyList.nil => rfl\n| LazyList.cons x xs =>\n  simp only [append, Thunk.get, LazyList.bind]\n  have := append_bind xs.get ys f\n  simp only [Thunk.get] at this\n  rw [this, append_assoc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 xs : LazyList \u03b1 ys : Thunk (LazyList \u03b1) f : \u03b1 \u2192 LazyList \u03b2 \u22a2 LazyList.bind (append nil ys) f = append (LazyList.bind nil f) { fn := fun x => LazyList.bind (Thunk.get ys) f } ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 xs\u271d : LazyList \u03b1 ys : Thunk (LazyList \u03b1) f : \u03b1 \u2192 LazyList \u03b2 x : \u03b1 xs : Thunk (LazyList \u03b1) \u22a2 LazyList.bind (append (cons x xs) ys) f =     append (LazyList.bind (cons x xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f } ** simp only [append, Thunk.get, LazyList.bind] ** \u03b1 : Type u_1 \u03b2 : Type u_2 xs\u271d : LazyList \u03b1 ys : Thunk (LazyList \u03b1) f : \u03b1 \u2192 LazyList \u03b2 x : \u03b1 xs : Thunk (LazyList \u03b1) \u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =     append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })       { fn := fun x => LazyList.bind (Thunk.fn ys ()) f } ** have := append_bind xs.get ys f ** \u03b1 : Type u_1 \u03b2 : Type u_2 xs\u271d : LazyList \u03b1 ys : Thunk (LazyList \u03b1) f : \u03b1 \u2192 LazyList \u03b2 x : \u03b1 xs : Thunk (LazyList \u03b1) this :   LazyList.bind (append (Thunk.get xs) ys) f =     append (LazyList.bind (Thunk.get xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f } \u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =     append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })       { fn := fun x => LazyList.bind (Thunk.fn ys ()) f } ** simp only [Thunk.get] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 xs\u271d : LazyList \u03b1 ys : Thunk (LazyList \u03b1) f : \u03b1 \u2192 LazyList \u03b2 x : \u03b1 xs : Thunk (LazyList \u03b1) this :   LazyList.bind (append (Thunk.fn xs ()) ys) f =     append (LazyList.bind (Thunk.fn xs ()) f) { fn := fun x => LazyList.bind (Thunk.fn ys ()) f } \u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =     append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })       { fn := fun x => LazyList.bind (Thunk.fn ys ()) f } ** rw [this, append_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.norm_condexpL2_coe_le ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } \u22a2 \u2016\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)\u2016 \u2264 \u2016f\u2016 ** rw [Lp.norm_def, Lp.norm_def, \u2190 lpMeas_coe] ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } \u22a2 ENNReal.toReal (snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc) \u2264 ENNReal.toReal (snorm (\u2191\u2191f) 2 \u03bc) ** refine' (ENNReal.toReal_le_toReal _ (Lp.snorm_ne_top _)).mpr (snorm_condexpL2_le hm f) ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E 2 } \u22a2 snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc \u2260 \u22a4 ** exact Lp.snorm_ne_top _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul ** p : \u2115 p_prime : Nat.Prime p m n : \u2124 k l : \u2115 hpm : \u2191(p ^ k) \u2223 m hpn : \u2191(p ^ l) \u2223 n hpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n hpm' : p ^ k \u2223 natAbs m hpn' : p ^ l \u2223 natAbs n \u22a2 p ^ (k + l + 1) \u2223 natAbs m * natAbs n ** rw [\u2190 Int.natAbs_mul] ** p : \u2115 p_prime : Nat.Prime p m n : \u2124 k l : \u2115 hpm : \u2191(p ^ k) \u2223 m hpn : \u2191(p ^ l) \u2223 n hpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n hpm' : p ^ k \u2223 natAbs m hpn' : p ^ l \u2223 natAbs n \u22a2 p ^ (k + l + 1) \u2223 natAbs (m * n) ** apply Int.coe_nat_dvd.1 <| Int.dvd_natAbs.2 hpmn ** p : \u2115 p_prime : Nat.Prime p m n : \u2124 k l : \u2115 hpm : \u2191(p ^ k) \u2223 m hpn : \u2191(p ^ l) \u2223 n hpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n hpm' : p ^ k \u2223 natAbs m hpn' : p ^ l \u2223 natAbs n hpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n hsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=   Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn' hsd1 : p ^ (k + 1) \u2223 natAbs m \u22a2 \u2191(p ^ (k + 1)) \u2223 m ** apply Int.dvd_natAbs.1 ** p : \u2115 p_prime : Nat.Prime p m n : \u2124 k l : \u2115 hpm : \u2191(p ^ k) \u2223 m hpn : \u2191(p ^ l) \u2223 n hpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n hpm' : p ^ k \u2223 natAbs m hpn' : p ^ l \u2223 natAbs n hpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n hsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=   Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn' hsd1 : p ^ (k + 1) \u2223 natAbs m \u22a2 \u2191(p ^ (k + 1)) \u2223 \u2191(natAbs m) ** apply Int.coe_nat_dvd.2 hsd1 ** p : \u2115 p_prime : Nat.Prime p m n : \u2124 k l : \u2115 hpm : \u2191(p ^ k) \u2223 m hpn : \u2191(p ^ l) \u2223 n hpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n hpm' : p ^ k \u2223 natAbs m hpn' : p ^ l \u2223 natAbs n hpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n hsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=   Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn' hsd2 : p ^ (l + 1) \u2223 natAbs n \u22a2 \u2191(p ^ (l + 1)) \u2223 n ** apply Int.dvd_natAbs.1 ** p : \u2115 p_prime : Nat.Prime p m n : \u2124 k l : \u2115 hpm : \u2191(p ^ k) \u2223 m hpn : \u2191(p ^ l) \u2223 n hpmn : \u2191(p ^ (k + l + 1)) \u2223 m * n hpm' : p ^ k \u2223 natAbs m hpn' : p ^ l \u2223 natAbs n hpmn' : p ^ (k + l + 1) \u2223 natAbs m * natAbs n hsd : p ^ (k + 1) \u2223 natAbs m \u2228 p ^ (l + 1) \u2223 natAbs n :=   Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn' hsd2 : p ^ (l + 1) \u2223 natAbs n \u22a2 \u2191(p ^ (l + 1)) \u2223 \u2191(natAbs n) ** apply Int.coe_nat_dvd.2 hsd2 ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.set_lintegral_deterministic ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 a : \u03b1 hg : Measurable g inst\u271d\u00b9 : MeasurableSingletonClass \u03b2 s : Set \u03b2 inst\u271d : Decidable (g a \u2208 s) \u22a2 \u222b\u207b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0 ** rw [kernel.deterministic_apply, set_lintegral_dirac f s] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.exists_insert_toList_zoom_node ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c' : RBColor l : RBNode \u03b1 v' : \u03b1 r : RBNode \u03b1 p : Path \u03b1 v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n e : zoom (cmp v) t Path.root = (node c' l v' r, p) \u22a2 \u2203 L R, toList t = L ++ v' :: R \u2227 toList (insert cmp t v) = L ++ v :: R ** refine \u27e8p.listL ++ l.toList, r.toList ++ p.listR, ?_\u27e9 ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering c' : RBColor l : RBNode \u03b1 v' : \u03b1 r : RBNode \u03b1 p : Path \u03b1 v : \u03b1 t : RBNode \u03b1 ht : Balanced t c n e : zoom (cmp v) t Path.root = (node c' l v' r, p) \u22a2 toList t = Path.listL p ++ toList l ++ v' :: (toList r ++ Path.listR p) \u2227     toList (insert cmp t v) = Path.listL p ++ toList l ++ v :: (toList r ++ Path.listR p) ** simp [\u2190 zoom_toList e, insert_toList_zoom_node ht e] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.ae_null_of_compProd_null ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 h : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0 \u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0 ** obtain \u27e8t, hst, mt, ht\u27e9 := exists_measurable_superset_of_null h ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 h : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0 t : Set (\u03b2 \u00d7 \u03b3) hst : s \u2286 t mt : MeasurableSet t ht : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) t = 0 \u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0 ** simp_rw [compProd_null a mt] at ht ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 h : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0 t : Set (\u03b2 \u00d7 \u03b3) hst : s \u2286 t mt : MeasurableSet t ht : (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)) =\u1d50[\u2191\u03ba a] 0 \u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0 ** rw [Filter.eventuallyLE_antisymm_iff] ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 h : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0 t : Set (\u03b2 \u00d7 \u03b3) hst : s \u2286 t mt : MeasurableSet t ht : (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t)) =\u1d50[\u2191\u03ba a] 0 \u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) \u2264\u1d50[\u2191\u03ba a] 0 \u2227 0 \u2264\u1d50[\u2191\u03ba a] fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) ** exact\n  \u27e8Filter.EventuallyLE.trans_eq\n      (Filter.eventually_of_forall fun x => (measure_mono (Set.preimage_mono hst) : _)) ht,\n    Filter.eventually_of_forall fun x => zero_le _\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.mem_support_bind_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 f : \u03b1 \u2192 PMF \u03b2 g : \u03b2 \u2192 PMF \u03b3 b : \u03b2 \u22a2 b \u2208 support (bind p f) \u2194 \u2203 a, a \u2208 support p \u2227 b \u2208 support (f a) ** simp only [support_bind, Set.mem_iUnion, Set.mem_setOf_eq, exists_prop] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.prehaar_le_index ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G U : Set G K : Compacts G hU : Set.Nonempty (interior U) \u22a2 prehaar (\u2191K\u2080) U K \u2264 \u2191(index \u2191K \u2191K\u2080) ** unfold prehaar ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G U : Set G K : Compacts G hU : Set.Nonempty (interior U) \u22a2 \u2191(index (\u2191K) U) / \u2191(index (\u2191K\u2080) U) \u2264 \u2191(index \u2191K \u2191K\u2080) ** rw [div_le_iff] <;> norm_cast ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G U : Set G K : Compacts G hU : Set.Nonempty (interior U) \u22a2 index (\u2191K) U \u2264 index \u2191K \u2191K\u2080 * index (\u2191K\u2080) U ** apply le_index_mul K\u2080 K hU ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G U : Set G K : Compacts G hU : Set.Nonempty (interior U) \u22a2 0 < index (\u2191K\u2080) U ** exact index_pos K\u2080 hU ** Qed",
        "informal": ""
    },
    {
        "formal": "map_finset_sup ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u2074 : SemilatticeSup \u03b1 inst\u271d\u00b3 : OrderBot \u03b1 s\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b2 f\u271d g\u271d : \u03b2 \u2192 \u03b1 a : \u03b1 inst\u271d\u00b2 : SemilatticeSup \u03b2 inst\u271d\u00b9 : OrderBot \u03b2 inst\u271d : SupBotHomClass F \u03b1 \u03b2 f : F s\u271d : Finset \u03b9 g : \u03b9 \u2192 \u03b1 i : \u03b9 s : Finset \u03b9 x\u271d : \u00aci \u2208 s h : \u2191f (sup s g) = sup s (\u2191f \u2218 g) \u22a2 \u2191f (sup (cons i s x\u271d) g) = sup (cons i s x\u271d) (\u2191f \u2218 g) ** rw [sup_cons, sup_cons, map_sup, h, Function.comp_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.DList.toList_empty ** \u03b1 : Type u \u22a2 toList empty = [] ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.mkMetric_top ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u22a2 (mkMetric fun x => \u22a4) = \u22a4 ** simp_rw [mkMetric, mkMetric', mkMetric'.pre, extend_top, boundedBy_top, eq_top_iff] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u22a2 \u22a4 \u2264 \u2a06 r, \u2a06 (_ : r > 0), \u22a4 ** rw [le_iSup_iff] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u22a2 \u2200 (b : OuterMeasure X), (\u2200 (i : \u211d\u22650\u221e), \u2a06 (_ : i > 0), \u22a4 \u2264 b) \u2192 \u22a4 \u2264 b ** intro b hb ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y b : OuterMeasure X hb : \u2200 (i : \u211d\u22650\u221e), \u2a06 (_ : i > 0), \u22a4 \u2264 b \u22a2 \u22a4 \u2264 b ** simpa using hb \u22a4 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_zero_left' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 \u22a2 setToFun \u03bc T hT f = 0 ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 hf : Integrable f \u22a2 setToFun \u03bc T hT f = 0 ** rw [setToFun_eq hT hf] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 hf : Integrable f \u22a2 \u2191(L1.setToL1 hT) (Integrable.toL1 f hf) = 0 ** exact L1.setToL1_zero_left' hT h_zero _ ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 hf : \u00acIntegrable f \u22a2 setToFun \u03bc T hT f = 0 ** exact setToFun_undef hT hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.analyticSet_iff_exists_polishSpace_range ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 \u22a2 AnalyticSet s \u2194 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s ** constructor ** case mp \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 \u22a2 AnalyticSet s \u2192 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s ** intro h ** case mp \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 h : AnalyticSet s \u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s ** rw [AnalyticSet] at h ** case mp \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 h : s = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = s \u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s ** cases' h with h h ** case mp.inl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 h : s = \u2205 \u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s ** refine' \u27e8Empty, inferInstance, inferInstance, Empty.elim, continuous_bot, _\u27e9 ** case mp.inl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 h : s = \u2205 \u22a2 range Empty.elim = s ** rw [h] ** case mp.inl \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 h : s = \u2205 \u22a2 range Empty.elim = \u2205 ** exact range_eq_empty _ ** case mp.inr \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 h : \u2203 f, Continuous f \u2227 range f = s \u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s ** exact \u27e8\u2115 \u2192 \u2115, inferInstance, inferInstance, h\u27e9 ** case mpr \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 \u22a2 (\u2203 \u03b2 h x f, Continuous f \u2227 range f = s) \u2192 AnalyticSet s ** rintro \u27e8\u03b2, h, h', f, f_cont, f_range\u27e9 ** case mpr.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 \u03b2 : Type h : TopologicalSpace \u03b2 h' : PolishSpace \u03b2 f : \u03b2 \u2192 \u03b1 f_cont : Continuous f f_range : range f = s \u22a2 AnalyticSet s ** rw [\u2190 f_range] ** case mpr.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 \u03b2 : Type h : TopologicalSpace \u03b2 h' : PolishSpace \u03b2 f : \u03b2 \u2192 \u03b1 f_cont : Continuous f f_range : range f = s \u22a2 AnalyticSet (range f) ** exact analyticSet_range_of_polishSpace f_cont ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_ne_top ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) \u22a2 TendstoInMeasure \u03bc f l g ** refine' TendstoInMeasure.congr (fun i => (hf i).ae_eq_mk.symm) hg.ae_eq_mk.symm _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) \u22a2 TendstoInMeasure \u03bc (fun i => AEStronglyMeasurable.mk (f i) (_ : AEStronglyMeasurable (f i) \u03bc)) l     (AEStronglyMeasurable.mk g hg) ** refine' tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable\n  hp_ne_zero hp_ne_top (fun i => (hf i).stronglyMeasurable_mk) hg.stronglyMeasurable_mk _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) \u22a2 Tendsto     (fun n =>       snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc)     l (\ud835\udcdd 0) ** have : (fun n => snorm ((hf n).mk (f n) - hg.mk g) p \u03bc) = fun n => snorm (f n - g) p \u03bc := by\n  ext1 n; refine' snorm_congr_ae (EventuallyEq.sub (hf n).ae_eq_mk.symm hg.ae_eq_mk.symm) ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) this :   (fun n =>       snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc) =     fun n => snorm (f n - g) p \u03bc \u22a2 Tendsto     (fun n =>       snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc)     l (\ud835\udcdd 0) ** rw [this] ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) this :   (fun n =>       snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc) =     fun n => snorm (f n - g) p \u03bc \u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) ** exact hfg ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) \u22a2 (fun n =>       snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc) =     fun n => snorm (f n - g) p \u03bc ** ext1 n ** case h \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), AEStronglyMeasurable (f n) \u03bc hg : AEStronglyMeasurable g \u03bc l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) n : \u03b9 \u22a2 snorm (AEStronglyMeasurable.mk (f n) (_ : AEStronglyMeasurable (f n) \u03bc) - AEStronglyMeasurable.mk g hg) p \u03bc =     snorm (f n - g) p \u03bc ** refine' snorm_congr_ae (EventuallyEq.sub (hf n).ae_eq_mk.symm hg.ae_eq_mk.symm) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.norm_toL1_eq_lintegral_norm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192 \u03b2 hf : Integrable f \u22a2 \u2016toL1 f hf\u2016 = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc) ** rw [norm_toL1, lintegral_norm_eq_lintegral_edist] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.FinMeasSupp.mul ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : Zero \u03b2\u271d inst\u271d\u00b9 : Zero \u03b3 \u03bc : Measure \u03b1 f\u271d : \u03b1 \u2192\u209b \u03b2\u271d \u03b2 : Type u_5 inst\u271d : MonoidWithZero \u03b2 f g : \u03b1 \u2192\u209b \u03b2 hf : SimpleFunc.FinMeasSupp f \u03bc hg : SimpleFunc.FinMeasSupp g \u03bc \u22a2 SimpleFunc.FinMeasSupp (f * g) \u03bc ** rw [mul_eq_map\u2082] ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 inst\u271d\u00b2 : Zero \u03b2\u271d inst\u271d\u00b9 : Zero \u03b3 \u03bc : Measure \u03b1 f\u271d : \u03b1 \u2192\u209b \u03b2\u271d \u03b2 : Type u_5 inst\u271d : MonoidWithZero \u03b2 f g : \u03b1 \u2192\u209b \u03b2 hf : SimpleFunc.FinMeasSupp f \u03bc hg : SimpleFunc.FinMeasSupp g \u03bc \u22a2 SimpleFunc.FinMeasSupp (map (fun p => p.1 * p.2) (pair f g)) \u03bc ** exact hf.map\u2082 hg (zero_mul 0) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** obtain \u27e8G, G_mble, G_nn, g_eq_G\u27e9 : \u2203 G : \u211d \u2192 \u211d, Measurable G \u2227 0 \u2264 G\n    \u2227 g =\u1d50[volume.restrict (Ioi 0)] G := by\n  refine' AEMeasurable.exists_measurable_nonneg _ g_nn\n  exact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.aemeasurable ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have g_eq_G_on : \u2200 t, g =\u1d50[volume.restrict (Ioc 0 t)] G := fun t =>\n  ae_mono (Measure.restrict_mono Ioc_subset_Ioi_self le_rfl) g_eq_G ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have G_intble : \u2200 t > 0, IntervalIntegrable G volume 0 t := by\n  refine' fun t t_pos => \u27e8(g_intble t t_pos).1.congr_fun_ae (g_eq_G_on t), _\u27e9\n  rw [Ioc_eq_empty_of_le t_pos.lt.le]\n  exact integrableOn_empty ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** obtain \u27e8F, F_mble, F_nn, f_eq_F\u27e9 : \u2203 F : \u03b1 \u2192 \u211d, Measurable F \u2227 0 \u2264 F \u2227 f =\u1d50[\u03bc] F := by\n  refine \u27e8fun \u03c9 \u21a6 max (f_mble.mk f \u03c9) 0, f_mble.measurable_mk.max measurable_const,\n      fun \u03c9 \u21a6 le_max_right _ _, ?_\u27e9\n  filter_upwards [f_mble.ae_eq_mk, f_nn] with \u03c9 h\u03c9 h'\u03c9\n  rw [\u2190 h\u03c9]\n  exact (max_eq_left h'\u03c9).symm ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have eq\u2081 :\n  (\u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)) =\n    \u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 F a} * ENNReal.ofReal (G t) := by\n  apply lintegral_congr_ae\n  filter_upwards [g_eq_G] with t ht\n  rw [ht]\n  congr 1\n  apply measure_congr\n  filter_upwards [f_eq_F] with a ha using by simp [setOf, ha] ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have eq\u2082 : \u2200\u1d50 \u03c9 \u2202\u03bc,\n    ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) = ENNReal.ofReal (\u222b t in (0)..F \u03c9, G t) := by\n  filter_upwards [f_eq_F] with \u03c9 f\u03c9_nn\n  rw [f\u03c9_nn]\n  congr 1\n  refine' intervalIntegral.integral_congr_ae _\n  have f\u03c9_nn : 0 \u2264 F \u03c9 := F_nn \u03c9\n  rw [uIoc_of_le f\u03c9_nn, \u2190\n    ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : \u211d) (F \u03c9)))]\n  exact g_eq_G_on (F \u03c9) ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) eq\u2082 : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t) \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** simp_rw [lintegral_congr_ae eq\u2082, eq\u2081] ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) eq\u2082 : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t) \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F a, G t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) ** exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable \u03bc F_nn F_mble\n        G_intble G_mble (fun t _ => G_nn t) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t \u22a2 \u2203 G, Measurable G \u2227 0 \u2264 G \u2227 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G ** refine' AEMeasurable.exists_measurable_nonneg _ g_nn ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t \u22a2 AEMeasurable g ** exact aemeasurable_Ioi_of_forall_Ioc fun t ht => (g_intble t ht).1.1.aemeasurable ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G \u22a2 \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t ** refine' fun t t_pos => \u27e8(g_intble t t_pos).1.congr_fun_ae (g_eq_G_on t), _\u27e9 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G t : \u211d t_pos : t > 0 \u22a2 IntegrableOn G (Ioc t 0) ** rw [Ioc_eq_empty_of_le t_pos.lt.le] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G t : \u211d t_pos : t > 0 \u22a2 IntegrableOn G \u2205 ** exact integrableOn_empty ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t \u22a2 \u2203 F, Measurable F \u2227 0 \u2264 F \u2227 f =\u1da0[ae \u03bc] F ** refine \u27e8fun \u03c9 \u21a6 max (f_mble.mk f \u03c9) 0, f_mble.measurable_mk.max measurable_const,\n    fun \u03c9 \u21a6 le_max_right _ _, ?_\u27e9 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t \u22a2 f =\u1da0[ae \u03bc] fun \u03c9 => max (AEMeasurable.mk f f_mble \u03c9) 0 ** filter_upwards [f_mble.ae_eq_mk, f_nn] with \u03c9 h\u03c9 h'\u03c9 ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t \u03c9 : \u03b1 h\u03c9 : f \u03c9 = AEMeasurable.mk f f_mble \u03c9 h'\u03c9 : OfNat.ofNat 0 \u03c9 \u2264 f \u03c9 \u22a2 f \u03c9 = max (AEMeasurable.mk f f_mble \u03c9) 0 ** rw [\u2190 h\u03c9] ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t \u03c9 : \u03b1 h\u03c9 : f \u03c9 = AEMeasurable.mk f f_mble \u03c9 h'\u03c9 : OfNat.ofNat 0 \u03c9 \u2264 f \u03c9 \u22a2 f \u03c9 = max (f \u03c9) 0 ** exact (max_eq_left h'\u03c9).symm ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) ** apply lintegral_congr_ae ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F \u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)) =\u1da0[ae (Measure.restrict volume (Ioi 0))] fun a =>     \u2191\u2191\u03bc {a_1 | a \u2264 F a_1} * ENNReal.ofReal (G a) ** filter_upwards [g_eq_G] with t ht ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F t : \u211d ht : g t = G t \u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) ** rw [ht] ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F t : \u211d ht : g t = G t \u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (G t) = \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) ** congr 1 ** case h.e_a \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F t : \u211d ht : g t = G t \u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t \u2264 F a} ** apply measure_congr ** case h.e_a.H \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F t : \u211d ht : g t = G t \u22a2 {a | t \u2264 f a} =\u1da0[ae \u03bc] {a | t \u2264 F a} ** filter_upwards [f_eq_F] with a ha using by simp [setOf, ha] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F t : \u211d ht : g t = G t a : \u03b1 ha : f a = F a \u22a2 setOf (fun a => t \u2264 f a) a = setOf (fun a => t \u2264 F a) a ** simp [setOf, ha] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) \u22a2 \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t) ** filter_upwards [f_eq_F] with \u03c9 f\u03c9_nn ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) \u03c9 : \u03b1 f\u03c9_nn : f \u03c9 = F \u03c9 \u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t) ** rw [f\u03c9_nn] ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) \u03c9 : \u03b1 f\u03c9_nn : f \u03c9 = F \u03c9 \u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, g t) = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..F \u03c9, G t) ** congr 1 ** case h.e_r \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) \u03c9 : \u03b1 f\u03c9_nn : f \u03c9 = F \u03c9 \u22a2 \u222b (t : \u211d) in 0 ..F \u03c9, g t = \u222b (t : \u211d) in 0 ..F \u03c9, G t ** refine' intervalIntegral.integral_congr_ae _ ** case h.e_r \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) \u03c9 : \u03b1 f\u03c9_nn : f \u03c9 = F \u03c9 \u22a2 \u2200\u1d50 (x : \u211d), x \u2208 \u0399 0 (F \u03c9) \u2192 g x = G x ** have f\u03c9_nn : 0 \u2264 F \u03c9 := F_nn \u03c9 ** case h.e_r \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) \u03c9 : \u03b1 f\u03c9_nn\u271d : f \u03c9 = F \u03c9 f\u03c9_nn : 0 \u2264 F \u03c9 \u22a2 \u2200\u1d50 (x : \u211d), x \u2208 \u0399 0 (F \u03c9) \u2192 g x = G x ** rw [uIoc_of_le f\u03c9_nn, \u2190\n  ae_restrict_iff' (measurableSet_Ioc : MeasurableSet (Ioc (0 : \u211d) (F \u03c9)))] ** case h.e_r \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t G : \u211d \u2192 \u211d G_mble : Measurable G G_nn : 0 \u2264 G g_eq_G : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] G g_eq_G_on : \u2200 (t : \u211d), g =\u1da0[ae (Measure.restrict volume (Ioc 0 t))] G G_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable G volume 0 t F : \u03b1 \u2192 \u211d F_mble : Measurable F F_nn : 0 \u2264 F f_eq_F : f =\u1da0[ae \u03bc] F eq\u2081 :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 F a} * ENNReal.ofReal (G t) \u03c9 : \u03b1 f\u03c9_nn\u271d : f \u03c9 = F \u03c9 f\u03c9_nn : 0 \u2264 F \u03c9 \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioc 0 (F \u03c9)), g x = G x ** exact g_eq_G_on (F \u03c9) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul\u2080' ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e hg : AEMeasurable g \u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc ** let f' := hf.mk f ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e hg : AEMeasurable g f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf \u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc ** have : \u03bc.withDensity f = \u03bc.withDensity f' := withDensity_congr_ae hf.ae_eq_mk ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e hg : AEMeasurable g f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf this : withDensity \u03bc f = withDensity \u03bc f' \u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc ** rw [this] at hg \u22a2 ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' \u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f' = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc ** let g' := hg.mk g ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg \u22a2 \u222b\u207b (a : \u03b1), (f' * g') a \u2202\u03bc = \u222b\u207b (a : \u03b1), (f' * g) a \u2202\u03bc ** apply lintegral_congr_ae ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg \u22a2 (fun a => (f' * g') a) =\u1da0[ae \u03bc] fun a => (f' * g) a ** apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x \u2260 0 } ** case h.ht \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x ** have Z := hg.ae_eq_mk ** case h.ht \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg Z : g =\u1da0[ae (withDensity \u03bc f')] AEMeasurable.mk g hg \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x ** rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z ** case h.ht \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg Z : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x ** filter_upwards [Z] ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg Z : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x \u22a2 \u2200 (a : \u03b1), g a = AEMeasurable.mk g hg a \u2192 (f' * g') a = (f' * g) a ** intro x hx ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg Z : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | AEMeasurable.mk f hf x \u2260 0}, g x = AEMeasurable.mk g hg x x : \u03b1 hx : g x = AEMeasurable.mk g hg x \u22a2 (f' * g') x = (f' * g) x ** simp only [hx, Pi.mul_apply] ** case h.htc \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}\u1d9c, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x ** have M : MeasurableSet { x : \u03b1 | f' x \u2260 0 }\u1d9c :=\n  (hf.measurable_mk (measurableSet_singleton 0).compl).compl ** case h.htc \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg M : MeasurableSet {x | f' x \u2260 0}\u1d9c \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc {x | f' x \u2260 0}\u1d9c, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x ** filter_upwards [ae_restrict_mem M] ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg M : MeasurableSet {x | f' x \u2260 0}\u1d9c \u22a2 \u2200 (a : \u03b1), a \u2208 {x | f' x \u2260 0}\u1d9c \u2192 (f' * g') a = (f' * g) a ** intro x hx ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg M : MeasurableSet {x | f' x \u2260 0}\u1d9c x : \u03b1 hx : x \u2208 {x | f' x \u2260 0}\u1d9c \u22a2 (f' * g') x = (f' * g) x ** simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg M : MeasurableSet {x | f' x \u2260 0}\u1d9c x : \u03b1 hx : AEMeasurable.mk f hf x = 0 \u22a2 (f' * g') x = (f' * g) x ** simp only [hx, zero_mul, Pi.mul_apply] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg \u22a2 \u222b\u207b (a : \u03b1), (f' * g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc ** apply lintegral_congr_ae ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg \u22a2 (fun a => (f' * g) a) =\u1da0[ae \u03bc] fun a => (f * g) a ** filter_upwards [hf.ae_eq_mk] ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg \u22a2 \u2200 (a : \u03b1), f a = AEMeasurable.mk f hf a \u2192 (f' * g) a = (f * g) a ** intro x hx ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f g : \u03b1 \u2192 \u211d\u22650\u221e f' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk f hf hg : AEMeasurable g this : withDensity \u03bc f = withDensity \u03bc f' g' : \u03b1 \u2192 \u211d\u22650\u221e := AEMeasurable.mk g hg x : \u03b1 hx : f x = AEMeasurable.mk f hf x \u22a2 (f' * g) x = (f * g) x ** simp only [hx, Pi.mul_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.is_left_invariant_haarContent ** G : Type u_1 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : TopologicalSpace G inst\u271d\u00b9 : TopologicalGroup G inst\u271d : T2Space G K\u2080 : PositiveCompacts G g : G K : Compacts G \u22a2 (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =     (fun s => \u2191(Content.toFun (haarContent K\u2080) s)) K ** simpa only [ENNReal.coe_eq_coe, \u2190 NNReal.coe_eq, haarContent_apply] using\n  is_left_invariant_chaar g K ** Qed",
        "informal": ""
    },
    {
        "formal": "Setoid.mapOfSurjective_eq_map ** \u03b1 : Type u_1 \u03b2 : Type u_2 r : Setoid \u03b1 f : \u03b1 \u2192 \u03b2 h : ker f \u2264 r hf : Surjective f \u22a2 map r f = mapOfSurjective r f h hf ** rw [\u2190 eqvGen_of_setoid (mapOfSurjective r f h hf)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 r : Setoid \u03b1 f : \u03b1 \u2192 \u03b2 h : ker f \u2264 r hf : Surjective f \u22a2 map r f = EqvGen.Setoid Setoid.r ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.map_iInf_comap ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 \u22a2 \u2191(map f) (\u2a05 i, \u2191(comap f) (m i)) = \u2a05 i, \u2191(map f) (\u2191(comap f) (m i)) ** refine' (map_iInf_le _ _).antisymm fun s => _ ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b2 \u22a2 \u2191(\u2a05 i, \u2191(map f) (\u2191(comap f) (m i))) s \u2264 \u2191(\u2191(map f) (\u2a05 i, \u2191(comap f) (m i))) s ** simp only [map_apply, comap_apply, iInf_apply, le_iInf_iff] ** \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b2 \u22a2 \u2200 (i : \u2115 \u2192 Set \u03b1),     f \u207b\u00b9' s \u2286 iUnion i \u2192       \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' (f \u207b\u00b9' t n)) \u2264 \u2211' (n : \u2115), \u2a05 i_2, \u2191(m i_2) (f '' i n) ** refine' fun t ht => iInf_le_of_le (fun n => f '' t n \u222a (range f)\u1d9c) (iInf_le_of_le _ _) ** case refine'_1 \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t \u22a2 s \u2286 \u22c3 n, f '' t n \u222a (range f)\u1d9c ** rw [\u2190 iUnion_union, Set.union_comm, \u2190 inter_subset, \u2190 image_iUnion, \u2190\n  image_preimage_eq_inter_range] ** case refine'_1 \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t \u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' \u22c3 i, t i ** exact image_subset _ ht ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t \u22a2 \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' (f \u207b\u00b9' (fun n => f '' t n \u222a (range f)\u1d9c) n)) \u2264 \u2211' (n : \u2115), \u2a05 i, \u2191(m i) (f '' t n) ** refine' ENNReal.tsum_le_tsum fun n => iInf_mono fun i => (m i).mono _ ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t n : \u2115 i : \u03b9 \u22a2 f '' (f \u207b\u00b9' (fun n => f '' t n \u222a (range f)\u1d9c) n) \u2286 f '' t n ** simp only [preimage_union, preimage_compl, preimage_range, compl_univ, union_empty,\n  image_subset_iff] ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Sort u_2 \u03b2 : Type u_3 inst\u271d : Nonempty \u03b9 f : \u03b1 \u2192 \u03b2 m : \u03b9 \u2192 OuterMeasure \u03b2 s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t n : \u2115 i : \u03b9 \u22a2 f \u207b\u00b9' (f '' t n) \u2286 f \u207b\u00b9' (f '' t n) ** exact subset_refl _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_finset ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (x : \u03b1) in \u2191s, f x \u2202\u03bc = \u2211 x in s, f x * \u2191\u2191\u03bc {x} ** simp only [lintegral_countable _ s.countable_toSet, \u2190 Finset.tsum_subtype'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.not_mem_map_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 n : \u2115 a a' : \u03b1 b : \u03b2 v : Vector \u03b1 0 f : \u03b1 \u2192 \u03b2 \u22a2 \u00acb \u2208 toList (map f v) ** simpa only [Vector.eq_nil v, Vector.map_nil, Vector.toList_nil] using List.not_mem_nil b ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.univ_pi_Iio_ae_eq_Iic ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u2074 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i) f : (i : \u03b9) \u2192 \u03b1 i \u22a2 (Set.pi univ fun i => Iio (f i)) =\u1da0[ae (Measure.pi \u03bc)] Iic f ** rw [\u2190 pi_univ_Iic] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u2074 : Fintype \u03b9 m : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i) f : (i : \u03b9) \u2192 \u03b1 i \u22a2 (Set.pi univ fun i => Iio (f i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Iic (f i) ** exact pi_Iio_ae_eq_pi_Iic ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.mem_core_res ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 f\u271d : \u03b1 \u2192. \u03b2 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 t : Set \u03b2 x : \u03b1 \u22a2 x \u2208 core (res f s) t \u2194 x \u2208 s \u2192 f x \u2208 t ** simp [mem_core, mem_res] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_div_left_eq_self ** \ud835\udd5c : Type u_1 M : Type u_2 \u03b1 : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst\u271d\u2079 : MeasurableSpace G inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E inst\u271d\u2075 : NormedAddCommGroup F \u03bc\u271d : Measure G f\u271d : G \u2192 E g : G inst\u271d\u2074 : Group G inst\u271d\u00b3 : MeasurableMul G inst\u271d\u00b2 : MeasurableInv G f : G \u2192 E \u03bc : Measure G inst\u271d\u00b9 : IsInvInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bc x' : G \u22a2 \u222b (x : G), f (x' / x) \u2202\u03bc = \u222b (x : G), f x \u2202\u03bc ** simp_rw [div_eq_mul_inv] ** \ud835\udd5c : Type u_1 M : Type u_2 \u03b1 : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst\u271d\u2079 : MeasurableSpace G inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E inst\u271d\u2075 : NormedAddCommGroup F \u03bc\u271d : Measure G f\u271d : G \u2192 E g : G inst\u271d\u2074 : Group G inst\u271d\u00b3 : MeasurableMul G inst\u271d\u00b2 : MeasurableInv G f : G \u2192 E \u03bc : Measure G inst\u271d\u00b9 : IsInvInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bc x' : G \u22a2 \u222b (x : G), f (x' * x\u207b\u00b9) \u2202\u03bc = \u222b (x : G), f x \u2202\u03bc ** rw [integral_inv_eq_self (fun x => f (x' * x)) \u03bc, integral_mul_left_eq_self f x'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_le_one_iff ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 encard s \u2264 1 \u2194 \u2200 (a b : \u03b1), a \u2208 s \u2192 b \u2208 s \u2192 a = b ** rw [encard_le_one_iff_eq, or_iff_not_imp_left, \u2190Ne.def, \u2190nonempty_iff_ne_empty] ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 (Set.Nonempty s \u2192 \u2203 x, s = {x}) \u2194 \u2200 (a b : \u03b1), a \u2208 s \u2192 b \u2208 s \u2192 a = b ** refine' \u27e8fun h a b has hbs \u21a6 _,\n  fun h \u27e8x, hx\u27e9 \u21a6 \u27e8x, ((singleton_subset_iff.2 hx).antisymm' (fun y hy \u21a6 h _ _ hy hx))\u27e9\u27e9 ** \u03b1 : Type u_1 s t : Set \u03b1 h : Set.Nonempty s \u2192 \u2203 x, s = {x} a b : \u03b1 has : a \u2208 s hbs : b \u2208 s \u22a2 a = b ** obtain \u27e8x, rfl\u27e9 := h \u27e8_, has\u27e9 ** case intro \u03b1 : Type u_1 t : Set \u03b1 a b x : \u03b1 h : Set.Nonempty {x} \u2192 \u2203 x_1, {x} = {x_1} has : a \u2208 {x} hbs : b \u2208 {x} \u22a2 a = b ** rw [(has : a = x), (hbs : b = x)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd \u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2 ** have h :=\n  @exists_seq_tendsto_sSup _ _ _ _ _ (measurableLEEval \u03bd \u03bc)\n    \u27e80, 0, zero_mem_measurableLE, by simp\u27e9 (OrderTop.bddAbove _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd h :   \u2203 u,     Monotone u \u2227       Filter.Tendsto u Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) \u2227 \u2200 (n : \u2115), u n \u2208 measurableLEEval \u03bd \u03bc \u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2 ** choose g _ hg\u2082 f hf\u2081 hf\u2082 using h ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2 ** set \u03be := \u2a06 (n) (k) (_ : k \u2264 n), f k with h\u03be ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2 ** have h\u03bem : Measurable \u03be := by\n  convert measurable_iSup fun n => (iSup_mem_measurableLE _ hf\u2081 n).1\n  refine Option.ext fun x => ?_; simp [h\u03be] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2 ** set \u03bc\u2081 := \u03bc - \u03bd.withDensity \u03be with h\u03bc\u2081 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be \u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2 ** have hle : \u03bd.withDensity \u03be \u2264 \u03bc := by\n  intro B hB\n  rw [h\u03be, withDensity_apply _ hB]\n  simp_rw [iSup_apply]\n  rw [lintegral_iSup (fun i => (iSup_mem_measurableLE _ hf\u2081 i).1) (iSup_monotone _)]\n  exact iSup_le fun i => (iSup_mem_measurableLE _ hf\u2081 i).2 B hB ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc \u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2 ** have : IsFiniteMeasure (\u03bd.withDensity \u03be) := by\n  refine' isFiniteMeasure_withDensity _\n  have hle' := hle univ MeasurableSet.univ\n  rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at hle'\n  exact ne_top_of_le_ne_top (measure_ne_top _ _) hle' ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) \u22a2 \u2203 p, Measurable p.2 \u2227 p.1 \u27c2\u2098 \u03bd \u2227 \u03bc = p.1 + withDensity \u03bd p.2 ** refine' \u27e8\u27e8\u03bc\u2081, \u03be\u27e9, h\u03bem, _, _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd \u22a2 (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) 0 = 0 ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k \u22a2 sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd ** have :=\n  @lintegral_tendsto_of_tendsto_of_monotone _ _ \u03bd (fun n => \u2a06 (k) (_ : k \u2264 n), f k)\n    (\u2a06 (n) (k) (_ : k \u2264 n), f k) ?_ ?_ ?_ ** case refine_4 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k this :   Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop     (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd)) \u22a2 sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd ** refine' tendsto_nhds_unique _ this ** case refine_4 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k this :   Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop     (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd)) \u22a2 Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop     (nhds (sSup (measurableLEEval \u03bd \u03bc))) ** refine' tendsto_of_tendsto_of_tendsto_of_le_of_le hg\u2082 tendsto_const_nhds _ _ ** case refine_4.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k this :   Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop     (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd)) \u22a2 g \u2264 fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd ** intro n ** case refine_4.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k this :   Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop     (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd)) n : \u2115 \u22a2 g n \u2264 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) n ** rw [\u2190 hf\u2082 n] ** case refine_4.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k this :   Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop     (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd)) n : \u2115 \u22a2 (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) \u2264 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) n ** apply lintegral_mono ** case refine_4.refine'_1.hfg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k this :   Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop     (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd)) n : \u2115 \u22a2 (fun a => f n a) \u2264 fun a => iSup (fun k => \u2a06 (_ : k \u2264 n), f k) a ** simp only [iSup_apply, iSup_le_le f n n le_rfl] ** case refine_4.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k this :   Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop     (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd)) \u22a2 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) \u2264 fun x => sSup (measurableLEEval \u03bd \u03bc) ** intro n ** case refine_4.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k this :   Filter.Tendsto (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) Filter.atTop     (nhds (\u222b\u207b (x : \u03b1), iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd)) n : \u2115 \u22a2 (fun n => \u222b\u207b (x : \u03b1), iSup (fun k => \u2a06 (_ : k \u2264 n), f k) x \u2202\u03bd) n \u2264 (fun x => sSup (measurableLEEval \u03bd \u03bc)) n ** exact le_sSup \u27e8\u2a06 (k : \u2115) (_ : k \u2264 n), f k, iSup_mem_measurableLE' _ hf\u2081 _, rfl\u27e9 ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k \u22a2 \u2200 (n : \u2115), AEMeasurable ((fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n) ** intro n ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k n : \u2115 \u22a2 AEMeasurable ((fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n) ** refine' Measurable.aemeasurable _ ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k n : \u2115 \u22a2 Measurable ((fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n) ** convert (iSup_mem_measurableLE _ hf\u2081 n).1 ** case h.e'_5.h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k n : \u2115 x\u271d : \u03b1 \u22a2 (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x\u271d = \u2a06 k, \u2a06 (_ : k \u2264 n), f k x\u271d ** refine Option.ext fun x => ?_ ** case h.e'_5.h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k n : \u2115 x\u271d : \u03b1 x : \u211d\u22650 \u22a2 x \u2208 (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x\u271d \u2194 x \u2208 \u2a06 k, \u2a06 (_ : k \u2264 n), f k x\u271d ** simp ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, Monotone fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x ** refine' Filter.eventually_of_forall fun a => _ ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k a : \u03b1 \u22a2 Monotone fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n a ** simp [iSup_monotone' f _] ** case refine_3 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd,     Filter.Tendsto (fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n x) Filter.atTop       (nhds (iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) x)) ** refine' Filter.eventually_of_forall fun a => _ ** case refine_3 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k a : \u03b1 \u22a2 Filter.Tendsto (fun n => (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) n a) Filter.atTop     (nhds (iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a)) ** simp [tendsto_atTop_iSup (iSup_monotone' f a)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u22a2 Measurable \u03be ** convert measurable_iSup fun n => (iSup_mem_measurableLE _ hf\u2081 n).1 ** case h.e'_5.h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd x\u271d : \u03b1 \u22a2 \u03be x\u271d = \u2a06 i, \u2a06 k, \u2a06 (_ : k \u2264 i), f k x\u271d ** refine Option.ext fun x => ?_ ** case h.e'_5.h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd x\u271d : \u03b1 x : \u211d\u22650 \u22a2 x \u2208 \u03be x\u271d \u2194 x \u2208 \u2a06 i, \u2a06 k, \u2a06 (_ : k \u2264 i), f k x\u271d ** simp [h\u03be] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be \u22a2 withDensity \u03bd \u03be \u2264 \u03bc ** intro B hB ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be B : Set \u03b1 hB : MeasurableSet B \u22a2 \u2191\u2191(withDensity \u03bd \u03be) B \u2264 \u2191\u2191\u03bc B ** rw [h\u03be, withDensity_apply _ hB] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be B : Set \u03b1 hB : MeasurableSet B \u22a2 \u222b\u207b (a : \u03b1) in B, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd \u2264 \u2191\u2191\u03bc B ** simp_rw [iSup_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be B : Set \u03b1 hB : MeasurableSet B \u22a2 \u222b\u207b (a : \u03b1) in B, \u2a06 i, \u2a06 i_1, \u2a06 (_ : i_1 \u2264 i), f i_1 a \u2202\u03bd \u2264 \u2191\u2191\u03bc B ** rw [lintegral_iSup (fun i => (iSup_mem_measurableLE _ hf\u2081 i).1) (iSup_monotone _)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be B : Set \u03b1 hB : MeasurableSet B \u22a2 \u2a06 n, \u222b\u207b (a : \u03b1) in B, \u2a06 k, \u2a06 (_ : k \u2264 n), f k a \u2202\u03bd \u2264 \u2191\u2191\u03bc B ** exact iSup_le fun i => (iSup_mem_measurableLE _ hf\u2081 i).2 B hB ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc \u22a2 IsFiniteMeasure (withDensity \u03bd \u03be) ** refine' isFiniteMeasure_withDensity _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc \u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4 ** have hle' := hle univ MeasurableSet.univ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc hle' : \u2191\u2191(withDensity \u03bd \u03be) univ \u2264 \u2191\u2191\u03bc univ \u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4 ** rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at hle' ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc hle' : \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc univ \u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4 ** exact ne_top_of_le_ne_top (measure_ne_top _ _) hle' ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) \u22a2 (\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd ** by_contra h ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u22a2 False ** obtain \u27e8\u03b5, h\u03b5\u2081, E, hE\u2081, hE\u2082, hE\u2083\u27e9 := exists_positive_of_not_mutuallySingular \u03bc\u2081 \u03bd h ** case refine'_1.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 : VectorMeasure.restrict 0 E \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc\u2081 - toSignedMeasure (\u03b5 \u2022 \u03bd)) E \u22a2 False ** simp_rw [h\u03bc\u2081] at hE\u2083 ** case refine'_1.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E \u22a2 False ** have h\u03bele : \u2200 A, MeasurableSet A \u2192 (\u222b\u207b a in A, \u03be a \u2202\u03bd) \u2264 \u03bc A := by\n  intro A hA; rw [h\u03be]\n  simp_rw [iSup_apply]\n  rw [lintegral_iSup (fun n => (iSup_mem_measurableLE _ hf\u2081 n).1) (iSup_monotone _)]\n  exact iSup_le fun n => (iSup_mem_measurableLE _ hf\u2081 n).2 A hA ** case refine'_1.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) \u22a2 False ** have h\u03be\u03b5 : (\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e)) \u2208 measurableLE \u03bd \u03bc := by\n  refine' \u27e8Measurable.add h\u03bem (Measurable.indicator measurable_const hE\u2081), fun A hA => _\u27e9\n  have :\n    (\u222b\u207b a in A, (\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e)) a \u2202\u03bd) =\n      (\u222b\u207b a in A \u2229 E, \u03b5 + \u03be a \u2202\u03bd) + \u222b\u207b a in A \\ E, \u03be a \u2202\u03bd := by\n    simp only [lintegral_add_left measurable_const, lintegral_add_left h\u03bem,\n      set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE\u2081, Pi.add_apply,\n      lintegral_indicator _ hE\u2081, restrict_apply hE\u2081]\n    rw [inter_comm, add_comm]\n  rw [this, \u2190 measure_inter_add_diff A hE\u2081]\n  exact add_le_add (h\u03b5\u2082 A hA) (h\u03bele (A \\ E) (hA.diff hE\u2081)) ** case refine'_1.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) h\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc \u22a2 False ** have : (\u222b\u207b a, \u03be a + E.indicator (fun _ => (\u03b5 : \u211d\u22650\u221e)) a \u2202\u03bd) \u2264 sSup (measurableLEEval \u03bd \u03bc) :=\n  le_sSup \u27e8\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e), h\u03be\u03b5, rfl\u27e9 ** case refine'_1.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) h\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc this : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc) \u22a2 False ** refine' not_lt.2 this _ ** case refine'_1.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) h\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc this : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc) \u22a2 sSup (measurableLEEval \u03bd \u03bc) < \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd ** rw [h\u03be\u2081, lintegral_add_left h\u03bem, lintegral_indicator _ hE\u2081, set_lintegral_const] ** case refine'_1.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) h\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc this : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc) \u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd < \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd + \u2191\u03b5 * \u2191\u2191\u03bd E ** refine' ENNReal.lt_add_right _ (ENNReal.mul_pos_iff.2 \u27e8ENNReal.coe_pos.2 h\u03b5\u2081, hE\u2082\u27e9).ne' ** case refine'_1.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) h\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc this : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc) \u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4 ** have := measure_ne_top (\u03bd.withDensity \u03be) univ ** case refine'_1.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d\u00b9 : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) h\u03be\u03b5 : (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc this\u271d : \u222b\u207b (a : \u03b1), \u03be a + indicator E (fun x => \u2191\u03b5) a \u2202\u03bd \u2264 sSup (measurableLEEval \u03bd \u03bc) this : \u2191\u2191(withDensity \u03bd \u03be) univ \u2260 \u22a4 \u22a2 \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd \u2260 \u22a4 ** rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E \u22a2 \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A ** intro A hA ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A ** rw [h\u03be] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (a : \u03b1) in A, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd \u2264 \u2191\u2191\u03bc A ** simp_rw [iSup_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (a : \u03b1) in A, \u2a06 i, \u2a06 i_1, \u2a06 (_ : i_1 \u2264 i), f i_1 a \u2202\u03bd \u2264 \u2191\u2191\u03bc A ** rw [lintegral_iSup (fun n => (iSup_mem_measurableLE _ hf\u2081 n).1) (iSup_monotone _)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E A : Set \u03b1 hA : MeasurableSet A \u22a2 \u2a06 n, \u222b\u207b (a : \u03b1) in A, \u2a06 k, \u2a06 (_ : k \u2264 n), f k a \u2202\u03bd \u2264 \u2191\u2191\u03bc A ** exact iSup_le fun n => (iSup_mem_measurableLE _ hf\u2081 n).2 A hA ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A \u22a2 \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) ** intro A hA ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) ** have := subset_le_of_restrict_le_restrict _ _ hE\u2081 hE\u2083 (inter_subset_right A E) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A this : \u21910 (A \u2229 E) \u2264 \u2191(toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) (A \u2229 E) \u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) ** rwa [zero_apply, toSignedMeasure_sub_apply (hA.inter hE\u2081),\n  Measure.sub_apply (hA.inter hE\u2081) hle,\n  ENNReal.toReal_sub_of_le _ (ne_of_lt (measure_lt_top _ _)), sub_nonneg, le_sub_iff_add_le,\n  \u2190 ENNReal.toReal_add, ENNReal.toReal_le_toReal, Measure.coe_smul, Pi.smul_apply,\n  withDensity_apply _ (hA.inter hE\u2081), show \u03b5 \u2022 \u03bd (A \u2229 E) = (\u03b5 : \u211d\u22650\u221e) * \u03bd (A \u2229 E) by rfl,\n  \u2190 set_lintegral_const, \u2190 lintegral_add_left measurable_const] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A this : \u03b5 \u2022 \u2191\u2191\u03bd (A \u2229 E) + \u222b\u207b (a : \u03b1) in A \u2229 E, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) \u22a2 \u03b5 \u2022 \u2191\u2191\u03bd (A \u2229 E) = \u2191\u03b5 * \u2191\u2191\u03bd (A \u2229 E) ** rfl ** case ha \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A this : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E)) \u22a2 \u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) \u2260 \u22a4 ** rw [Ne.def, ENNReal.add_eq_top, not_or] ** case ha \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A this : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E)) \u22a2 \u00ac\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) = \u22a4 \u2227 \u00ac\u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) = \u22a4 ** exact \u27e8ne_of_lt (measure_lt_top _ _), ne_of_lt (measure_lt_top _ _)\u27e9 ** case hb \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A this : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) + \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E)) \u22a2 \u2191\u2191\u03bc (A \u2229 E) \u2260 \u22a4 ** exact ne_of_lt (measure_lt_top _ _) ** case ha \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A this : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E)) + ENNReal.toReal (\u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E)) \u22a2 \u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E) \u2260 \u22a4 ** exact ne_of_lt (measure_lt_top _ _) ** case hb \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A this : ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E)) + ENNReal.toReal (\u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E)) \u22a2 \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) \u2260 \u22a4 ** exact ne_of_lt (measure_lt_top _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A this : 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E) - \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) - ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E)) \u22a2 \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E) \u2264 \u2191\u2191\u03bc (A \u2229 E) ** rw [withDensity_apply _ (hA.inter hE\u2081)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A A : Set \u03b1 hA : MeasurableSet A this : 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (A \u2229 E) - \u2191\u2191(withDensity \u03bd \u03be) (A \u2229 E)) - ENNReal.toReal (\u2191\u2191(\u03b5 \u2022 \u03bd) (A \u2229 E)) \u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) ** exact h\u03bele (A \u2229 E) (hA.inter hE\u2081) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) \u22a2 (\u03be + indicator E fun x => \u2191\u03b5) \u2208 measurableLE \u03bd \u03bc ** refine' \u27e8Measurable.add h\u03bem (Measurable.indicator measurable_const hE\u2081), fun A hA => _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (x : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) x \u2202\u03bd \u2264 \u2191\u2191\u03bc A ** have :\n  (\u222b\u207b a in A, (\u03be + E.indicator fun _ => (\u03b5 : \u211d\u22650\u221e)) a \u2202\u03bd) =\n    (\u222b\u207b a in A \u2229 E, \u03b5 + \u03be a \u2202\u03bd) + \u222b\u207b a in A \\ E, \u03be a \u2202\u03bd := by\n  simp only [lintegral_add_left measurable_const, lintegral_add_left h\u03bem,\n    set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE\u2081, Pi.add_apply,\n    lintegral_indicator _ hE\u2081, restrict_apply hE\u2081]\n  rw [inter_comm, add_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) A : Set \u03b1 hA : MeasurableSet A this :   \u222b\u207b (a : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) a \u2202\u03bd = \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd \u22a2 \u222b\u207b (x : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) x \u2202\u03bd \u2264 \u2191\u2191\u03bc A ** rw [this, \u2190 measure_inter_add_diff A hE\u2081] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this\u271d : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) A : Set \u03b1 hA : MeasurableSet A this :   \u222b\u207b (a : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) a \u2202\u03bd = \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd \u22a2 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) + \u2191\u2191\u03bc (A \\ E) ** exact add_le_add (h\u03b5\u2082 A hA) (h\u03bele (A \\ E) (hA.diff hE\u2081)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (a : \u03b1) in A, (\u03be + indicator E fun x => \u2191\u03b5) a \u2202\u03bd = \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd + \u222b\u207b (a : \u03b1) in A \\ E, \u03be a \u2202\u03bd ** simp only [lintegral_add_left measurable_const, lintegral_add_left h\u03bem,\n  set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE\u2081, Pi.add_apply,\n  lintegral_indicator _ hE\u2081, restrict_apply hE\u2081] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) h : \u00ac(\u03bc\u2081, \u03be).1 \u27c2\u2098 \u03bd \u03b5 : \u211d\u22650 h\u03b5\u2081 : 0 < \u03b5 E : Set \u03b1 hE\u2081 : MeasurableSet E hE\u2082 : 0 < \u2191\u2191\u03bd E hE\u2083 :   VectorMeasure.restrict 0 E \u2264     VectorMeasure.restrict (toSignedMeasure (\u03bc - withDensity \u03bd \u03be) - toSignedMeasure (\u03b5 \u2022 \u03bd)) E h\u03bele : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A, \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc A h\u03b5\u2082 : \u2200 (A : Set \u03b1), MeasurableSet A \u2192 \u222b\u207b (a : \u03b1) in A \u2229 E, \u2191\u03b5 + \u03be a \u2202\u03bd \u2264 \u2191\u2191\u03bc (A \u2229 E) A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (a : \u03b1) in A, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd + \u2191\u03b5 * \u2191\u2191\u03bd (E \u2229 A) =     \u2191\u03b5 * \u2191\u2191\u03bd (A \u2229 E) + \u222b\u207b (a : \u03b1) in A, iSup (fun n => \u2a06 k, \u2a06 (_ : k \u2264 n), f k) a \u2202\u03bd ** rw [inter_comm, add_comm] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) \u22a2 \u03bc = (\u03bc\u2081, \u03be).1 + withDensity \u03bd (\u03bc\u2081, \u03be).2 ** rw [h\u03bc\u2081] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) \u22a2 \u03bc = (\u03bc - withDensity \u03bd \u03be, \u03be).1 + withDensity \u03bd (\u03bc - withDensity \u03bd \u03be, \u03be).2 ** ext1 A hA ** case refine'_2.h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd g : \u2115 \u2192 \u211d\u22650\u221e h\u271d : Monotone g hg\u2082 : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval \u03bd \u03bc))) f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf\u2081 : \u2200 (n : \u2115), f n \u2208 measurableLE \u03bd \u03bc hf\u2082 : \u2200 (n : \u2115), (fun f => \u222b\u207b (x : \u03b1), f x \u2202\u03bd) (f n) = g n \u03be : \u03b1 \u2192 \u211d\u22650\u221e := \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be : \u03be = \u2a06 n, \u2a06 k, \u2a06 (_ : k \u2264 n), f k h\u03be\u2081 : sSup (measurableLEEval \u03bd \u03bc) = \u222b\u207b (a : \u03b1), \u03be a \u2202\u03bd h\u03bem : Measurable \u03be \u03bc\u2081 : Measure \u03b1 := \u03bc - withDensity \u03bd \u03be h\u03bc\u2081 : \u03bc\u2081 = \u03bc - withDensity \u03bd \u03be hle : withDensity \u03bd \u03be \u2264 \u03bc this : IsFiniteMeasure (withDensity \u03bd \u03be) A : Set \u03b1 hA : MeasurableSet A \u22a2 \u2191\u2191\u03bc A = \u2191\u2191((\u03bc - withDensity \u03bd \u03be, \u03be).1 + withDensity \u03bd (\u03bc - withDensity \u03bd \u03be, \u03be).2) A ** rw [Measure.coe_add, Pi.add_apply, Measure.sub_apply hA hle, add_comm,\n  add_tsub_cancel_of_le (hle A hA)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.normalize_num_den ** n : Int d : Nat z : d \u2260 0 n' : Int d' : Nat z' : d' \u2260 0 c : Nat.Coprime (Int.natAbs n') d' h : normalize n d = mk' n' d' \u22a2 \u2203 m, m \u2260 0 \u2227 n = n' * \u2191m \u2227 d = d' * m ** have := normalize_num_den' n d z ** n : Int d : Nat z : d \u2260 0 n' : Int d' : Nat z' : d' \u2260 0 c : Nat.Coprime (Int.natAbs n') d' h : normalize n d = mk' n' d' this : \u2203 d_1, d_1 \u2260 0 \u2227 n = (normalize n d).num * \u2191d_1 \u2227 d = (normalize n d).den * d_1 \u22a2 \u2203 m, m \u2260 0 \u2227 n = n' * \u2191m \u2227 d = d' * m ** rwa [h] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.exists_closedBall_covering_tsum_measure_le ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** obtain \u27e8u, su, u_open, \u03bcu\u27e9 : \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u03bc U \u2264 \u03bc s + \u03b5 / 2 :=\n  Set.exists_isOpen_le_add _ _\n    (by\n      simpa only [or_false_iff, Ne.def, ENNReal.div_eq_zero_iff, ENNReal.one_ne_top] using h\u03b5) ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** have : \u2200 x \u2208 s, \u2203 R > 0, ball x R \u2286 u := fun x hx =>\n  Metric.mem_nhds_iff.1 (u_open.mem_nhds (su hx)) ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 this : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 R, R > 0 \u2227 ball x R \u2286 u \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** choose! R hR using this ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** obtain \u27e8t0, r0, t0_count, t0s, hr0, \u03bct0, t0_disj\u27e9 :\n  \u2203 (t0 : Set \u03b1) (r0 : \u03b1 \u2192 \u211d), t0.Countable \u2227 t0 \u2286 s \u2227\n    (\u2200 x \u2208 t0, r0 x \u2208 f x \u2229 Ioo 0 (R x)) \u2227 \u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 \u2227\n      t0.PairwiseDisjoint fun x => closedBall x (r0 x) :=\n  exists_disjoint_closedBall_covering_ae \u03bc f s hf R fun x hx => (hR x hx).1 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** let s' := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) ** case intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** have s's : s' \u2286 s := diff_subset _ _ ** case intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** obtain \u27e8N, \u03c4, h\u03c4, H\u27e9 : \u2203 N \u03c4, 1 < \u03c4 \u2227 IsEmpty (Besicovitch.SatelliteConfig \u03b1 N \u03c4) :=\n  HasBesicovitchCovering.no_satelliteConfig ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** obtain \u27e8v, s'v, v_open, \u03bcv\u27e9 : \u2203 v, v \u2287 s' \u2227 IsOpen v \u2227 \u03bc v \u2264 \u03bc s' + \u03b5 / 2 / N :=\n  Set.exists_isOpen_le_add _ _\n    (by\n      simp only [h\u03b5, ENNReal.nat_ne_top, WithTop.mul_eq_top_iff, Ne.def, ENNReal.div_eq_zero_iff,\n        ENNReal.one_ne_top, not_false_iff, and_false_iff, false_and_iff, or_self_iff]) ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** have : \u2200 x \u2208 s', \u2203 r1 \u2208 f x \u2229 Ioo (0 : \u211d) 1, closedBall x r1 \u2286 v := by\n  intro x hx\n  rcases Metric.mem_nhds_iff.1 (v_open.mem_nhds (s'v hx)) with \u27e8r, rpos, hr\u27e9\n  rcases hf x (s's hx) (min r 1) (lt_min rpos zero_lt_one) with \u27e8R', hR'\u27e9\n  exact\n    \u27e8R', \u27e8hR'.1, hR'.2.1, hR'.2.2.trans_le (min_le_right _ _)\u27e9,\n      Subset.trans (closedBall_subset_ball (hR'.2.2.trans_le (min_le_left _ _))) hr\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N this : \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** choose! r1 hr1 using this ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** let q : BallPackage s' \u03b1 :=\n  { c := fun x => x\n    r := fun x => r1 x\n    rpos := fun x => (hr1 x.1 x.2).1.2.1\n    r_bound := 1\n    r_le := fun x => (hr1 x.1 x.2).1.2.2.le } ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** obtain \u27e8S, S_disj, hS\u27e9 :\n  \u2203 S : Fin N \u2192 Set s',\n    (\u2200 i : Fin N, (S i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) \u2227\n      range q.c \u2286 \u22c3 i : Fin N, \u22c3 j \u2208 S i, ball (q.c j) (q.r j) :=\n  exist_disjoint_covering_families h\u03c4 H q ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** have S_count : \u2200 i, (S i).Countable := by\n  intro i\n  apply (S_disj i).countable_of_nonempty_interior fun j _ => ?_\n  have : (ball (j : \u03b1) (r1 j)).Nonempty := nonempty_ball.2 (q.rpos _)\n  exact this.mono ball_subset_interior_closedBall ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** let r x := if x \u2208 s' then r1 x else r0 x ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** have r_t0 : \u2200 x \u2208 t0, r x = r0 x := by\n  intro x hx\n  have : \u00acx \u2208 s' := by\n    simp only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_lt, not_le,\n      mem_diff, not_forall]\n    intro _\n    refine' \u27e8x, hx, _\u27e9\n    rw [dist_self]\n    exact (hr0 x hx).2.1.le\n  simp only [if_neg this] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 \u2203 t r,     Set.Countable t \u2227       t \u2286 s \u2227         (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227           s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** refine' \u27e8t0 \u222a \u22c3 i : Fin N, ((\u2191) : s' \u2192 \u03b1) '' S i, r, _, _, _, _, _\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) \u22a2 \u03b5 / 2 \u2260 0 ** simpa only [or_false_iff, Ne.def, ENNReal.div_eq_zero_iff, ENNReal.one_ne_top] using h\u03b5 ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) \u22a2 \u03b5 / 2 / \u2191N \u2260 0 ** simp only [h\u03b5, ENNReal.nat_ne_top, WithTop.mul_eq_top_iff, Ne.def, ENNReal.div_eq_zero_iff,\n  ENNReal.one_ne_top, not_false_iff, and_false_iff, false_and_iff, or_self_iff] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N \u22a2 \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v ** intro x hx ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N x : \u03b1 hx : x \u2208 s' \u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v ** rcases Metric.mem_nhds_iff.1 (v_open.mem_nhds (s'v hx)) with \u27e8r, rpos, hr\u27e9 ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N x : \u03b1 hx : x \u2208 s' r : \u211d rpos : r > 0 hr : ball x r \u2286 v \u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v ** rcases hf x (s's hx) (min r 1) (lt_min rpos zero_lt_one) with \u27e8R', hR'\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N x : \u03b1 hx : x \u2208 s' r : \u211d rpos : r > 0 hr : ball x r \u2286 v R' : \u211d hR' : R' \u2208 f x \u2229 Ioo 0 (min r 1) \u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v ** exact\n  \u27e8R', \u27e8hR'.1, hR'.2.1, hR'.2.2.trans_le (min_le_right _ _)\u27e9,\n    Subset.trans (closedBall_subset_ball (hR'.2.2.trans_le (min_le_left _ _))) hr\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) \u22a2 \u2200 (i : Fin N), Set.Countable (S i) ** intro i ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) i : Fin N \u22a2 Set.Countable (S i) ** apply (S_disj i).countable_of_nonempty_interior fun j _ => ?_ ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) i : Fin N j : \u2191s' x\u271d : j \u2208 S i \u22a2 Set.Nonempty (interior (closedBall (BallPackage.c q j) (BallPackage.r q j))) ** have : (ball (j : \u03b1) (r1 j)).Nonempty := nonempty_ball.2 (q.rpos _) ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) i : Fin N j : \u2191s' x\u271d : j \u2208 S i this : Set.Nonempty (ball (\u2191j) (r1 \u2191j)) \u22a2 Set.Nonempty (interior (closedBall (BallPackage.c q j) (BallPackage.r q j))) ** exact this.mono ball_subset_interior_closedBall ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x \u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x ** intro x hx ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x x : \u03b1 hx : x \u2208 t0 \u22a2 r x = r0 x ** have : \u00acx \u2208 s' := by\n  simp only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_lt, not_le,\n    mem_diff, not_forall]\n  intro _\n  refine' \u27e8x, hx, _\u27e9\n  rw [dist_self]\n  exact (hr0 x hx).2.1.le ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x x : \u03b1 hx : x \u2208 t0 this : \u00acx \u2208 s' \u22a2 r x = r0 x ** simp only [if_neg this] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x x : \u03b1 hx : x \u2208 t0 \u22a2 \u00acx \u2208 s' ** simp only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_lt, not_le,\n  mem_diff, not_forall] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x x : \u03b1 hx : x \u2208 t0 \u22a2 x \u2208 s \u2192 \u2203 x_1, x_1 \u2208 t0 \u2227 dist x x_1 \u2264 r0 x_1 ** intro _ ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x x : \u03b1 hx : x \u2208 t0 a\u271d : x \u2208 s \u22a2 \u2203 x_1, x_1 \u2208 t0 \u2227 dist x x_1 \u2264 r0 x_1 ** refine' \u27e8x, hx, _\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x x : \u03b1 hx : x \u2208 t0 a\u271d : x \u2208 s \u22a2 dist x x \u2264 r0 x ** rw [dist_self] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x x : \u03b1 hx : x \u2208 t0 a\u271d : x \u2208 s \u22a2 0 \u2264 r0 x ** exact (hr0 x hx).2.1.le ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 Set.Countable (t0 \u222a \u22c3 i, Subtype.val '' S i) ** exact t0_count.union (countable_iUnion fun i => (S_count i).image _) ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 t0 \u222a \u22c3 i, Subtype.val '' S i \u2286 s ** simp only [t0s, true_and_iff, union_subset_iff, image_subset_iff, iUnion_subset_iff] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 \u2200 (i : Fin N), S i \u2286 (fun a => \u2191a) \u207b\u00b9' s ** intro i x _ ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x i : Fin N x : { x // x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) } a\u271d : x \u2208 S i \u22a2 x \u2208 (fun a => \u2191a) \u207b\u00b9' s ** exact s's x.2 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i \u2192 r x \u2208 f x ** intro x hx ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i \u22a2 r x \u2208 f x ** cases hx with\n| inl hx =>\n  rw [r_t0 x hx]\n  exact (hr0 _ hx).1\n| inr hx =>\n  have h'x : x \u2208 s' := by\n    simp only [mem_iUnion, mem_image] at hx\n    rcases hx with \u27e8i, y, _, rfl\u27e9\n    exact y.2\n  simp only [if_pos h'x, (hr1 x h'x).1.1] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 t0 \u22a2 r x \u2208 f x ** rw [r_t0 x hx] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 t0 \u22a2 r0 x \u2208 f x ** exact (hr0 _ hx).1 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 \u22c3 i, Subtype.val '' S i \u22a2 r x \u2208 f x ** have h'x : x \u2208 s' := by\n  simp only [mem_iUnion, mem_image] at hx\n  rcases hx with \u27e8i, y, _, rfl\u27e9\n  exact y.2 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 \u22c3 i, Subtype.val '' S i h'x : x \u2208 s' \u22a2 r x \u2208 f x ** simp only [if_pos h'x, (hr1 x h'x).1.1] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 \u22c3 i, Subtype.val '' S i \u22a2 x \u2208 s' ** simp only [mem_iUnion, mem_image] at hx ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : \u2203 i x_1, x_1 \u2208 S i \u2227 \u2191x_1 = x \u22a2 x \u2208 s' ** rcases hx with \u27e8i, y, _, rfl\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x i : Fin N y : { x // x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) } left\u271d : y \u2208 S i \u22a2 \u2191y \u2208 s' ** exact y.2 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_4 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 s \u2286 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x) ** intro x hx ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_4 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s \u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x) ** by_cases h'x : x \u2208 s' ** case pos \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' \u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x) ** obtain \u27e8i, y, ySi, xy\u27e9 : \u2203 (i : Fin N) (y : \u21a5s'), y \u2208 S i \u2227 x \u2208 ball (y : \u03b1) (r1 y) := by\n  have A : x \u2208 range q.c := by\n    simpa only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,\n      mem_setOf_eq, Subtype.range_coe_subtype, mem_diff] using h'x\n  simpa only [mem_iUnion, mem_image, bex_def] using hS A ** case pos.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' i : Fin N y : \u2191s' ySi : y \u2208 S i xy : x \u2208 ball (\u2191y) (r1 \u2191y) \u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x) ** refine' mem_iUnion\u2082.2 \u27e8y, Or.inr _, _\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' \u22a2 \u2203 i y, y \u2208 S i \u2227 x \u2208 ball (\u2191y) (r1 \u2191y) ** have A : x \u2208 range q.c := by\n  simpa only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,\n    mem_setOf_eq, Subtype.range_coe_subtype, mem_diff] using h'x ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' A : x \u2208 range q.c \u22a2 \u2203 i y, y \u2208 S i \u2227 x \u2208 ball (\u2191y) (r1 \u2191y) ** simpa only [mem_iUnion, mem_image, bex_def] using hS A ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' \u22a2 x \u2208 range q.c ** simpa only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,\n  mem_setOf_eq, Subtype.range_coe_subtype, mem_diff] using h'x ** case pos.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' i : Fin N y : \u2191s' ySi : y \u2208 S i xy : x \u2208 ball (\u2191y) (r1 \u2191y) \u22a2 \u2191y \u2208 \u22c3 i, Subtype.val '' S i ** simp only [mem_iUnion, mem_image] ** case pos.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' i : Fin N y : \u2191s' ySi : y \u2208 S i xy : x \u2208 ball (\u2191y) (r1 \u2191y) \u22a2 \u2203 i x, x \u2208 S i \u2227 \u2191x = \u2191y ** exact \u27e8i, y, ySi, rfl\u27e9 ** case pos.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' i : Fin N y : \u2191s' ySi : y \u2208 S i xy : x \u2208 ball (\u2191y) (r1 \u2191y) \u22a2 x \u2208 closedBall (\u2191y) (r \u2191y) ** have : (y : \u03b1) \u2208 s' := y.2 ** case pos.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' i : Fin N y : \u2191s' ySi : y \u2208 S i xy : x \u2208 ball (\u2191y) (r1 \u2191y) this : \u2191y \u2208 s' \u22a2 x \u2208 closedBall (\u2191y) (r \u2191y) ** simp only [if_pos this] ** case pos.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : x \u2208 s' i : Fin N y : \u2191s' ySi : y \u2208 S i xy : x \u2208 ball (\u2191y) (r1 \u2191y) this : \u2191y \u2208 s' \u22a2 x \u2208 closedBall (\u2191y) (r1 \u2191y) ** exact ball_subset_closedBall xy ** case neg \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : \u00acx \u2208 s' \u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x) ** obtain \u27e8y, yt0, hxy\u27e9 : \u2203 y : \u03b1, y \u2208 t0 \u2227 x \u2208 closedBall y (r0 y) := by\n  simpa [hx, -mem_closedBall] using h'x ** case neg.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : \u00acx \u2208 s' y : \u03b1 yt0 : y \u2208 t0 hxy : x \u2208 closedBall y (r0 y) \u22a2 x \u2208 \u22c3 x \u2208 t0 \u222a \u22c3 i, Subtype.val '' S i, closedBall x (r x) ** refine' mem_iUnion\u2082.2 \u27e8y, Or.inl yt0, _\u27e9 ** case neg.intro.intro \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : \u00acx \u2208 s' y : \u03b1 yt0 : y \u2208 t0 hxy : x \u2208 closedBall y (r0 y) \u22a2 x \u2208 closedBall y (r y) ** rwa [r_t0 _ yt0] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 s h'x : \u00acx \u2208 s' \u22a2 \u2203 y, y \u2208 t0 \u2227 x \u2208 closedBall y (r0 y) ** simpa [hx, -mem_closedBall] using h'x ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_5 \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 B : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N \u22a2 \u2211' (x : \u2191(t0 \u222a \u22c3 i, Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 ** calc\n  (\u2211' x : \u21a5(t0 \u222a \u22c3 i : Fin N, ((\u2191) : s' \u2192 \u03b1) '' S i), \u03bc (closedBall x (r x))) \u2264\n      (\u2211' x : t0, \u03bc (closedBall x (r x))) +\n        \u2211' x : \u22c3 i : Fin N, ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (closedBall x (r x)) :=\n    ENNReal.tsum_union_le (fun x => \u03bc (closedBall x (r x))) _ _\n  _ \u2264\n      (\u2211' x : t0, \u03bc (closedBall x (r x))) +\n        \u2211 i : Fin N, \u2211' x : ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (closedBall x (r x)) :=\n    (add_le_add le_rfl (ENNReal.tsum_iUnion_le (fun x => \u03bc (closedBall x (r x))) _))\n  _ \u2264 \u03bc s + \u03b5 / 2 + \u2211 i : Fin N, \u03b5 / 2 / N := by\n    refine' add_le_add A _\n    refine' Finset.sum_le_sum _\n    intro i _\n    exact B i\n  _ \u2264 \u03bc s + \u03b5 / 2 + \u03b5 / 2 := by\n    refine' add_le_add le_rfl _\n    simp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul, ENNReal.mul_div_le]\n  _ = \u03bc s + \u03b5 := by rw [add_assoc, ENNReal.add_halves] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) ** congr 1 ** case e_f \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 (fun x => \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x))) = fun x => \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) ** ext x ** case e_f.h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u2191t0 \u22a2 \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) ** rw [r_t0 x x.2] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191x) (r0 \u2191x)) ** haveI : Encodable t0 := t0_count.toEncodable ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x this : Encodable \u2191t0 \u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191x) (r0 \u2191x)) ** rw [measure_iUnion] ** case hn \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x this : Encodable \u2191t0 \u22a2 Pairwise (Disjoint on fun x => closedBall (\u2191x) (r0 \u2191x)) ** exact (pairwise_subtype_iff_pairwise_set _ _).2 t0_disj ** case h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x this : Encodable \u2191t0 \u22a2 \u2200 (i : \u2191t0), MeasurableSet (closedBall (\u2191i) (r0 \u2191i)) ** exact fun i => measurableSet_closedBall ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191x) (r0 \u2191x)) \u2264 \u2191\u2191\u03bc u ** apply measure_mono ** case h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 \u22c3 x, closedBall (\u2191x) (r0 \u2191x) \u2286 u ** simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff] ** case h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x \u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u2192 closedBall x (r0 x) \u2286 u ** intro x hx ** case h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x x : \u03b1 hx : x \u2208 t0 \u22a2 closedBall x (r0 x) \u2286 u ** apply Subset.trans (closedBall_subset_ball (hr0 x hx).2.2) (hR x (t0s hx)).2 ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N \u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** have : InjOn ((\u2191) : s' \u2192 \u03b1) (S i) := Subtype.val_injective.injOn _ ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N this : InjOn Subtype.val (S i) \u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** let F : S i \u2243 ((\u2191) : s' \u2192 \u03b1) '' S i := this.bijOn_image.equiv _ ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N this : InjOn Subtype.val (S i) F : \u2191(S i) \u2243 \u2191(Subtype.val '' S i) := BijOn.equiv Subtype.val (_ : BijOn Subtype.val (S i) (Subtype.val '' S i)) \u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** exact (F.tsum_eq fun x => \u03bc (closedBall x (r x))).symm ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N \u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) ** congr 1 ** case e_f \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N \u22a2 (fun x => \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))) = fun x => \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) ** ext x ** case e_f.h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N x : \u2191(S i) \u22a2 \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) ** have : (x : \u03b1) \u2208 s' := x.1.2 ** case e_f.h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N x : \u2191(S i) this : \u2191\u2191x \u2208 s' \u22a2 \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) ** simp only [if_pos this] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N \u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) ** haveI : Encodable (S i) := (S_count i).toEncodable ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N this : Encodable \u2191(S i) \u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) = \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) ** rw [measure_iUnion] ** case hn \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N this : Encodable \u2191(S i) \u22a2 Pairwise (Disjoint on fun x => closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) ** exact (pairwise_subtype_iff_pairwise_set _ _).2 (S_disj i) ** case h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N this : Encodable \u2191(S i) \u22a2 \u2200 (i_1 : \u2191(S i)), MeasurableSet (closedBall (\u2191\u2191i_1) (r1 \u2191\u2191i_1)) ** exact fun i => measurableSet_closedBall ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N \u22a2 \u2191\u2191\u03bc (\u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) \u2264 \u2191\u2191\u03bc v ** apply measure_mono ** case h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N \u22a2 \u22c3 x, closedBall (\u2191\u2191x) (r1 \u2191\u2191x) \u2286 v ** simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff] ** case h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N \u22a2 \u2200 (x : \u03b1) (h : x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)), { val := x, property := h } \u2208 S i \u2192 closedBall x (r1 x) \u2286 v ** intro x xs' _ ** case h \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N x : \u03b1 xs' : x \u2208 s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) h\u271d : { val := x, property := xs' } \u2208 S i \u22a2 closedBall x (r1 x) \u2286 v ** exact (hr1 x xs').2 ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N \u22a2 \u2191\u2191\u03bc v \u2264 \u03b5 / 2 / \u2191N ** have : \u03bc s' = 0 := \u03bct0 ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 i : Fin N this : \u2191\u2191\u03bc s' = 0 \u22a2 \u2191\u2191\u03bc v \u2264 \u03b5 / 2 / \u2191N ** rwa [this, zero_add] at \u03bcv ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 B : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N \u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) +       \u2211 i : Fin N, \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264     \u2191\u2191\u03bc s + \u03b5 / 2 + \u2211 i : Fin N, \u03b5 / 2 / \u2191N ** refine' add_le_add A _ ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 B : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N \u22a2 \u2211 i : Fin N, \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2211 i : Fin N, \u03b5 / 2 / \u2191N ** refine' Finset.sum_le_sum _ ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 B : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N \u22a2 \u2200 (i : Fin N), i \u2208 Finset.univ \u2192 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N ** intro i _ ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 B : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N i : Fin N a\u271d : i \u2208 Finset.univ \u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N ** exact B i ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 B : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N \u22a2 \u2191\u2191\u03bc s + \u03b5 / 2 + \u2211 i : Fin N, \u03b5 / 2 / \u2191N \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 + \u03b5 / 2 ** refine' add_le_add le_rfl _ ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 B : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N \u22a2 \u2211 i : Fin N, \u03b5 / 2 / \u2191N \u2264 \u03b5 / 2 ** simp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul, ENNReal.mul_div_le] ** \u03b1 : Type u_1 inst\u271d\u2076 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u2075 : SecondCountableTopology \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : OpensMeasurableSpace \u03b1 inst\u271d\u00b2 : HasBesicovitchCovering \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : Measure.OuterRegular \u03bc \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 f : \u03b1 \u2192 Set \u211d s : Set \u03b1 hf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4) u : Set \u03b1 su : u \u2287 s u_open : IsOpen u \u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 R : \u03b1 \u2192 \u211d hR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u t0 : Set \u03b1 r0 : \u03b1 \u2192 \u211d t0_count : Set.Countable t0 t0s : t0 \u2286 s hr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x) \u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 t0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x) s' : Set \u03b1 := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x) s's : s' \u2286 s N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 H : IsEmpty (SatelliteConfig \u03b1 N \u03c4) v : Set \u03b1 s'v : v \u2287 s' v_open : IsOpen v \u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N r1 : \u03b1 \u2192 \u211d hr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v q : BallPackage (\u2191s') \u03b1 :=   { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) } S : Fin N \u2192 Set \u2191s' S_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) hS : range q.c \u2286 \u22c3 i, \u22c3 j \u2208 S i, ball (BallPackage.c q j) (BallPackage.r q j) S_count : \u2200 (i : Fin N), Set.Countable (S i) r : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x r_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x A : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 B : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N \u22a2 \u2191\u2191\u03bc s + \u03b5 / 2 + \u03b5 / 2 = \u2191\u2191\u03bc s + \u03b5 ** rw [add_assoc, ENNReal.add_halves] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.drop_empty ** n : Nat \u22a2 drop \"\" n = \"\" ** rw [drop_eq, List.drop_nil] ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.sdiff_get_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : SDiff \u03b1 a b : Part \u03b1 hab : (a \\ b).Dom \u22a2 get (a \\ b) hab = get a (_ : a.Dom) \\ get b (_ : b.Dom) ** simp [sdiff_def] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : SDiff \u03b1 a b : Part \u03b1 hab : (a \\ b).Dom \u22a2 get (Part.bind a fun y => map (fun x => y \\ x) b) (_ : (Part.bind a fun y => map (fun x => y \\ x) b).Dom) =     get a (_ : a.Dom) \\ get b (_ : b.Dom) ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.mul_emod ** a b n : Int \u22a2 a * b % n = a % n * (b % n) % n ** conv => lhs; rw [\n  \u2190 emod_add_ediv a n, \u2190 emod_add_ediv' b n, Int.add_mul, Int.mul_add, Int.mul_add,\n  Int.mul_assoc, Int.mul_assoc, \u2190 Int.mul_add n _ _, add_mul_emod_self_left,\n  \u2190 Int.mul_assoc, add_mul_emod_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.Heap.realSize_tail? ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 \u22a2 tail? le s = some s' \u2192 realSize s = realSize s' + 1 ** simp only [Heap.tail?] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 \u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 realSize s = realSize s' + 1 ** intro eq ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 eq : Option.map (fun x => x.snd) (deleteMin le s) = some s' \u22a2 realSize s = realSize s' + 1 ** match eq\u2082 : s.deleteMin le, eq with\n| some (a, tl), rfl => exact realSize_deleteMin eq\u2082 ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 eq : Option.map (fun x => x.snd) (deleteMin le s) = some s' a : \u03b1 tl : Heap \u03b1 eq\u2082 : deleteMin le s = some (a, tl) \u22a2 realSize s = realSize ((fun x => x.snd) (a, tl)) + 1 ** exact realSize_deleteMin eq\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.div_get_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Div \u03b1 a b : Part \u03b1 hab : (a / b).Dom \u22a2 get (a / b) hab = get a (_ : a.Dom) / get b (_ : b.Dom) ** simp [div_def] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Div \u03b1 a b : Part \u03b1 hab : (a / b).Dom \u22a2 get (Part.bind a fun y => map (fun x => y / x) b) (_ : (Part.bind a fun y => map (fun x => y / x) b).Dom) =     get a (_ : a.Dom) / get b (_ : b.Dom) ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "Semiquot.pure_isPure ** \u03b1 : Type u_1 \u03b2 : Type u_2 a b : \u03b1 ab : b \u2208 pure a c : \u03b1 ac : c \u2208 pure a \u22a2 b = c ** rw [mem_pure] at ab ac ** \u03b1 : Type u_1 \u03b2 : Type u_2 a b : \u03b1 ab : b = a c : \u03b1 ac : c = a \u22a2 b = c ** rwa [\u2190ac] at ab ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.measurableSpace_iSup_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u_6 s t u : Set \u03b1 m : \u03b9 \u2192 MeasurableSpace \u03b1 \u22a2 \u2a06 n, m n = generateFrom {s | \u2203 n, MeasurableSet s} ** ext s ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u_6 s\u271d t u : Set \u03b1 m : \u03b9 \u2192 MeasurableSpace \u03b1 s : Set \u03b1 \u22a2 MeasurableSet s \u2194 MeasurableSet s ** rw [measurableSet_iSup] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u_6 s\u271d t u : Set \u03b1 m : \u03b9 \u2192 MeasurableSpace \u03b1 s : Set \u03b1 \u22a2 GenerateMeasurable {s | \u2203 i, MeasurableSet s} s \u2194 MeasurableSet s ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have f_nonneg : \u2200 \u03c9, 0 \u2264 f \u03c9 := fun \u03c9 \u21a6 f_nn \u03c9 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** by_cases H1 : g =\u1d50[volume.restrict (Ioi (0 : \u211d))] 0 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** by_cases H2 : \u2203 s > 0, 0 < \u222b t in (0)..s, g t \u2227 \u03bc {a : \u03b1 | s < f a} = \u221e ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u00ac\u2203 s, s > 0 \u2227 0 < \u222b (t : \u211d) in 0 ..s, g t \u2227 \u2191\u2191\u03bc {a | s < f a} = \u22a4 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** push_neg at H2 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have M_bdd : BddAbove {s : \u211d | g =\u1d50[volume.restrict (Ioc (0 : \u211d) s)] 0} := by\n  contrapose! H1\n  have : \u2200 (n : \u2115), g =\u1d50[volume.restrict (Ioc (0 : \u211d) n)] 0 := by\n    intro n\n    rcases not_bddAbove_iff.1 H1 n with \u27e8s, hs, ns\u27e9\n    exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right ns.le) hs\n  have Hg : g =\u1d50[volume.restrict (\u22c3 (n : \u2115), (Ioc (0 : \u211d) n))] 0 :=\n    (ae_restrict_iUnion_iff _ _).2 this\n  have : (\u22c3 (n : \u2115), (Ioc (0 : \u211d) n)) = Ioi 0 :=\n    iUnion_Ioc_eq_Ioi_self_iff.2 (fun x _ \u21a6 exists_nat_ge x)\n  rwa [this] at Hg ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** let M : \u211d := sSup {s : \u211d | g =\u1d50[volume.restrict (Ioc (0 : \u211d) s)] 0} ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have zero_mem : 0 \u2208 {s : \u211d | g =\u1d50[volume.restrict (Ioc (0 : \u211d) s)] 0} := by simpa using trivial ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have M_nonneg : 0 \u2264 M := le_csSup M_bdd zero_mem ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have hgM : g =\u1d50[volume.restrict (Ioc (0 : \u211d) M)] 0 := by\n  rw [\u2190 restrict_Ioo_eq_restrict_Ioc]\n  obtain \u27e8u, -, uM, ulim\u27e9 : \u2203 u, StrictMono u \u2227 (\u2200 (n : \u2115), u n < M) \u2227 Tendsto u atTop (\ud835\udcdd M) :=\n    exists_seq_strictMono_tendsto M\n  have I : \u2200 n, g =\u1d50[volume.restrict (Ioc (0 : \u211d) (u n))] 0 := by\n    intro n\n    obtain \u27e8s, hs, uns\u27e9 : \u2203 s, g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0 \u2227 u n < s :=\n      exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n)\n    exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right uns.le) hs\n  have : g =\u1d50[volume.restrict (\u22c3 n, Ioc (0 : \u211d) (u n))] 0 := (ae_restrict_iUnion_iff _ _).2 I\n  apply ae_restrict_of_ae_restrict_of_subset _ this\n  rintro x \u27e8x_pos, xM\u27e9\n  obtain \u27e8n, hn\u27e9 : \u2203 n, x < u n := ((tendsto_order.1 ulim).1 _ xM).exists\n  exact mem_iUnion.2 \u27e8n, \u27e8x_pos, hn.le\u27e9\u27e9 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** let \u03bd := \u03bc.restrict {a : \u03b1 | M < f a} ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have A : \u222b\u207b \u03c9, ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) \u2202\u03bc\n       = \u222b\u207b \u03c9, ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) \u2202\u03bd := by\n  have meas : MeasurableSet {a | M < f a} := measurableSet_lt measurable_const f_mble\n  have I : \u222b\u207b \u03c9 in {a | M < f a}\u1d9c, ENNReal.ofReal (\u222b t in (0).. f \u03c9, g t) \u2202\u03bc\n           = \u222b\u207b _ in {a | M < f a}\u1d9c, 0 \u2202\u03bc := by\n    apply set_lintegral_congr_fun meas.compl (eventually_of_forall (fun s hs \u21a6 ?_))\n    have : \u222b (t : \u211d) in (0)..f s, g t = \u222b (t : \u211d) in (0)..f s, 0 := by\n      simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]\n      apply integral_congr_ae\n      apply ae_mono (restrict_mono ?_ le_rfl) hgM\n      apply Ioc_subset_Ioc_right\n      simpa using hs\n    simp [this]\n  simp only [lintegral_const, zero_mul] at I\n  rw [\u2190 lintegral_add_compl _ meas, I, add_zero] ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have B : \u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n         = \u222b\u207b t in Ioi 0, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n  have B1 : \u222b\u207b t in Ioc 0 M, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n       = \u222b\u207b t in Ioc 0 M, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n    apply lintegral_congr_ae\n    filter_upwards [hgM] with t ht\n    simp [ht]\n  have B2 : \u222b\u207b t in Ioi M, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n            = \u222b\u207b t in Ioi M, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n    apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall (fun t ht \u21a6 ?_))\n    rw [Measure.restrict_apply (measurableSet_le measurable_const f_mble)]\n    congr 3\n    exact (inter_eq_left.2 (fun a ha \u21a6 (mem_Ioi.1 ht).trans_le ha)).symm\n  have I : Ioi (0 : \u211d) = Ioc (0 : \u211d) M \u222a Ioi M := (Ioc_union_Ioi_eq_Ioi M_nonneg).symm\n  have J : Disjoint (Ioc 0 M) (Ioi M) := Ioc_disjoint_Ioi le_rfl\n  rw [I, lintegral_union measurableSet_Ioi J, lintegral_union measurableSet_Ioi J, B1, B2] ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** rw [A, B] ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B :   \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** exact lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite\n  \u03bd f_nn f_mble g_intble g_mble g_nn ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have A : \u222b\u207b \u03c9, ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) \u2202\u03bc = 0 := by\n  have : \u2200 \u03c9, \u222b t in (0)..f \u03c9, g t = \u222b t in (0)..f \u03c9, 0 := by\n    intro \u03c9\n    simp_rw [intervalIntegral.integral_of_le (f_nonneg \u03c9)]\n    apply integral_congr_ae\n    exact ae_restrict_of_ae_restrict_of_subset Ioc_subset_Ioi_self H1\n  simp [this] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 A : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have B : \u222b\u207b t in Ioi 0, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) = 0 := by\n  have : (fun t \u21a6 \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t))\n    =\u1d50[volume.restrict (Ioi (0:\u211d))] 0 := by\n      filter_upwards [H1] with t ht using by simp [ht]\n  simp [lintegral_congr_ae this] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 A : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0 B : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = 0 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** rw [A, B] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0 ** have : \u2200 \u03c9, \u222b t in (0)..f \u03c9, g t = \u222b t in (0)..f \u03c9, 0 := by\n  intro \u03c9\n  simp_rw [intervalIntegral.integral_of_le (f_nonneg \u03c9)]\n  apply integral_congr_ae\n  exact ae_restrict_of_ae_restrict_of_subset Ioc_subset_Ioi_self H1 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 this : \u2200 (\u03c9 : \u03b1), \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, 0 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0 ** simp [this] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 \u22a2 \u2200 (\u03c9 : \u03b1), \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, 0 ** intro \u03c9 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 \u03c9 : \u03b1 \u22a2 \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (t : \u211d) in 0 ..f \u03c9, 0 ** simp_rw [intervalIntegral.integral_of_le (f_nonneg \u03c9)] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 \u03c9 : \u03b1 \u22a2 \u222b (t : \u211d) in Ioc 0 (f \u03c9), g t = \u222b (t : \u211d) in Ioc 0 (f \u03c9), 0 ** apply integral_congr_ae ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 \u03c9 : \u03b1 \u22a2 (fun a => g a) =\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] fun a => 0 ** exact ae_restrict_of_ae_restrict_of_subset Ioc_subset_Ioi_self H1 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 A : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0 \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = 0 ** have : (fun t \u21a6 \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t))\n  =\u1d50[volume.restrict (Ioi (0:\u211d))] 0 := by\n    filter_upwards [H1] with t ht using by simp [ht] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 A : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0 this : (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = 0 ** simp [lintegral_congr_ae this] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 A : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0 \u22a2 (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 ** filter_upwards [H1] with t ht using by simp [ht] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 A : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = 0 t : \u211d ht : g t = OfNat.ofNat 0 t \u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = OfNat.ofNat 0 t ** simp [ht] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2203 s, s > 0 \u2227 0 < \u222b (t : \u211d) in 0 ..s, g t \u2227 \u2191\u2191\u03bc {a | s < f a} = \u22a4 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** rcases H2 with \u27e8s, s_pos, hs, h's\u27e9 ** case pos.intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (t : \u211d) in 0 ..s, g t h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** rw [intervalIntegral.integral_of_le s_pos.le] at hs ** case pos.intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 B : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u22a4 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** rw [A, B] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 ** rw [eq_top_iff] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 I_pos : \u222b\u207b (a : \u211d) in Ioc 0 s, ENNReal.ofReal (g a) \u2260 0 \u22a2 \u22a4 = \u222b\u207b (t : \u211d) in Ioc 0 s, \u22a4 * ENNReal.ofReal (g t) ** rw [lintegral_const_mul, ENNReal.top_mul I_pos] ** case hf \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 I_pos : \u222b\u207b (a : \u211d) in Ioc 0 s, ENNReal.ofReal (g a) \u2260 0 \u22a2 Measurable fun t => ENNReal.ofReal (g t) ** exact ENNReal.measurable_ofReal.comp g_mble ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 \u22a2 \u222b\u207b (a : \u211d) in Ioc 0 s, ENNReal.ofReal (g a) \u2260 0 ** rw [\u2190 ofReal_integral_eq_lintegral_ofReal (g_intble s s_pos).1] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 \u22a2 ENNReal.ofReal (\u222b (x : \u211d) in Ioc 0 s, g x) \u2260 0 ** simpa only [not_lt, ge_iff_le, ne_eq, ENNReal.ofReal_eq_zero, not_le] using hs ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 \u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 s))] g ** filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht using g_nn _ ht.1 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 \u22a2 \u222b\u207b (t : \u211d) in Ioc 0 s, \u22a4 * ENNReal.ofReal (g t) \u2264 \u222b\u207b (t : \u211d) in Ioc 0 s, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** apply set_lintegral_mono' measurableSet_Ioc (fun x hx \u21a6 ?_) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 x : \u211d hx : x \u2208 Ioc 0 s \u22a2 \u22a4 * ENNReal.ofReal (g x) \u2264 \u2191\u2191\u03bc {a | x \u2264 f a} * ENNReal.ofReal (g x) ** rw [\u2190 h's] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 x : \u211d hx : x \u2208 Ioc 0 s \u22a2 \u2191\u2191\u03bc {a | s < f a} * ENNReal.ofReal (g x) \u2264 \u2191\u2191\u03bc {a | x \u2264 f a} * ENNReal.ofReal (g x) ** gcongr ** case bc \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 x : \u211d hx : x \u2208 Ioc 0 s \u22a2 \u2191\u2191\u03bc {a | s < f a} \u2264 \u2191\u2191\u03bc {a | x \u2264 f a} ** exact measure_mono (fun a ha \u21a6 hx.2.trans (le_of_lt ha)) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u22a4 ** rw [eq_top_iff] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 \u22a2 \u22a4 = \u222b\u207b (x : \u03b1) in {a | s < f a}, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2202\u03bc ** simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter,\n  h's, ne_eq, ENNReal.ofReal_eq_zero, not_le] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 \u22a2 \u22a4 = ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) * \u22a4 ** rw [ENNReal.mul_top] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 \u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2260 0 ** simpa [intervalIntegral.integral_of_le s_pos.le] using hs ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in {a | s < f a}, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2202\u03bc \u2264     \u222b\u207b (\u03c9 : \u03b1) in {a | s < f a}, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc ** apply set_lintegral_mono' (measurableSet_lt measurable_const f_mble) (fun a ha \u21a6 ?_) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 a : \u03b1 ha : a \u2208 {a | s < f a} \u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..s, g t) \u2264 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f a, g t) ** apply ENNReal.ofReal_le_ofReal ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 a : \u03b1 ha : a \u2208 {a | s < f a} \u22a2 \u222b (t : \u211d) in 0 ..s, g t \u2264 \u222b (t : \u211d) in 0 ..f a, g t ** apply intervalIntegral.integral_mono_interval le_rfl s_pos.le (le_of_lt ha) ** case h.hf \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 a : \u03b1 ha : a \u2208 {a | s < f a} \u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f a)))] fun x => g x ** filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht using g_nn _ ht.1 ** case h.hfi \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 s : \u211d s_pos : s > 0 hs : 0 < \u222b (x : \u211d) in Ioc 0 s, g x h's : \u2191\u2191\u03bc {a | s < f a} = \u22a4 A : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u22a4 a : \u03b1 ha : a \u2208 {a | s < f a} \u22a2 IntervalIntegrable (fun x => g x) volume 0 (f a) ** exact g_intble _ (s_pos.trans ha) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 \u22a2 BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} ** contrapose! H1 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 H1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 ** have : \u2200 (n : \u2115), g =\u1d50[volume.restrict (Ioc (0 : \u211d) n)] 0 := by\n  intro n\n  rcases not_bddAbove_iff.1 H1 n with \u27e8s, hs, ns\u27e9\n  exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right ns.le) hs ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 H1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} this : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0 \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 ** have Hg : g =\u1d50[volume.restrict (\u22c3 (n : \u2115), (Ioc (0 : \u211d) n))] 0 :=\n  (ae_restrict_iUnion_iff _ _).2 this ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 H1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} this : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0 Hg : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 \u2191n))] 0 \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 ** have : (\u22c3 (n : \u2115), (Ioc (0 : \u211d) n)) = Ioi 0 :=\n  iUnion_Ioc_eq_Ioi_self_iff.2 (fun x _ \u21a6 exists_nat_ge x) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 H1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} this\u271d : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0 Hg : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 \u2191n))] 0 this : \u22c3 n, Ioc 0 \u2191n = Ioi 0 \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 ** rwa [this] at Hg ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 H1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} \u22a2 \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0 ** intro n ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 H1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} n : \u2115 \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0 ** rcases not_bddAbove_iff.1 H1 n with \u27e8s, hs, ns\u27e9 ** case intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 H1 : \u00acBddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} n : \u2115 s : \u211d hs : s \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} ns : \u2191n < s \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 \u2191n))] 0 ** exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right ns.le) hs ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} \u22a2 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} ** simpa using trivial ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 ** rw [\u2190 restrict_Ioo_eq_restrict_Ioc] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0 ** obtain \u27e8u, -, uM, ulim\u27e9 : \u2203 u, StrictMono u \u2227 (\u2200 (n : \u2115), u n < M) \u2227 Tendsto u atTop (\ud835\udcdd M) :=\n  exists_seq_strictMono_tendsto M ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), u n < M ulim : Tendsto u atTop (\ud835\udcdd M) \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0 ** have I : \u2200 n, g =\u1d50[volume.restrict (Ioc (0 : \u211d) (u n))] 0 := by\n  intro n\n  obtain \u27e8s, hs, uns\u27e9 : \u2203 s, g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0 \u2227 u n < s :=\n    exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n)\n  exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right uns.le) hs ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), u n < M ulim : Tendsto u atTop (\ud835\udcdd M) I : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0 \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0 ** have : g =\u1d50[volume.restrict (\u22c3 n, Ioc (0 : \u211d) (u n))] 0 := (ae_restrict_iUnion_iff _ _).2 I ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), u n < M ulim : Tendsto u atTop (\ud835\udcdd M) I : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0 this : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0 \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioo 0 M))] 0 ** apply ae_restrict_of_ae_restrict_of_subset _ this ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), u n < M ulim : Tendsto u atTop (\ud835\udcdd M) I : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0 this : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0 \u22a2 Ioo 0 M \u2286 \u22c3 n, Ioc 0 (u n) ** rintro x \u27e8x_pos, xM\u27e9 ** case intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), u n < M ulim : Tendsto u atTop (\ud835\udcdd M) I : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0 this : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0 x : \u211d x_pos : 0 < x xM : x < M \u22a2 x \u2208 \u22c3 n, Ioc 0 (u n) ** obtain \u27e8n, hn\u27e9 : \u2203 n, x < u n := ((tendsto_order.1 ulim).1 _ xM).exists ** case intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), u n < M ulim : Tendsto u atTop (\ud835\udcdd M) I : \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0 this : g =\u1da0[ae (Measure.restrict volume (\u22c3 n, Ioc 0 (u n)))] 0 x : \u211d x_pos : 0 < x xM : x < M n : \u2115 hn : x < u n \u22a2 x \u2208 \u22c3 n, Ioc 0 (u n) ** exact mem_iUnion.2 \u27e8n, \u27e8x_pos, hn.le\u27e9\u27e9 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), u n < M ulim : Tendsto u atTop (\ud835\udcdd M) \u22a2 \u2200 (n : \u2115), g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0 ** intro n ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), u n < M ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0 ** obtain \u27e8s, hs, uns\u27e9 : \u2203 s, g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0 \u2227 u n < s :=\n  exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n) ** case intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), u n < M ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 s : \u211d hs : g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0 uns : u n < s \u22a2 g =\u1da0[ae (Measure.restrict volume (Ioc 0 (u n)))] 0 ** exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right uns.le) hs ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} \u22a2 SigmaFinite \u03bd ** obtain \u27e8u, -, uM, ulim\u27e9 : \u2203 u, StrictAnti u \u2227 (\u2200 (n : \u2115), M < u n) \u2227 Tendsto u atTop (\ud835\udcdd M) :=\n  exists_seq_strictAnti_tendsto M ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) s : FiniteSpanningSetsIn \u03bd univ :=   { set := fun n => {a | f a \u2264 M} \u222a {a | u n < f a}, set_mem := (_ : \u2115 \u2192 True),     finite := (_ : \u2200 (n : \u2115), \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4),     spanning := (_ : \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i = univ) } \u22a2 SigmaFinite \u03bd ** exact \u27e8\u27e8s\u27e9\u27e9 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) \u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i) < \u22a4 ** intro n ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 \u22a2 \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4 ** have I : \u03bd {a | f a \u2264 M} = 0 := by\n  rw [Measure.restrict_apply (measurableSet_le f_mble measurable_const)]\n  convert measure_empty\n  rw [\u2190 disjoint_iff_inter_eq_empty]\n  exact disjoint_left.mpr (fun a ha \u21a6 by simpa using ha) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 \u22a2 \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4 ** have J : \u03bc {a | u n < f a} < \u221e := by\n  rw [lt_top_iff_ne_top]\n  apply H2 _ (M_nonneg.trans_lt (uM n))\n  by_contra H3\n  rw [not_lt, intervalIntegral.integral_of_le (M_nonneg.trans (uM n).le)] at H3\n  have g_nn_ae : \u2200\u1d50 t \u2202(volume.restrict (Ioc 0 (u n))), 0 \u2264 g t := by\n    filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs using g_nn _ hs.1\n  have Ig : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0 :=\n    le_antisymm H3 (integral_nonneg_of_ae g_nn_ae)\n  have J : \u2200\u1d50 t \u2202(volume.restrict (Ioc 0 (u n))), g t = 0 :=\n    (integral_eq_zero_iff_of_nonneg_ae g_nn_ae\n      (g_intble (u n) (M_nonneg.trans_lt (uM n))).1).1 Ig\n  have : u n \u2264 M := le_csSup M_bdd J\n  exact lt_irrefl _ (this.trans_lt (uM n)) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 J : \u2191\u2191\u03bc {a | u n < f a} < \u22a4 \u22a2 \u2191\u2191\u03bd ((fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) n) < \u22a4 ** refine lt_of_le_of_lt (measure_union_le _ _) ?_ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 J : \u2191\u2191\u03bc {a | u n < f a} < \u22a4 \u22a2 \u2191\u2191\u03bd {a | f a \u2264 M} + \u2191\u2191\u03bd {a | u n < f a} < \u22a4 ** rw [I, zero_add] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 J : \u2191\u2191\u03bc {a | u n < f a} < \u22a4 \u22a2 \u2191\u2191\u03bd {a | u n < f a} < \u22a4 ** apply lt_of_le_of_lt _ J ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 J : \u2191\u2191\u03bc {a | u n < f a} < \u22a4 \u22a2 \u2191\u2191\u03bd {a | u n < f a} \u2264 \u2191\u2191\u03bc {a | u n < f a} ** exact restrict_le_self _ (measurableSet_lt measurable_const f_mble) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 \u22a2 \u2191\u2191\u03bd {a | f a \u2264 M} = 0 ** rw [Measure.restrict_apply (measurableSet_le f_mble measurable_const)] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 \u22a2 \u2191\u2191\u03bc ({a | f a \u2264 M} \u2229 {a | M < f a}) = 0 ** convert measure_empty ** case h.e'_2.h.e'_3 \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 \u22a2 {a | f a \u2264 M} \u2229 {a | M < f a} = \u2205 ** rw [\u2190 disjoint_iff_inter_eq_empty] ** case h.e'_2.h.e'_3 \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 \u22a2 Disjoint {a | f a \u2264 M} {a | M < f a} ** exact disjoint_left.mpr (fun a ha \u21a6 by simpa using ha) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 a : \u03b1 ha : a \u2208 {a | f a \u2264 M} \u22a2 \u00aca \u2208 {a | M < f a} ** simpa using ha ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 \u22a2 \u2191\u2191\u03bc {a | u n < f a} < \u22a4 ** rw [lt_top_iff_ne_top] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 \u22a2 \u2191\u2191\u03bc {a | u n < f a} \u2260 \u22a4 ** apply H2 _ (M_nonneg.trans_lt (uM n)) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 \u22a2 0 < \u222b (t : \u211d) in 0 ..u n, g t ** by_contra H3 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 H3 : \u00ac0 < \u222b (t : \u211d) in 0 ..u n, g t \u22a2 False ** rw [not_lt, intervalIntegral.integral_of_le (M_nonneg.trans (uM n).le)] at H3 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 H3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0 \u22a2 False ** have g_nn_ae : \u2200\u1d50 t \u2202(volume.restrict (Ioc 0 (u n))), 0 \u2264 g t := by\n  filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs using g_nn _ hs.1 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 H3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0 g_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t \u22a2 False ** have Ig : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0 :=\n  le_antisymm H3 (integral_nonneg_of_ae g_nn_ae) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 H3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0 g_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t Ig : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0 \u22a2 False ** have J : \u2200\u1d50 t \u2202(volume.restrict (Ioc 0 (u n))), g t = 0 :=\n  (integral_eq_zero_iff_of_nonneg_ae g_nn_ae\n    (g_intble (u n) (M_nonneg.trans_lt (uM n))).1).1 Ig ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 H3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0 g_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t Ig : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0 J : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), g t = 0 \u22a2 False ** have : u n \u2264 M := le_csSup M_bdd J ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 H3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0 g_nn_ae : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t Ig : \u222b (t : \u211d) in Ioc 0 (u n), g t = 0 J : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), g t = 0 this : u n \u2264 M \u22a2 False ** exact lt_irrefl _ (this.trans_lt (uM n)) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) n : \u2115 I : \u2191\u2191\u03bd {a | f a \u2264 M} = 0 H3 : \u222b (x : \u211d) in Ioc 0 (u n), g x \u2264 0 \u22a2 \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioc 0 (u n)), 0 \u2264 g t ** filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs using g_nn _ hs.1 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) \u22a2 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i = univ ** apply eq_univ_iff_forall.2 (fun a \u21a6 ?_) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) a : \u03b1 \u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i ** rcases le_or_lt (f a) M with ha|ha ** case inl \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) a : \u03b1 ha : f a \u2264 M \u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i ** exact mem_iUnion.2 \u27e80, Or.inl ha\u27e9 ** case inr \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) a : \u03b1 ha : M < f a \u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i ** obtain \u27e8n, hn\u27e9 : \u2203 n, u n < f a := ((tendsto_order.1 ulim).2 _ ha).exists ** case inr.intro \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} u : \u2115 \u2192 \u211d uM : \u2200 (n : \u2115), M < u n ulim : Tendsto u atTop (\ud835\udcdd M) a : \u03b1 ha : M < f a n : \u2115 hn : u n < f a \u22a2 a \u2208 \u22c3 i, (fun n => {a | f a \u2264 M} \u222a {a | u n < f a}) i ** exact mem_iUnion.2 \u27e8n, Or.inr hn\u27e9 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd ** have meas : MeasurableSet {a | M < f a} := measurableSet_lt measurable_const f_mble ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd ** have I : \u222b\u207b \u03c9 in {a | M < f a}\u1d9c, ENNReal.ofReal (\u222b t in (0).. f \u03c9, g t) \u2202\u03bc\n         = \u222b\u207b _ in {a | M < f a}\u1d9c, 0 \u2202\u03bc := by\n  apply set_lintegral_congr_fun meas.compl (eventually_of_forall (fun s hs \u21a6 ?_))\n  have : \u222b (t : \u211d) in (0)..f s, g t = \u222b (t : \u211d) in (0)..f s, 0 := by\n    simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]\n    apply integral_congr_ae\n    apply ae_mono (restrict_mono ?_ le_rfl) hgM\n    apply Ioc_subset_Ioc_right\n    simpa using hs\n  simp [this] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} I : \u222b\u207b (\u03c9 : \u03b1) in {a | M < f a}\u1d9c, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {a | M < f a}\u1d9c, 0 \u2202\u03bc \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd ** simp only [lintegral_const, zero_mul] at I ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} I :   \u222b\u207b (\u03c9 : \u03b1) in {a | sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} < f a}\u1d9c,       ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     0 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd ** rw [\u2190 lintegral_add_compl _ meas, I, add_zero] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} \u22a2 \u222b\u207b (\u03c9 : \u03b1) in {a | M < f a}\u1d9c, ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {a | M < f a}\u1d9c, 0 \u2202\u03bc ** apply set_lintegral_congr_fun meas.compl (eventually_of_forall (fun s hs \u21a6 ?_)) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} s : \u03b1 hs : s \u2208 {a | M < f a}\u1d9c \u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f s, g t) = 0 ** have : \u222b (t : \u211d) in (0)..f s, g t = \u222b (t : \u211d) in (0)..f s, 0 := by\n  simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]\n  apply integral_congr_ae\n  apply ae_mono (restrict_mono ?_ le_rfl) hgM\n  apply Ioc_subset_Ioc_right\n  simpa using hs ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this\u271d : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} s : \u03b1 hs : s \u2208 {a | M < f a}\u1d9c this : \u222b (t : \u211d) in 0 ..f s, g t = \u222b (t : \u211d) in 0 ..f s, 0 \u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f s, g t) = 0 ** simp [this] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} s : \u03b1 hs : s \u2208 {a | M < f a}\u1d9c \u22a2 \u222b (t : \u211d) in 0 ..f s, g t = \u222b (t : \u211d) in 0 ..f s, 0 ** simp_rw [intervalIntegral.integral_of_le (f_nonneg s)] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} s : \u03b1 hs : s \u2208 {a | M < f a}\u1d9c \u22a2 \u222b (t : \u211d) in Ioc 0 (f s), g t = \u222b (t : \u211d) in Ioc 0 (f s), 0 ** apply integral_congr_ae ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} s : \u03b1 hs : s \u2208 {a | M < f a}\u1d9c \u22a2 (fun a => g a) =\u1da0[ae (Measure.restrict volume (Ioc 0 (f s)))] fun a => 0 ** apply ae_mono (restrict_mono ?_ le_rfl) hgM ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} s : \u03b1 hs : s \u2208 {a | M < f a}\u1d9c \u22a2 Ioc 0 (f s) \u2286 Ioc 0 M ** apply Ioc_subset_Ioc_right ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd meas : MeasurableSet {a | M < f a} s : \u03b1 hs : s \u2208 {a | M < f a}\u1d9c \u22a2 f s \u2264 M ** simpa using hs ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have B1 : \u222b\u207b t in Ioc 0 M, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n     = \u222b\u207b t in Ioc 0 M, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n  apply lintegral_congr_ae\n  filter_upwards [hgM] with t ht\n  simp [ht] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B1 :   \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have B2 : \u222b\u207b t in Ioi M, \u03bc {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t)\n          = \u222b\u207b t in Ioi M, \u03bd {a : \u03b1 | t \u2264 f a} * ENNReal.ofReal (g t) := by\n  apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall (fun t ht \u21a6 ?_))\n  rw [Measure.restrict_apply (measurableSet_le measurable_const f_mble)]\n  congr 3\n  exact (inter_eq_left.2 (fun a ha \u21a6 (mem_Ioi.1 ht).trans_le ha)).symm ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B1 :   \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) B2 :   \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have I : Ioi (0 : \u211d) = Ioc (0 : \u211d) M \u222a Ioi M := (Ioc_union_Ioi_eq_Ioi M_nonneg).symm ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B1 :   \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) B2 :   \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) I : Ioi 0 = Ioc 0 M \u222a Ioi M \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have J : Disjoint (Ioc 0 M) (Ioi M) := Ioc_disjoint_Ioi le_rfl ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B1 :   \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) B2 :   \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) I : Ioi 0 = Ioc 0 M \u222a Ioi M J : Disjoint (Ioc 0 M) (Ioi M) \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** rw [I, lintegral_union measurableSet_Ioi J, lintegral_union measurableSet_Ioi J, B1, B2] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd \u22a2 \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** apply lintegral_congr_ae ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd \u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a)) =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] fun a =>     \u2191\u2191\u03bd {a_1 | a \u2264 f a_1} * ENNReal.ofReal (g a) ** filter_upwards [hgM] with t ht ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd t : \u211d ht : g t = OfNat.ofNat 0 t \u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** simp [ht] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B1 :   \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioi M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall (fun t ht \u21a6 ?_)) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B1 :   \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) t : \u211d ht : t \u2208 Ioi M \u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) ** rw [Measure.restrict_apply (measurableSet_le measurable_const f_mble)] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B1 :   \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) t : \u211d ht : t \u2208 Ioi M \u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) = \u2191\u2191\u03bc ({a | t \u2264 f a} \u2229 {a | M < f a}) * ENNReal.ofReal (g t) ** congr 3 ** case e_a.e_a \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t f_nonneg : \u2200 (\u03c9 : \u03b1), 0 \u2264 f \u03c9 H1 : \u00acg =\u1da0[ae (Measure.restrict volume (Ioi 0))] 0 H2 : \u2200 (s : \u211d), s > 0 \u2192 0 < \u222b (t : \u211d) in 0 ..s, g t \u2192 \u2191\u2191\u03bc {a | s < f a} \u2260 \u22a4 M_bdd : BddAbove {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M : \u211d := sSup {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} zero_mem : 0 \u2208 {s | g =\u1da0[ae (Measure.restrict volume (Ioc 0 s))] 0} M_nonneg : 0 \u2264 M hgM : g =\u1da0[ae (Measure.restrict volume (Ioc 0 M))] 0 \u03bd : Measure \u03b1 := Measure.restrict \u03bc {a | M < f a} this : SigmaFinite \u03bd A :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bd B1 :   \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) =     \u222b\u207b (t : \u211d) in Ioc 0 M, \u2191\u2191\u03bd {a | t \u2264 f a} * ENNReal.ofReal (g t) t : \u211d ht : t \u2208 Ioi M \u22a2 {a | t \u2264 f a} = {a | t \u2264 f a} \u2229 {a | M < f a} ** exact (inter_eq_left.2 (fun a ha \u21a6 (mem_Ioi.1 ht).trans_le ha)).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.mass_zero_iff ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u22a2 mass \u03bc = 0 \u2194 \u03bc = 0 ** refine' \u27e8fun \u03bc_mass => _, fun h\u03bc => by simp only [h\u03bc, zero_mass]\u27e9 ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bc_mass : mass \u03bc = 0 \u22a2 \u03bc = 0 ** apply toMeasure_injective ** case a \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bc_mass : mass \u03bc = 0 \u22a2 \u2191\u03bc = \u21910 ** apply Measure.measure_univ_eq_zero.mp ** case a \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 \u03bc_mass : mass \u03bc = 0 \u22a2 \u2191\u2191\u2191\u03bc univ = 0 ** rwa [\u2190 ennreal_mass, ENNReal.coe_eq_zero] ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 h\u03bc : \u03bc = 0 \u22a2 mass \u03bc = 0 ** simp only [h\u03bc, zero_mass] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_interval_sub_interval_comm ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hcd : IntervalIntegrable f \u03bc c d hac : IntervalIntegrable f \u03bc a c \u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc - \u222b (x : \u211d) in c..d, f x \u2202\u03bc = \u222b (x : \u211d) in a..c, f x \u2202\u03bc - \u222b (x : \u211d) in b..d, f x \u2202\u03bc ** simp only [sub_eq_add_neg, \u2190 integral_symm,\n  integral_interval_add_interval_comm hab hcd.symm (hac.trans hcd)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.ModEq.cancel_right_div_gcd ** m n a b c d : \u2124 hm : 0 < m h : a * c \u2261 b * c [ZMOD m] \u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)] ** letI d := gcd m c ** m n a b c d\u271d : \u2124 hm : 0 < m h : a * c \u2261 b * c [ZMOD m] d : \u2115 := gcd m c \u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)] ** have hmd := gcd_dvd_left m c ** m n a b c d\u271d : \u2124 hm : 0 < m h : a * c \u2261 b * c [ZMOD m] d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m \u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)] ** have hcd := gcd_dvd_right m c ** m n a b c d\u271d : \u2124 hm : 0 < m h : a * c \u2261 b * c [ZMOD m] d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)] ** rw [modEq_iff_dvd] at h \u22a2 ** m n a b c d\u271d : \u2124 hm : 0 < m h : m \u2223 b * c - a * c d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 m / \u2191(gcd m c) \u2223 b - a ** refine Int.dvd_of_dvd_mul_right_of_gcd_one (?_ : m / d \u2223 c / d * (b - a)) ?_ ** case refine_1 m n a b c d\u271d : \u2124 hm : 0 < m h : m \u2223 b * c - a * c d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 m / \u2191d \u2223 c / \u2191d * (b - a) ** rw [mul_comm, \u2190 Int.mul_ediv_assoc (b - a) hcd, sub_mul] ** case refine_1 m n a b c d\u271d : \u2124 hm : 0 < m h : m \u2223 b * c - a * c d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 m / \u2191d \u2223 (b * c - a * c) / \u2191(gcd m c) ** exact Int.ediv_dvd_ediv hmd h ** case refine_2 m n a b c d\u271d : \u2124 hm : 0 < m h : m \u2223 b * c - a * c d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 gcd (m / \u2191(gcd m c)) (c / \u2191d) = 1 ** rw [gcd_div hmd hcd, natAbs_ofNat, Nat.div_self (gcd_pos_of_ne_zero_left c hm.ne')] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.sum_fintype ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Fintype \u03b9 \u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } \u22a2 kernel.sum \u03ba = \u2211 i : \u03b9, \u03ba i ** ext a s hs ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : Fintype \u03b9 \u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(kernel.sum \u03ba) a) s = \u2191\u2191(\u2191(\u2211 i : \u03b9, \u03ba i) a) s ** simp only [sum_apply' \u03ba a hs, finset_sum_apply' _ \u03ba a s, tsum_fintype] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.nsmul_univ ** F : Type u_1 \u03b1\u271d : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b9 : Monoid \u03b1\u271d s t : Set \u03b1\u271d a : \u03b1\u271d m n\u271d : \u2115 \u03b1 : Type u_5 inst\u271d : AddMonoid \u03b1 n : \u2115 x\u271d : n + 2 \u2260 0 \u22a2 (n + 2) \u2022 univ = univ ** rw [succ_nsmul, nsmul_univ n.succ_ne_zero, univ_add_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Pos.addChar_right_comm ** p : Pos c\u2081 c\u2082 : Char \u22a2 p + c\u2081 + c\u2082 = p + c\u2082 + c\u2081 ** apply ext ** case h p : Pos c\u2081 c\u2082 : Char \u22a2 (p + c\u2081 + c\u2082).byteIdx = (p + c\u2082 + c\u2081).byteIdx ** repeat rw [pos_add_char] ** case h p : Pos c\u2081 c\u2082 : Char \u22a2 p.byteIdx + csize c\u2081 + csize c\u2082 = p.byteIdx + csize c\u2082 + csize c\u2081 ** apply Nat.add_right_comm ** case h p : Pos c\u2081 c\u2082 : Char \u22a2 p.byteIdx + csize c\u2081 + csize c\u2082 = (p + c\u2082).byteIdx + csize c\u2081 ** rw [pos_add_char] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.tsum_prob_mem_Ioi_lt_top ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X \u22a2 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4 ** suffices : \u2200 K : \u2115, \u2211 j in range K, \u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi (j : \u211d)} \u2264 ENNReal.ofReal (\ud835\udd3c[X] + 1) ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X this : \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1) \u22a2 \u2211' (j : \u2115), \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} < \u22a4  case this \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X \u22a2 \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1) ** exact (le_of_tendsto_of_tendsto (ENNReal.tendsto_nat_tsum _) tendsto_const_nhds\n  (eventually_of_forall this)).trans_lt ENNReal.ofReal_lt_top ** case this \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X \u22a2 \u2200 (K : \u2115), \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1) ** intro K ** case this \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 A :   Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop     (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j})) \u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1) ** apply le_of_tendsto_of_tendsto A tendsto_const_nhds ** case this \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 A :   Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop     (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j})) \u22a2 (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) \u2264\u1da0[atTop] fun x => ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1) ** filter_upwards [Ici_mem_atTop K] with N hN ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 A :   Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop     (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j})) N : \u2115 hN : N \u2208 Set.Ici K \u22a2 \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N} \u2264 ENNReal.ofReal ((\u222b (a : \u03a9), X a) + 1) ** exact sum_prob_mem_Ioc_le hint hnonneg hN ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K : \u2115 \u22a2 Tendsto (fun N => \u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191j \u2191N}) atTop     (\ud835\udcdd (\u2211 j in range K, \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191j})) ** refine' tendsto_finset_sum _ fun i _ => _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K i : \u2115 x\u271d : i \u2208 range K this : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} \u22a2 Tendsto (fun N => \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) atTop (\ud835\udcdd (\u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i})) ** rw [this] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K i : \u2115 x\u271d : i \u2208 range K this : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} \u22a2 Tendsto (fun N => \u2191\u2191\u2119 {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) atTop (\ud835\udcdd (\u2191\u2191\u2119 (\u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}))) ** apply tendsto_measure_iUnion ** case hm \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K i : \u2115 x\u271d : i \u2208 range K this : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} \u22a2 Monotone fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} ** intro m n hmn x hx ** case hm \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K i : \u2115 x\u271d : i \u2208 range K this : {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} m n : \u2115 hmn : m \u2264 n x : \u03a9 hx : x \u2208 (fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) m \u22a2 x \u2208 (fun N => {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N}) n ** exact \u27e8hx.1, hx.2.trans (Nat.cast_le.2 hmn)\u27e9 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K i : \u2115 x\u271d : i \u2208 range K \u22a2 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} = \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} ** apply Set.Subset.antisymm _ _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K i : \u2115 x\u271d : i \u2208 range K \u22a2 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} \u2286 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} ** intro \u03c9 h\u03c9 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K i : \u2115 x\u271d : i \u2208 range K \u03c9 : \u03a9 h\u03c9 : \u03c9 \u2208 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} \u22a2 \u03c9 \u2208 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} ** obtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, X \u03c9 \u2264 N := exists_nat_ge (X \u03c9) ** case intro \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K i : \u2115 x\u271d : i \u2208 range K \u03c9 : \u03a9 h\u03c9 : \u03c9 \u2208 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} N : \u2115 hN : X \u03c9 \u2264 \u2191N \u22a2 \u03c9 \u2208 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} ** exact Set.mem_iUnion.2 \u27e8N, h\u03c9, hN\u27e9 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hint : Integrable X hnonneg : 0 \u2264 X K i : \u2115 x\u271d : i \u2208 range K \u22a2 \u22c3 N, {\u03c9 | X \u03c9 \u2208 Set.Ioc \u2191i \u2191N} \u2286 {\u03c9 | X \u03c9 \u2208 Set.Ioi \u2191i} ** simp (config := {contextual := true}) only [Set.mem_Ioc, Set.mem_Ioi,\n  Set.iUnion_subset_iff, Set.setOf_subset_setOf, imp_true_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_mem\u2112p ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** have hp_pos_real : 0 < p.toReal := ENNReal.toReal_pos hp_pos hp_ne_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** have hf_snorm := Mem\u2112p.snorm_lt_top hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : snorm (\u2191f) p \u03bc < \u22a4 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** rw [snorm_eq_snorm' hp_pos hp_ne_top, f.snorm'_eq, \u2190\n  @ENNReal.lt_rpow_one_div_iff _ _ (1 / p.toReal) (by simp [hp_pos_real]),\n  @ENNReal.top_rpow_of_pos (1 / (1 / p.toReal)) (by simp [hp_pos_real]),\n  ENNReal.sum_lt_top_iff] at hf_snorm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** by_cases hyf : y \u2208 f.range ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : y \u2208 SimpleFunc.range f \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : \u00acy \u2208 SimpleFunc.range f \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : y \u2208 SimpleFunc.range f \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** specialize hf_snorm y hyf ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hyf : y \u2208 SimpleFunc.range f hf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** rw [ENNReal.mul_lt_top_iff] at hf_snorm ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hyf : y \u2208 SimpleFunc.range f hf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p < \u22a4 \u2227 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 \u2228 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** cases hf_snorm with\n| inl hf_snorm => exact hf_snorm.2\n| inr hf_snorm =>\n  cases hf_snorm with\n  | inl hf_snorm =>\n    refine' absurd _ hy_ne\n    simpa [hp_pos_real] using hf_snorm\n  | inr hf_snorm => simp [hf_snorm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : (\u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y})) ^ (1 / ENNReal.toReal p) < \u22a4 \u22a2 0 < 1 / ENNReal.toReal p ** simp [hp_pos_real] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ^ (1 / (1 / ENNReal.toReal p)) \u22a2 0 < 1 / (1 / ENNReal.toReal p) ** simp [hp_pos_real] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : \u00acy \u2208 SimpleFunc.range f \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** suffices h_empty : f \u207b\u00b9' {y} = \u2205 ** case h_empty \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : \u00acy \u2208 SimpleFunc.range f \u22a2 \u2191f \u207b\u00b9' {y} = \u2205 ** ext1 x ** case h_empty.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : \u00acy \u2208 SimpleFunc.range f x : \u03b1 \u22a2 x \u2208 \u2191f \u207b\u00b9' {y} \u2194 x \u2208 \u2205 ** rw [Set.mem_preimage, Set.mem_singleton_iff, mem_empty_iff_false, iff_false_iff] ** case h_empty.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : \u00acy \u2208 SimpleFunc.range f x : \u03b1 \u22a2 \u00ac\u2191f x = y ** refine' fun hxy => hyf _ ** case h_empty.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : \u00acy \u2208 SimpleFunc.range f x : \u03b1 hxy : \u2191f x = y \u22a2 y \u2208 SimpleFunc.range f ** rw [mem_range, Set.mem_range] ** case h_empty.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : \u00acy \u2208 SimpleFunc.range f x : \u03b1 hxy : \u2191f x = y \u22a2 \u2203 y_1, \u2191f y_1 = y ** exact \u27e8x, hxy\u27e9 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : \u00acy \u2208 SimpleFunc.range f h_empty : \u2191f \u207b\u00b9' {y} = \u2205 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** rw [h_empty, measure_empty] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4 hyf : \u00acy \u2208 SimpleFunc.range f h_empty : \u2191f \u207b\u00b9' {y} = \u2205 \u22a2 0 < \u22a4 ** exact ENNReal.coe_lt_top ** case pos.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hyf : y \u2208 SimpleFunc.range f hf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p < \u22a4 \u2227 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** exact hf_snorm.2 ** case pos.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hyf : y \u2208 SimpleFunc.range f hf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** cases hf_snorm with\n| inl hf_snorm =>\n  refine' absurd _ hy_ne\n  simpa [hp_pos_real] using hf_snorm\n| inr hf_snorm => simp [hf_snorm] ** case pos.inr.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hyf : y \u2208 SimpleFunc.range f hf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** refine' absurd _ hy_ne ** case pos.inr.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hyf : y \u2208 SimpleFunc.range f hf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u22a2 y = 0 ** simpa [hp_pos_real] using hf_snorm ** case pos.inr.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 f : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p y : E hy_ne : y \u2260 0 hp_pos_real : 0 < ENNReal.toReal p hyf : y \u2208 SimpleFunc.range f hf_snorm : \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** simp [hf_snorm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FinStronglyMeasurable.inf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : SemilatticeInf \u03b2 inst\u271d : ContinuousInf \u03b2 hf : FinStronglyMeasurable f \u03bc hg : FinStronglyMeasurable g \u03bc \u22a2 FinStronglyMeasurable (f \u2293 g) \u03bc ** refine'\n  \u27e8fun n => hf.approx n \u2293 hg.approx n, fun n => _, fun x =>\n    (hf.tendsto_approx x).inf_right_nhds (hg.tendsto_approx x)\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : SemilatticeInf \u03b2 inst\u271d : ContinuousInf \u03b2 hf : FinStronglyMeasurable f \u03bc hg : FinStronglyMeasurable g \u03bc n : \u2115 \u22a2 \u2191\u2191\u03bc (support \u2191((fun n => FinStronglyMeasurable.approx hf n \u2293 FinStronglyMeasurable.approx hg n) n)) < \u22a4 ** refine' (measure_mono (support_inf _ _)).trans_lt _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : SemilatticeInf \u03b2 inst\u271d : ContinuousInf \u03b2 hf : FinStronglyMeasurable f \u03bc hg : FinStronglyMeasurable g \u03bc n : \u2115 \u22a2 \u2191\u2191\u03bc       ((support fun x => \u2191(FinStronglyMeasurable.approx hf n) x) \u222a         support fun x => \u2191(FinStronglyMeasurable.approx hg n) x) <     \u22a4 ** exact measure_union_lt_top_iff.mpr \u27e8hf.fin_support_approx n, hg.fin_support_approx n\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.finset_sup_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 K : Type u_5 inst\u271d\u00b9 : SemilatticeSup \u03b2 inst\u271d : OrderBot \u03b2 f : \u03b3 \u2192 \u03b1 \u2192\u209b \u03b2 s : Finset \u03b3 a : \u03b1 \u22a2 \u2191(Finset.sup s f) a = Finset.sup s fun c => \u2191(f c) a ** refine' Finset.induction_on s rfl _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 K : Type u_5 inst\u271d\u00b9 : SemilatticeSup \u03b2 inst\u271d : OrderBot \u03b2 f : \u03b3 \u2192 \u03b1 \u2192\u209b \u03b2 s : Finset \u03b3 a : \u03b1 \u22a2 \u2200 \u2983a_1 : \u03b3\u2984 {s : Finset \u03b3},     \u00aca_1 \u2208 s \u2192       (\u2191(Finset.sup s f) a = Finset.sup s fun c => \u2191(f c) a) \u2192         \u2191(Finset.sup (insert a_1 s) f) a = Finset.sup (insert a_1 s) fun c => \u2191(f c) a ** intro a s _ ih ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 K : Type u_5 inst\u271d\u00b9 : SemilatticeSup \u03b2 inst\u271d : OrderBot \u03b2 f : \u03b3 \u2192 \u03b1 \u2192\u209b \u03b2 s\u271d : Finset \u03b3 a\u271d\u00b9 : \u03b1 a : \u03b3 s : Finset \u03b3 a\u271d : \u00aca \u2208 s ih : \u2191(Finset.sup s f) a\u271d\u00b9 = Finset.sup s fun c => \u2191(f c) a\u271d\u00b9 \u22a2 \u2191(Finset.sup (insert a s) f) a\u271d\u00b9 = Finset.sup (insert a s) fun c => \u2191(f c) a\u271d\u00b9 ** rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih] ** Qed",
        "informal": ""
    },
    {
        "formal": "StieltjesFunction.measure_Icc ** f : StieltjesFunction a b : \u211d \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a) ** rcases le_or_lt a b with (hab | hab) ** case inl f : StieltjesFunction a b : \u211d hab : a \u2264 b \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a) ** have A : Disjoint {a} (Ioc a b) := by simp ** case inl f : StieltjesFunction a b : \u211d hab : a \u2264 b A : Disjoint {a} (Ioc a b) \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a) ** simp [\u2190 Icc_union_Ioc_eq_Icc le_rfl hab, -singleton_union, \u2190 ENNReal.ofReal_add,\n  f.mono.leftLim_le, measure_union A measurableSet_Ioc, f.mono hab] ** f : StieltjesFunction a b : \u211d hab : a \u2264 b \u22a2 Disjoint {a} (Ioc a b) ** simp ** case inr f : StieltjesFunction a b : \u211d hab : b < a \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Icc a b) = ofReal (\u2191f b - leftLim (\u2191f) a) ** simp only [hab, measure_empty, Icc_eq_empty, not_le] ** case inr f : StieltjesFunction a b : \u211d hab : b < a \u22a2 0 = ofReal (\u2191f b - leftLim (\u2191f) a) ** symm ** case inr f : StieltjesFunction a b : \u211d hab : b < a \u22a2 ofReal (\u2191f b - leftLim (\u2191f) a) = 0 ** simp [ENNReal.ofReal_eq_zero, f.mono.le_leftLim hab] ** Qed",
        "informal": ""
    },
    {
        "formal": "aemeasurable_uIoc_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 inst\u271d\u00b9 : MeasurableSpace \u03b4 f\u271d g : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 inst\u271d : LinearOrder \u03b1 f : \u03b1 \u2192 \u03b2 a b : \u03b1 \u22a2 AEMeasurable f \u2194 AEMeasurable f \u2227 AEMeasurable f ** rw [uIoc_eq_union, aemeasurable_union_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.prod_eq_iff_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 a : \u03b1 b : \u03b2 ht : Set.Nonempty t \u22a2 s \u00d7\u02e2 t = s\u2081 \u00d7\u02e2 t \u2194 s = s\u2081 ** simp_rw [prod_eq_prod_iff, ht.ne_empty, and_true_iff, or_iff_left_iff_imp,\n  or_false_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 a : \u03b1 b : \u03b2 ht : Set.Nonempty t \u22a2 s = \u2205 \u2227 s\u2081 = \u2205 \u2192 s = s\u2081 ** rintro \u27e8rfl, rfl\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 a : \u03b1 b : \u03b2 ht : Set.Nonempty t \u22a2 \u2205 = \u2205 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.pred_succ ** \u03b1 : Type u_1 p : PosNum \u22a2 toZNumNeg (PosNum.pred' (succ' (pos p))) = ZNum.neg p ** rw [PosNum.pred'_succ'] ** \u03b1 : Type u_1 p : PosNum \u22a2 toZNumNeg (pos p) = ZNum.neg p ** rfl ** \u03b1 : Type u_1 p : PosNum \u22a2 ZNum.succ (ZNum.pred (ZNum.pos p)) = ZNum.pos p ** rw [ZNum.pred, \u2190 toZNum_succ, Num.succ, PosNum.succ'_pred', toZNum] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.pair_preimage_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192\u209b \u03b2 g : \u03b1 \u2192\u209b \u03b3 b : \u03b2 c : \u03b3 \u22a2 \u2191(pair f g) \u207b\u00b9' {(b, c)} = \u2191f \u207b\u00b9' {b} \u2229 \u2191g \u207b\u00b9' {c} ** rw [\u2190 singleton_prod_singleton] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192\u209b \u03b2 g : \u03b1 \u2192\u209b \u03b3 b : \u03b2 c : \u03b3 \u22a2 \u2191(pair f g) \u207b\u00b9' {b} \u00d7\u02e2 {c} = \u2191f \u207b\u00b9' {b} \u2229 \u2191g \u207b\u00b9' {c} ** exact pair_preimage _ _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_bdd_condexp_of_ae_bdd ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R ** by_cases hnm : m \u2264 m0 ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R  case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : \u00acm \u2264 m0 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R ** swap ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R ** by_cases hfint : Integrable f \u03bc ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R  case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : \u00acIntegrable f \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R ** swap ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R ** by_contra h ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : \u00ac\u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R \u22a2 False ** change \u03bc _ \u2260 0 at h ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : \u2191\u2191\u03bc {x | (fun x => |(\u03bc[f|m]) x| \u2264 \u2191R) x}\u1d9c \u2260 0 \u22a2 False ** simp only [\u2190 zero_lt_iff, Set.compl_def, Set.mem_setOf_eq, not_le] at h ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 False ** suffices (\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}).toReal * \u2191R < (\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}).toReal * \u2191R by\n  exact this.ne rfl ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R < ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R ** refine' lt_of_lt_of_le (set_integral_gt_gt R.coe_nonneg _ _ h.ne.symm) _ ** case pos.refine'_3 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R ** refine' (set_integral_abs_condexp_le _ _).trans _ ** case pos.refine'_3.refine'_2 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, |f x| \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R ** simp only [\u2190 smul_eq_mul, \u2190 set_integral_const, NNReal.val_eq_coe, IsROrC.ofReal_real_eq_id,\n  id.def] ** case pos.refine'_3.refine'_2 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, |f x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, \u2191R \u2202\u03bc ** refine' set_integral_mono_ae hfint.abs.integrableOn _ _ ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : \u00acm \u2264 m0 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R ** simp_rw [condexp_of_not_le hnm, Pi.zero_apply, abs_zero] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : \u00acm \u2264 m0 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, 0 \u2264 \u2191R ** refine' eventually_of_forall fun _ => R.coe_nonneg ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : \u00acIntegrable f \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |(\u03bc[f|m]) x| \u2264 \u2191R ** simp_rw [condexp_undef hfint] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : \u00acIntegrable f \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |OfNat.ofNat 0 x| \u2264 \u2191R ** filter_upwards [hbdd] with x hx ** case h \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : \u00acIntegrable f x : \u03b1 hx : |f x| \u2264 \u2191R \u22a2 |OfNat.ofNat 0 x| \u2264 \u2191R ** rw [Pi.zero_apply, abs_zero] ** case h \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : \u00acIntegrable f x : \u03b1 hx : |f x| \u2264 \u2191R \u22a2 0 \u2264 \u2191R ** exact (abs_nonneg _).trans hx ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} this : ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R < ENNReal.toReal (\u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|}) * \u2191R \u22a2 False ** exact this.ne rfl ** case pos.refine'_1 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 Measurable fun x => |(\u03bc[f|m]) x| ** simp_rw [\u2190 Real.norm_eq_abs] ** case pos.refine'_1 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 Measurable fun x => \u2016(\u03bc[f|m]) x\u2016 ** exact (stronglyMeasurable_condexp.mono hnm).measurable.norm ** case pos.refine'_2 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 IntegrableOn (fun x => |(\u03bc[f|m]) x|) {x | \u2191R < |(\u03bc[f|m]) x|} ** exact integrable_condexp.abs.integrableOn ** case pos.refine'_3.refine'_1 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 MeasurableSet {x | \u2191R < |(\u03bc[f|m]) x|} ** simp_rw [\u2190 Real.norm_eq_abs] ** case pos.refine'_3.refine'_1 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 MeasurableSet {x | \u2191R < \u2016(\u03bc[f|m]) x\u2016} ** exact @measurableSet_lt _ _ _ _ _ m _ _ _ _ _ measurable_const\n  stronglyMeasurable_condexp.norm.measurable ** case pos.refine'_3.refine'_2.refine'_1 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 IntegrableOn (fun x => \u2191R) {x | \u2191R < |(\u03bc[f|m]) x|} ** refine' \u27e8aestronglyMeasurable_const, lt_of_le_of_lt _\n  (integrable_condexp.integrableOn : IntegrableOn (\u03bc[f|m]) {x | \u2191R < |(\u03bc[f|m]) x|} \u03bc).2\u27e9 ** case pos.refine'_3.refine'_2.refine'_1 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 \u222b\u207b (a : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, \u2191\u2016(fun x => \u2191R) a\u2016\u208a \u2202\u03bc \u2264     \u222b\u207b (a : \u03b1) in {x | \u2191R < |(\u03bc[f|m]) x|}, \u2191\u2016(\u03bc[f|m]) a\u2016\u208a \u2202\u03bc ** refine' set_lintegral_mono (Measurable.nnnorm _).coe_nnreal_ennreal\n  (stronglyMeasurable_condexp.mono hnm).measurable.nnnorm.coe_nnreal_ennreal fun x hx => _ ** case pos.refine'_3.refine'_2.refine'_1.refine'_1 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 Measurable fun a => (fun x => \u2191R) a ** exact measurable_const ** case pos.refine'_3.refine'_2.refine'_1.refine'_2 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} x : \u03b1 hx : x \u2208 {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 \u2191\u2016(fun x => \u2191R) x\u2016\u208a \u2264 \u2191\u2016(\u03bc[f|m]) x\u2016\u208a ** rw [ENNReal.coe_le_coe, Real.nnnorm_of_nonneg R.coe_nonneg] ** case pos.refine'_3.refine'_2.refine'_1.refine'_2 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} x : \u03b1 hx : x \u2208 {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 { val := \u2191R, property := (_ : 0 \u2264 \u2191R) } \u2264 \u2016(\u03bc[f|m]) x\u2016\u208a ** exact Subtype.mk_le_mk.2 (le_of_lt hx) ** case pos.refine'_3.refine'_2.refine'_2 \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 R : \u211d\u22650 f : \u03b1 \u2192 \u211d hbdd : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, |f x| \u2264 \u2191R hnm : m \u2264 m0 hfint : Integrable f h : 0 < \u2191\u2191\u03bc {x | \u2191R < |(\u03bc[f|m]) x|} \u22a2 (fun x => |f x|) \u2264\u1d50[\u03bc] fun x => \u2191R ** exact hbdd ** Qed",
        "informal": ""
    },
    {
        "formal": "TypeVec.splitFun_inj ** n : \u2115 \u03b1 : TypeVec.{u_1} (n + 1) \u03b1' : TypeVec.{u_2} (n + 1) f f' : drop \u03b1 \u27f9 drop \u03b1' g g' : last \u03b1 \u2192 last \u03b1' H : splitFun f g = splitFun f' g' \u22a2 f = f' \u2227 g = g' ** rw [\u2190 dropFun_splitFun f g, H, \u2190 lastFun_splitFun f g, H] ** n : \u2115 \u03b1 : TypeVec.{u_1} (n + 1) \u03b1' : TypeVec.{u_2} (n + 1) f f' : drop \u03b1 \u27f9 drop \u03b1' g g' : last \u03b1 \u2192 last \u03b1' H : splitFun f g = splitFun f' g' \u22a2 dropFun (splitFun f' g') = f' \u2227 lastFun (splitFun f' g') = g' ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ite_inter ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s\u271d s\u2081\u271d s\u2082\u271d t\u271d t\u2081 t\u2082 u t s\u2081 s\u2082 s : Set \u03b1 \u22a2 Set.ite t (s\u2081 \u2229 s) (s\u2082 \u2229 s) = Set.ite t s\u2081 s\u2082 \u2229 s ** rw [ite_inter_inter, ite_same] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.of_nat_toZNumNeg ** \u03b1 : Type u_1 n : \u2115 \u22a2 Num.toZNumNeg \u2191n = -\u2191n ** rw [\u2190 of_nat_toZNum, Num.zneg_toZNum] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero' ** \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc \u22a2 \u2191\u2191f =\u1d50[\u03bc] 0 ** let f_meas : lpMeas E' \ud835\udd5c m p \u03bc := \u27e8f, hf_meas\u27e9 ** \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } \u22a2 \u2191\u2191f =\u1d50[\u03bc] 0 ** have hf_f_meas : f =\u1d50[\u03bc] f_meas := by simp only [Subtype.coe_mk]; rfl ** \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas \u22a2 \u2191\u2191f =\u1d50[\u03bc] 0 ** refine' hf_f_meas.trans _ ** \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas \u22a2 \u2191\u2191\u2191f_meas =\u1d50[\u03bc] 0 ** refine' lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero hm f_meas hp_ne_zero hp_ne_top _ _ ** \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } \u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas ** simp only [Subtype.coe_mk] ** \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } \u22a2 \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191f ** rfl ** case refine'_1 \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191\u2191f_meas) s ** intro s hs h\u03bcs ** case refine'_1 \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 IntegrableOn (\u2191\u2191\u2191f_meas) s ** have hfg_restrict : f =\u1d50[\u03bc.restrict s] f_meas := ae_restrict_of_ae hf_f_meas ** case refine'_1 \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 hfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas \u22a2 IntegrableOn (\u2191\u2191\u2191f_meas) s ** rw [IntegrableOn, integrable_congr hfg_restrict.symm] ** case refine'_1 \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 hfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas \u22a2 Integrable \u2191\u2191f ** exact hf_int_finite s hs h\u03bcs ** case refine'_2 \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f_meas x \u2202\u03bc = 0 ** intro s hs h\u03bcs ** case refine'_2 \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f_meas x \u2202\u03bc = 0 ** have hfg_restrict : f =\u1d50[\u03bc.restrict s] f_meas := ae_restrict_of_ae hf_f_meas ** case refine'_2 \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 hfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191f_meas x \u2202\u03bc = 0 ** rw [integral_congr_ae hfg_restrict.symm] ** case refine'_2 \u03b1 : Type u_1 E' : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : IsROrC \ud835\udd5c inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2075 : CompleteSpace E' inst\u271d\u2074 : NormedSpace \u211d E' inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' hm : m \u2264 m0 f : { x // x \u2208 Lp E' p } hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191f) s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc = 0 hf_meas : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc f_meas : { x // x \u2208 lpMeas E' \ud835\udd5c m p \u03bc } := { val := f, property := hf_meas } hf_f_meas : \u2191\u2191f =\u1d50[\u03bc] \u2191\u2191\u2191f_meas s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 hfg_restrict : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191f_meas \u22a2 \u222b (a : \u03b1) in s, \u2191\u2191f a \u2202\u03bc = 0 ** exact hf_zero s hs h\u03bcs ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.lintegral_Lp_add_le ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** have hp_pos : 0 < p := lt_of_lt_of_le zero_lt_one hp1 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** by_cases hf_top : \u222b\u207b a, f a ^ p \u2202\u03bc = \u22a4 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** by_cases hg_top : \u222b\u207b a, g a ^ p \u2202\u03bc = \u22a4 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** by_cases h1 : p = 1 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** have hp1_lt : 1 < p := by\n  refine' lt_of_le_of_ne hp1 _\n  symm\n  exact h1 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** have hpq := Real.isConjugateExponent_conjugateExponent hp1_lt ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p hpq : Real.IsConjugateExponent p (Real.conjugateExponent p) \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** by_cases h0 : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) = 0 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p hpq : Real.IsConjugateExponent p (Real.conjugateExponent p) h0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** have htop : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4 := by\n  rw [\u2190 Ne.def] at hf_top hg_top\n  rw [\u2190 lt_top_iff_ne_top] at hf_top hg_top \u22a2\n  exact lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top hf hf_top hg_top hp1 ** case neg \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p hpq : Real.IsConjugateExponent p (Real.conjugateExponent p) h0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0 htop : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** exact lintegral_Lp_add_le_aux hpq hf hf_top hg hg_top h0 htop ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** simp [hf_top, hp_pos] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** simp [hg_top, hp_pos] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : p = 1 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** refine' le_of_eq _ ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : p = 1 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** simp_rw [h1, one_div_one, ENNReal.rpow_one] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : p = 1 \u22a2 \u222b\u207b (a : \u03b1), (f + g) a \u2202\u03bc = \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc ** exact lintegral_add_left' hf _ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 \u22a2 1 < p ** refine' lt_of_le_of_ne hp1 _ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 \u22a2 1 \u2260 p ** symm ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 \u22a2 p \u2260 1 ** exact h1 ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p hpq : Real.IsConjugateExponent p (Real.conjugateExponent p) h0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** rw [h0, @ENNReal.zero_rpow_of_pos (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])] ** case pos \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p hpq : Real.IsConjugateExponent p (Real.conjugateExponent p) h0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0 \u22a2 0 \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** exact zero_le _ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p hpq : Real.IsConjugateExponent p (Real.conjugateExponent p) h0 : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0 \u22a2 0 < 1 / p ** simp [lt_of_lt_of_le zero_lt_one hp1] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4 hg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc = \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p hpq : Real.IsConjugateExponent p (Real.conjugateExponent p) h0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 ** rw [\u2190 Ne.def] at hf_top hg_top ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p hpq : Real.IsConjugateExponent p (Real.conjugateExponent p) h0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 ** rw [\u2190 lt_top_iff_ne_top] at hf_top hg_top \u22a2 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hp1 : 1 \u2264 p hp_pos : 0 < p hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc < \u22a4 hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc < \u22a4 h1 : \u00acp = 1 hp1_lt : 1 < p hpq : Real.IsConjugateExponent p (Real.conjugateExponent p) h0 : \u00ac\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc < \u22a4 ** exact lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top hf hf_top hg_top hp1 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f = \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc ** refine' le_antisymm (lintegral_withDensity_le_lintegral_mul \u03bc f_meas g) _ ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a \u2202withDensity \u03bc f ** rw [\u2190 iSup_lintegral_measurable_le_eq_lintegral, \u2190 iSup_lintegral_measurable_le_eq_lintegral] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => (f * g) a), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc \u2264     \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202withDensity \u03bc f ** refine' iSup\u2082_le fun i i_meas => iSup_le fun hi => _ ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a \u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202withDensity \u03bc f ** have A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g := by\n  intro x\n  dsimp\n  rw [mul_comm, \u2190 div_eq_mul_inv]\n  exact div_le_of_le_mul' (hi x) ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g \u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => g a), \u222b\u207b (a : \u03b1), g_1 a \u2202withDensity \u03bc f ** refine' le_iSup_of_le (fun x => (f x)\u207b\u00b9 * i x) (le_iSup_of_le (f_meas.inv.mul i_meas) _) ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g \u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264     \u2a06 (_ : (fun x => (f x)\u207b\u00b9 * i x) \u2264 fun a => g a), \u222b\u207b (a : \u03b1), (fun x => (f x)\u207b\u00b9 * i x) a \u2202withDensity \u03bc f ** refine' le_iSup_of_le A _ ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g \u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (fun x => (f x)\u207b\u00b9 * i x) a \u2202withDensity \u03bc f ** rw [lintegral_withDensity_eq_lintegral_mul _ f_meas (f_meas.inv.mul i_meas)] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g \u22a2 \u222b\u207b (a : \u03b1), i a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), (f * fun a => (f a)\u207b\u00b9 * i a) a \u2202\u03bc ** apply lintegral_mono_ae ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, i a \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) a ** filter_upwards [hf] ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g \u22a2 \u2200 (a : \u03b1), f a < \u22a4 \u2192 i a \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) a ** intro x h'x ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g x : \u03b1 h'x : f x < \u22a4 \u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x ** rcases eq_or_ne (f x) 0 with (hx | hx) ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a \u22a2 (fun x => (f x)\u207b\u00b9 * i x) \u2264 g ** intro x ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a x : \u03b1 \u22a2 (fun x => (f x)\u207b\u00b9 * i x) x \u2264 g x ** dsimp ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a x : \u03b1 \u22a2 (f x)\u207b\u00b9 * i x \u2264 g x ** rw [mul_comm, \u2190 div_eq_mul_inv] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a x : \u03b1 \u22a2 i x / f x \u2264 g x ** exact div_le_of_le_mul' (hi x) ** case h.inl \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g x : \u03b1 h'x : f x < \u22a4 hx : f x = 0 \u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x ** have := hi x ** case h.inl \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g x : \u03b1 h'x : f x < \u22a4 hx : f x = 0 this : i x \u2264 (fun a => (f * g) a) x \u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x ** simp only [hx, zero_mul, Pi.mul_apply, nonpos_iff_eq_zero] at this ** case h.inl \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g x : \u03b1 h'x : f x < \u22a4 hx : f x = 0 this : i x = 0 \u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x ** simp [this] ** case h.inr \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g x : \u03b1 h'x : f x < \u22a4 hx : f x \u2260 0 \u22a2 i x \u2264 (f * fun a => (f a)\u207b\u00b9 * i a) x ** apply le_of_eq _ ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g x : \u03b1 h'x : f x < \u22a4 hx : f x \u2260 0 \u22a2 i x = (f * fun a => (f a)\u207b\u00b9 * i a) x ** dsimp ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc\u271d \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e f_meas : Measurable f hf : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < \u22a4 g i : \u03b1 \u2192 \u211d\u22650\u221e i_meas : Measurable i hi : i \u2264 fun a => (f * g) a A : (fun x => (f x)\u207b\u00b9 * i x) \u2264 g x : \u03b1 h'x : f x < \u22a4 hx : f x \u2260 0 \u22a2 i x = f x * ((f x)\u207b\u00b9 * i x) ** rw [\u2190 mul_assoc, ENNReal.mul_inv_cancel hx h'x.ne, one_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.volume_val ** \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 s : Set \u211d \u22a2 \u2191\u2191volume s = \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) s ** simp [volume_eq_stieltjes_id] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace E f : \u03b2 \u2192 E hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 fmeas : Measurable f inst\u271d : SeparableSpace \u2191(Set.range f \u222a {0}) hf : Mem\u2112p f p n : \u2115 \u22a2 0 \u2208 Set.range f \u222a {0} ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : BorelSpace E f : \u03b2 \u2192 E hp : Fact (1 \u2264 p) hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 fmeas : Measurable f inst\u271d : SeparableSpace \u2191(Set.range f \u222a {0}) hf : Mem\u2112p f p \u22a2 Tendsto     (fun n =>       Mem\u2112p.toLp \u2191(approxOn f fmeas (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) n)         (_ : Mem\u2112p (\u2191(approxOn f fmeas (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) n)) p))     atTop (\ud835\udcdd (Mem\u2112p.toLp f hf)) ** simpa only [Lp.tendsto_Lp_iff_tendsto_\u2112p''] using\n  tendsto_approxOn_range_Lp_snorm hp_ne_top fmeas hf.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.hasPDF_iff ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d \u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 map X \u2119 \u226a volume ** by_cases hX : Measurable X ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Measurable X \u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 map X \u2119 \u226a volume ** rw [Real.hasPDF_iff_of_measurable hX, iff_and_self] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d hX : Measurable X \u22a2 map X \u2119 \u226a volume \u2192 Measurable X ** exact fun _ => hX ** case neg \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc : Measure E inst\u271d : IsFiniteMeasure \u2119 X : \u03a9 \u2192 \u211d hX : \u00acMeasurable X \u22a2 HasPDF X \u2119 \u2194 Measurable X \u2227 map X \u2119 \u226a volume ** exact \u27e8fun h => False.elim (hX h.pdf'.1), fun h => False.elim (hX h.1)\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.continuousOn_primitive_Icc ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E inst\u271d : NoAtoms \u03bc h_int : IntegrableOn f (Icc a b) \u22a2 ContinuousOn (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) (Icc a b) ** have aux : (fun x => \u222b t in Icc a x, f t \u2202\u03bc) = fun x => \u222b t in Ioc a x, f t \u2202\u03bc := by\n  ext x\n  exact integral_Icc_eq_integral_Ioc ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E inst\u271d : NoAtoms \u03bc h_int : IntegrableOn f (Icc a b) aux : (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) = fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc \u22a2 ContinuousOn (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) (Icc a b) ** rw [aux] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E inst\u271d : NoAtoms \u03bc h_int : IntegrableOn f (Icc a b) aux : (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) = fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc \u22a2 ContinuousOn (fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc) (Icc a b) ** exact continuousOn_primitive h_int ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E inst\u271d : NoAtoms \u03bc h_int : IntegrableOn f (Icc a b) \u22a2 (fun x => \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc) = fun x => \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc ** ext x ** case h \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E a b b\u2080 b\u2081 b\u2082 : \u211d \u03bc : Measure \u211d f g : \u211d \u2192 E inst\u271d : NoAtoms \u03bc h_int : IntegrableOn f (Icc a b) x : \u211d \u22a2 \u222b (t : \u211d) in Icc a x, f t \u2202\u03bc = \u222b (t : \u211d) in Ioc a x, f t \u2202\u03bc ** exact integral_Icc_eq_integral_Ioc ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.measure_condCdf_univ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 \u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ = 1 ** rw [\u2190 ENNReal.ofReal_one, \u2190 sub_zero (1 : \u211d)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 \u22a2 \u2191\u2191(StieltjesFunction.measure (condCdf \u03c1 a)) univ = ENNReal.ofReal (1 - 0) ** exact StieltjesFunction.measure_univ _ (tendsto_condCdf_atBot \u03c1 a) (tendsto_condCdf_atTop \u03c1 a) ** Qed",
        "informal": ""
    },
    {
        "formal": "ManyOneDegree.add_le ** \u03b1 : Type u inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Inhabited \u03b1 \u03b2 : Type v inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Inhabited \u03b2 \u03b3 : Type w inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Inhabited \u03b3 d\u2081 d\u2082 d\u2083 : ManyOneDegree \u22a2 d\u2081 + d\u2082 \u2264 d\u2083 \u2194 d\u2081 \u2264 d\u2083 \u2227 d\u2082 \u2264 d\u2083 ** induction d\u2081 using ManyOneDegree.ind_on ** case h \u03b1 : Type u inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Inhabited \u03b1 \u03b2 : Type v inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Inhabited \u03b2 \u03b3 : Type w inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Inhabited \u03b3 d\u2082 d\u2083 : ManyOneDegree p\u271d : Set \u2115 \u22a2 of p\u271d + d\u2082 \u2264 d\u2083 \u2194 of p\u271d \u2264 d\u2083 \u2227 d\u2082 \u2264 d\u2083 ** induction d\u2082 using ManyOneDegree.ind_on ** case h.h \u03b1 : Type u inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Inhabited \u03b1 \u03b2 : Type v inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Inhabited \u03b2 \u03b3 : Type w inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Inhabited \u03b3 d\u2083 : ManyOneDegree p\u271d\u00b9 p\u271d : Set \u2115 \u22a2 of p\u271d\u00b9 + of p\u271d \u2264 d\u2083 \u2194 of p\u271d\u00b9 \u2264 d\u2083 \u2227 of p\u271d \u2264 d\u2083 ** induction d\u2083 using ManyOneDegree.ind_on ** case h.h.h \u03b1 : Type u inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Inhabited \u03b1 \u03b2 : Type v inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Inhabited \u03b2 \u03b3 : Type w inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Inhabited \u03b3 p\u271d\u00b2 p\u271d\u00b9 p\u271d : Set \u2115 \u22a2 of p\u271d\u00b2 + of p\u271d\u00b9 \u2264 of p\u271d \u2194 of p\u271d\u00b2 \u2264 of p\u271d \u2227 of p\u271d\u00b9 \u2264 of p\u271d ** simpa only [\u2190 add_of, of_le_of] using disjoin_le ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.noncommProd_commute ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b) y : \u03b2 h : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x) \u22a2 Commute y (noncommProd s f comm) ** apply Multiset.noncommProd_commute ** case h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b) y : \u03b2 h : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x) \u22a2 \u2200 (x : \u03b2), x \u2208 Multiset.map f s.val \u2192 Commute y x ** intro y ** case h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b) y\u271d : \u03b2 h : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y\u271d (f x) y : \u03b2 \u22a2 y \u2208 Multiset.map f s.val \u2192 Commute y\u271d y ** rw [Multiset.mem_map] ** case h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b) y\u271d : \u03b2 h : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y\u271d (f x) y : \u03b2 \u22a2 (\u2203 a, a \u2208 s.val \u2227 f a = y) \u2192 Commute y\u271d y ** rintro \u27e8x, \u27e8hx, rfl\u27e9\u27e9 ** case h.intro.intro F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : Monoid \u03b3 s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191s fun a b => Commute (f a) (f b) y : \u03b2 h : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y (f x) x : \u03b1 hx : x \u2208 s.val \u22a2 Commute y (f x) ** exact h x hx ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurablySeparable_range_of_disjoint ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) \u22a2 MeasurablySeparable (range f) (range g) ** by_contra hfg ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) \u22a2 False ** have I : \u2200 n x y, \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n    \u2203 x' y', x' \u2208 cylinder x n \u2227 y' \u2208 cylinder y n \u2227\n    \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) := by\n  intro n x y\n  contrapose!\n  intro H\n  rw [\u2190 iUnion_cylinder_update x n, \u2190 iUnion_cylinder_update y n, image_iUnion, image_iUnion]\n  refine' MeasurablySeparable.iUnion fun i j => _\n  exact H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) \u22a2 False ** let A :=\n  { p : \u2115 \u00d7 (\u2115 \u2192 \u2115) \u00d7 (\u2115 \u2192 \u2115) //\n    \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } \u22a2 False ** have : \u2200 p : A, \u2203 q : A,\n    q.1.1 = p.1.1 + 1 \u2227 q.1.2.1 \u2208 cylinder p.1.2.1 p.1.1 \u2227 q.1.2.2 \u2208 cylinder p.1.2.2 p.1.1 := by\n  rintro \u27e8\u27e8n, x, y\u27e9, hp\u27e9\n  rcases I n x y hp with \u27e8x', y', hx', hy', h'\u27e9\n  exact \u27e8\u27e8\u27e8n + 1, x', y'\u27e9, h'\u27e9, rfl, hx', hy'\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } this : \u2200 (p : A), \u2203 q, (\u2191q).1 = (\u2191p).1 + 1 \u2227 (\u2191q).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 \u2227 (\u2191q).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 \u22a2 False ** choose F hFn hFx hFy using this ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 \u22a2 False ** let p0 : A := \u27e8\u27e80, fun _ => 0, fun _ => 0\u27e9, by simp [hfg]\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } \u22a2 False ** let p : \u2115 \u2192 A := fun n => F^[n] p0 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 \u22a2 False ** have prec : \u2200 n, p (n + 1) = F (p n) := fun n => by simp only [iterate_succ', Function.comp] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m \u22a2 False ** set x : \u2115 \u2192 \u2115 := fun n => (p (n + 1)).1.2.1 n with hx ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n \u22a2 False ** set y : \u2115 \u2192 \u2115 := fun n => (p (n + 1)).1.2.2 n with hy ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u22a2 False ** obtain \u27e8u, v, u_open, v_open, xu, yv, huv\u27e9 :\n  \u2203 u v : Set \u03b1, IsOpen u \u2227 IsOpen v \u2227 f x \u2208 u \u2227 g y \u2208 v \u2227 Disjoint u v := by\n  apply t2_separation\n  exact disjoint_iff_forall_ne.1 h (mem_range_self _) (mem_range_self _) ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v \u22a2 False ** letI : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u22a2 False ** obtain \u27e8\u03b5x, \u03b5xpos, h\u03b5x\u27e9 : \u2203 (\u03b5x : \u211d), \u03b5x > 0 \u2227 Metric.ball x \u03b5x \u2286 f \u207b\u00b9' u := by\n  apply Metric.mem_nhds_iff.1\n  exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu) ** case intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u22a2 False ** obtain \u27e8\u03b5y, \u03b5ypos, h\u03b5y\u27e9 : \u2203 (\u03b5y : \u211d), \u03b5y > 0 \u2227 Metric.ball y \u03b5y \u2286 g \u207b\u00b9' v := by\n  apply Metric.mem_nhds_iff.1\n  exact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv) ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v \u22a2 False ** obtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, (1 / 2 : \u211d) ^ n < min \u03b5x \u03b5y :=\n  exists_pow_lt_of_lt_one (lt_min \u03b5xpos \u03b5ypos) (by norm_num) ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y B : MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u22a2 False ** exact M n B ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) \u22a2 \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) ** intro n x y ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) n : \u2115 x y : \u2115 \u2192 \u2115 \u22a2 \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192     \u2203 x' y',       x' \u2208 cylinder x n \u2227 y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) ** contrapose! ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) n : \u2115 x y : \u2115 \u2192 \u2115 \u22a2 (\u2200 (x' y' : \u2115 \u2192 \u2115),       x' \u2208 cylinder x n \u2192         y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))) \u2192     MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) ** intro H ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) n : \u2115 x y : \u2115 \u2192 \u2115 H :   \u2200 (x' y' : \u2115 \u2192 \u2115),     x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) \u22a2 MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) ** rw [\u2190 iUnion_cylinder_update x n, \u2190 iUnion_cylinder_update y n, image_iUnion, image_iUnion] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) n : \u2115 x y : \u2115 \u2192 \u2115 H :   \u2200 (x' y' : \u2115 \u2192 \u2115),     x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) \u22a2 MeasurablySeparable (\u22c3 i, f '' cylinder (update x n i) (n + 1)) (\u22c3 i, g '' cylinder (update y n i) (n + 1)) ** refine' MeasurablySeparable.iUnion fun i j => _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) n : \u2115 x y : \u2115 \u2192 \u2115 H :   \u2200 (x' y' : \u2115 \u2192 \u2115),     x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) i j : \u2115 \u22a2 MeasurablySeparable (f '' cylinder (update x n i) (n + 1)) (g '' cylinder (update y n j) (n + 1)) ** exact H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } \u22a2 \u2200 (p : A), \u2203 q, (\u2191q).1 = (\u2191p).1 + 1 \u2227 (\u2191q).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 \u2227 (\u2191q).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 ** rintro \u27e8\u27e8n, x, y\u27e9, hp\u27e9 ** case mk.mk.mk \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } n : \u2115 x y : \u2115 \u2192 \u2115 hp : \u00acMeasurablySeparable (f '' cylinder (n, x, y).2.1 (n, x, y).1) (g '' cylinder (n, x, y).2.2 (n, x, y).1) \u22a2 \u2203 q,     (\u2191q).1 = (\u2191{ val := (n, x, y), property := hp }).1 + 1 \u2227       (\u2191q).2.1 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.1 (\u2191{ val := (n, x, y), property := hp }).1 \u2227         (\u2191q).2.2 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.2 (\u2191{ val := (n, x, y), property := hp }).1 ** rcases I n x y hp with \u27e8x', y', hx', hy', h'\u27e9 ** case mk.mk.mk.intro.intro.intro.intro \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } n : \u2115 x y : \u2115 \u2192 \u2115 hp : \u00acMeasurablySeparable (f '' cylinder (n, x, y).2.1 (n, x, y).1) (g '' cylinder (n, x, y).2.2 (n, x, y).1) x' y' : \u2115 \u2192 \u2115 hx' : x' \u2208 cylinder x n hy' : y' \u2208 cylinder y n h' : \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) \u22a2 \u2203 q,     (\u2191q).1 = (\u2191{ val := (n, x, y), property := hp }).1 + 1 \u2227       (\u2191q).2.1 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.1 (\u2191{ val := (n, x, y), property := hp }).1 \u2227         (\u2191q).2.2 \u2208 cylinder (\u2191{ val := (n, x, y), property := hp }).2.2 (\u2191{ val := (n, x, y), property := hp }).1 ** exact \u27e8\u27e8\u27e8n + 1, x', y'\u27e9, h'\u27e9, rfl, hx', hy'\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 \u22a2 \u00acMeasurablySeparable (f '' cylinder (0, fun x => 0, fun x => 0).2.1 (0, fun x => 0, fun x => 0).1)       (g '' cylinder (0, fun x => 0, fun x => 0).2.2 (0, fun x => 0, fun x => 0).1) ** simp [hfg] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 n : \u2115 \u22a2 p (n + 1) = F (p n) ** simp only [iterate_succ', Function.comp] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) \u22a2 \u2200 (n : \u2115), (\u2191(p n)).1 = n ** intro n ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) n : \u2115 \u22a2 (\u2191(p n)).1 = n ** induction' n with n IH ** case zero \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) \u22a2 (\u2191(p Nat.zero)).1 = Nat.zero ** rfl ** case succ \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) n : \u2115 IH : (\u2191(p n)).1 = n \u22a2 (\u2191(p (Nat.succ n))).1 = Nat.succ n ** simp only [prec, hFn, IH] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n \u22a2 \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m ** intro m ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n m : \u2115 \u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m ** apply Nat.le_induction ** case succ \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n m : \u2115 \u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m \u2192 (\u2191(p (n + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m ** intro n hmn IH ** case succ \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m \u22a2 (\u2191(p (n + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m ** have I : (F (p n)).val.snd.fst m = (p n).val.snd.fst m := by\n  apply hFx (p n) m\n  rw [pn_fst]\n  exact hmn ** case succ \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I\u271d :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m I : (\u2191(F (p n))).2.1 m = (\u2191(p n)).2.1 m \u22a2 (\u2191(p (n + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m ** rw [prec, I, IH] ** case base \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n m : \u2115 \u22a2 (\u2191(p (m + 1))).2.1 m = (\u2191(p (m + 1))).2.1 m ** rfl ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m \u22a2 (\u2191(F (p n))).2.1 m = (\u2191(p n)).2.1 m ** apply hFx (p n) m ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m \u22a2 m < (\u2191(p n)).1 ** rw [pn_fst] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m \u22a2 m < n ** exact hmn ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m \u22a2 \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m ** intro m ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m m : \u2115 \u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m ** apply Nat.le_induction ** case succ \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m m : \u2115 \u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m \u2192 (\u2191(p (n + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m ** intro n hmn IH ** case succ \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m \u22a2 (\u2191(p (n + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m ** have I : (F (p n)).val.snd.snd m = (p n).val.snd.snd m := by\n  apply hFy (p n) m\n  rw [pn_fst]\n  exact hmn ** case succ \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I\u271d :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m I : (\u2191(F (p n))).2.2 m = (\u2191(p n)).2.2 m \u22a2 (\u2191(p (n + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m ** rw [prec, I, IH] ** case base \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m m : \u2115 \u22a2 (\u2191(p (m + 1))).2.2 m = (\u2191(p (m + 1))).2.2 m ** rfl ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m \u22a2 (\u2191(F (p n))).2.2 m = (\u2191(p n)).2.2 m ** apply hFy (p n) m ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m \u22a2 m < (\u2191(p n)).1 ** rw [pn_fst] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m m n : \u2115 hmn : m + 1 \u2264 n IH : (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m \u22a2 m < n ** exact hmn ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n \u22a2 \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) ** intro n ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n n : \u2115 \u22a2 \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) ** convert (p n).2 using 3 ** case h.e'_1.h.e'_3.h.e'_4 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n n : \u2115 \u22a2 cylinder x n = cylinder (\u2191(p n)).2.1 (\u2191(p n)).1 ** rw [pn_fst, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff] ** case h.e'_1.h.e'_3.h.e'_4 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n n : \u2115 \u22a2 \u2200 (i : \u2115), i < n \u2192 x i = (\u2191(p n)).2.1 i ** intro i hi ** case h.e'_1.h.e'_3.h.e'_4 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n n i : \u2115 hi : i < n \u22a2 x i = (\u2191(p n)).2.1 i ** rw [hx] ** case h.e'_1.h.e'_3.h.e'_4 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n n i : \u2115 hi : i < n \u22a2 (fun n => (\u2191(p (n + 1))).2.1 n) i = (\u2191(p n)).2.1 i ** exact (Ix i n hi).symm ** case h.e'_1.h.e'_4.h.e'_4 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n n : \u2115 \u22a2 cylinder y n = cylinder (\u2191(p n)).2.2 (\u2191(p n)).1 ** rw [pn_fst, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff] ** case h.e'_1.h.e'_4.h.e'_4 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n n : \u2115 \u22a2 \u2200 (i : \u2115), i < n \u2192 y i = (\u2191(p n)).2.2 i ** intro i hi ** case h.e'_1.h.e'_4.h.e'_4 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n n i : \u2115 hi : i < n \u22a2 y i = (\u2191(p n)).2.2 i ** rw [hy] ** case h.e'_1.h.e'_4.h.e'_4 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n n i : \u2115 hi : i < n \u22a2 (fun n => (\u2191(p (n + 1))).2.2 n) i = (\u2191(p n)).2.2 i ** exact (Iy i n hi).symm ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 f x \u2208 u \u2227 g y \u2208 v \u2227 Disjoint u v ** apply t2_separation ** case h \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u22a2 f x \u2260 g y ** exact disjoint_iff_forall_ne.1 h (mem_range_self _) (mem_range_self _) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u22a2 \u2203 \u03b5x, \u03b5x > 0 \u2227 ball x \u03b5x \u2286 f \u207b\u00b9' u ** apply Metric.mem_nhds_iff.1 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u22a2 f \u207b\u00b9' u \u2208 \ud835\udcdd x ** exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u22a2 \u2203 \u03b5y, \u03b5y > 0 \u2227 ball y \u03b5y \u2286 g \u207b\u00b9' v ** apply Metric.mem_nhds_iff.1 ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u22a2 g \u207b\u00b9' v \u2208 \ud835\udcdd y ** exact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv) ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v \u22a2 1 / 2 < 1 ** norm_num ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y \u22a2 MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) ** refine' \u27e8u, _, _, u_open.measurableSet\u27e9 ** case refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y \u22a2 f '' cylinder x n \u2286 u ** rw [image_subset_iff] ** case refine'_1 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y \u22a2 cylinder x n \u2286 f \u207b\u00b9' u ** apply Subset.trans _ h\u03b5x ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y \u22a2 cylinder x n \u2286 ball x \u03b5x ** intro z hz ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y z : \u2115 \u2192 \u2115 hz : z \u2208 cylinder x n \u22a2 z \u2208 ball x \u03b5x ** rw [mem_cylinder_iff_dist_le] at hz ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y z : \u2115 \u2192 \u2115 hz : dist z x \u2264 (1 / 2) ^ n \u22a2 z \u2208 ball x \u03b5x ** exact hz.trans_lt (hn.trans_le (min_le_left _ _)) ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y \u22a2 Disjoint (g '' cylinder y n) u ** refine' Disjoint.mono_left _ huv.symm ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y \u22a2 g '' cylinder y n \u2264 v ** change g '' cylinder y n \u2286 v ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y \u22a2 g '' cylinder y n \u2286 v ** rw [image_subset_iff] ** case refine'_2 \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y \u22a2 cylinder y n \u2286 g \u207b\u00b9' v ** apply Subset.trans _ h\u03b5y ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y \u22a2 cylinder y n \u2286 ball y \u03b5y ** intro z hz ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y z : \u2115 \u2192 \u2115 hz : z \u2208 cylinder y n \u22a2 z \u2208 ball y \u03b5y ** rw [mem_cylinder_iff_dist_le] at hz ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : T2Space \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : OpensMeasurableSpace \u03b1 f g : (\u2115 \u2192 \u2115) \u2192 \u03b1 hf : Continuous f hg : Continuous g h : Disjoint (range f) (range g) hfg : \u00acMeasurablySeparable (range f) (range g) I :   \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),     \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192       \u2203 x' y',         x' \u2208 cylinder x n \u2227           y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) A : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) } F : A \u2192 A hFn : \u2200 (p : A), (\u2191(F p)).1 = (\u2191p).1 + 1 hFx : \u2200 (p : A), (\u2191(F p)).2.1 \u2208 cylinder (\u2191p).2.1 (\u2191p).1 hFy : \u2200 (p : A), (\u2191(F p)).2.2 \u2208 cylinder (\u2191p).2.2 (\u2191p).1 p0 : A :=   { val := (0, fun x => 0, fun x => 0),     property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) } p : \u2115 \u2192 A := fun n => F^[n] p0 prec : \u2200 (n : \u2115), p (n + 1) = F (p n) pn_fst : \u2200 (n : \u2115), (\u2191(p n)).1 = n Ix : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.1 m = (\u2191(p (m + 1))).2.1 m Iy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 (\u2191(p n)).2.2 m = (\u2191(p (m + 1))).2.2 m x : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.1 n hx : x = fun n => (\u2191(p (n + 1))).2.1 n y : \u2115 \u2192 \u2115 := fun n => (\u2191(p (n + 1))).2.2 n hy : y = fun n => (\u2191(p (n + 1))).2.2 n M : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) u v : Set \u03b1 u_open : IsOpen u v_open : IsOpen v xu : f x \u2208 u yv : g y \u2208 v huv : Disjoint u v this : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat \u03b5x : \u211d \u03b5xpos : \u03b5x > 0 h\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u \u03b5y : \u211d \u03b5ypos : \u03b5y > 0 h\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v n : \u2115 hn : (1 / 2) ^ n < min \u03b5x \u03b5y z : \u2115 \u2192 \u2115 hz : dist z y \u2264 (1 / 2) ^ n \u22a2 z \u2208 ball y \u03b5y ** exact hz.trans_lt (hn.trans_le (min_le_right _ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) \u22a2 TendstoInMeasure \u03bc f l g ** intro \u03b5 h\u03b5 ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) l (\ud835\udcdd 0) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0) ** replace hfg := ENNReal.Tendsto.const_mul\n  (Tendsto.ennrpow_const p.toReal hfg) (Or.inr <| @ENNReal.ofReal_ne_top (1 / \u03b5 ^ p.toReal)) ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   Tendsto (fun b => ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f b - g) p \u03bc ^ ENNReal.toReal p) l     (\ud835\udcdd (ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * 0 ^ ENNReal.toReal p)) \u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0) ** simp only [mul_zero,\n  ENNReal.zero_rpow_of_pos (ENNReal.toReal_pos hp_ne_zero hp_ne_top)] at hfg ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg : Tendsto (fun b => ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f b - g) p \u03bc ^ ENNReal.toReal p) l (\ud835\udcdd 0) \u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0) ** rw [ENNReal.tendsto_nhds_zero] at hfg \u22a2 ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u22a2 \u2200 (\u03b5_1 : \u211d\u22650\u221e), \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b5 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5_1 ** intro \u03b4 h\u03b4 ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u03b4 : \u211d\u22650\u221e h\u03b4 : \u03b4 > 0 \u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b5 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b4 ** refine' (hfg \u03b4 h\u03b4).mono fun n hn => _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u03b4 : \u211d\u22650\u221e h\u03b4 : \u03b4 > 0 n : \u03b9 hn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4 \u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 \u03b4 ** refine' le_trans _ hn ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u03b4 : \u211d\u22650\u221e h\u03b4 : \u03b4 > 0 n : \u03b9 hn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4 \u22a2 \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p ** rw [ENNReal.ofReal_div_of_pos (Real.rpow_pos_of_pos h\u03b5 _), ENNReal.ofReal_one, mul_comm,\n  mul_one_div, ENNReal.le_div_iff_mul_le _ (Or.inl ENNReal.ofReal_ne_top), mul_comm] ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u03b4 : \u211d\u22650\u221e h\u03b4 : \u03b4 > 0 n : \u03b9 hn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4 \u22a2 ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) * \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 snorm (f n - g) p \u03bc ^ ENNReal.toReal p ** rw [\u2190 ENNReal.ofReal_rpow_of_pos h\u03b5] ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u03b4 : \u211d\u22650\u221e h\u03b4 : \u03b4 > 0 n : \u03b9 hn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4 \u22a2 ENNReal.ofReal \u03b5 ^ ENNReal.toReal p * \u2191\u2191\u03bc {x | \u03b5 \u2264 dist (f n x) (g x)} \u2264 snorm (f n - g) p \u03bc ^ ENNReal.toReal p ** convert mul_meas_ge_le_pow_snorm' \u03bc hp_ne_zero hp_ne_top ((hf n).sub hg).aestronglyMeasurable\n    (ENNReal.ofReal \u03b5) ** case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u03b4 : \u211d\u22650\u221e h\u03b4 : \u03b4 > 0 n : \u03b9 hn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4 x\u271d : \u03b1 \u22a2 \u03b5 \u2264 dist (f n x\u271d) (g x\u271d) \u2194 ENNReal.ofReal \u03b5 \u2264 \u2191\u2016(f n - g) x\u271d\u2016\u208a ** rw [dist_eq_norm, \u2190 ENNReal.ofReal_le_ofReal_iff (norm_nonneg _), ofReal_norm_eq_coe_nnnorm] ** case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u03b4 : \u211d\u22650\u221e h\u03b4 : \u03b4 > 0 n : \u03b9 hn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4 x\u271d : \u03b1 \u22a2 ENNReal.ofReal \u03b5 \u2264 \u2191\u2016f n x\u271d - g x\u271d\u2016\u208a \u2194 ENNReal.ofReal \u03b5 \u2264 \u2191\u2016(f n - g) x\u271d\u2016\u208a ** exact Iff.rfl ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u03b4 : \u211d\u22650\u221e h\u03b4 : \u03b4 > 0 n : \u03b9 hn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4 \u22a2 ENNReal.ofReal (\u03b5 ^ ENNReal.toReal p) \u2260 0 \u2228 snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2260 0 ** rw [Ne, ENNReal.ofReal_eq_zero, not_le] ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup E p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g l : Filter \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 hfg :   \u2200 (\u03b5_1 : \u211d\u22650\u221e),     \u03b5_1 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f x - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b5_1 \u03b4 : \u211d\u22650\u221e h\u03b4 : \u03b4 > 0 n : \u03b9 hn : ENNReal.ofReal (1 / \u03b5 ^ ENNReal.toReal p) * snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2264 \u03b4 \u22a2 0 < \u03b5 ^ ENNReal.toReal p \u2228 snorm (f n - g) p \u03bc ^ ENNReal.toReal p \u2260 0 ** exact Or.inl (Real.rpow_pos_of_pos h\u03b5 _) ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.castHom_bijective ** n : \u2115 R : Type u_1 inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : CharP R n inst\u271d : Fintype R h : Fintype.card R = n \u22a2 Bijective \u2191(castHom (_ : n \u2223 n) R) ** haveI : NeZero n :=\n  \u27e8by\n    intro hn\n    rw [hn] at h\n    exact (Fintype.card_eq_zero_iff.mp h).elim' 0\u27e9 ** n : \u2115 R : Type u_1 inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : CharP R n inst\u271d : Fintype R h : Fintype.card R = n this : NeZero n \u22a2 Bijective \u2191(castHom (_ : n \u2223 n) R) ** rw [Fintype.bijective_iff_injective_and_card, ZMod.card, h, eq_self_iff_true, and_true_iff] ** n : \u2115 R : Type u_1 inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : CharP R n inst\u271d : Fintype R h : Fintype.card R = n this : NeZero n \u22a2 Injective \u2191(castHom (_ : n \u2223 n) R) ** apply ZMod.castHom_injective ** n : \u2115 R : Type u_1 inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : CharP R n inst\u271d : Fintype R h : Fintype.card R = n \u22a2 n \u2260 0 ** intro hn ** n : \u2115 R : Type u_1 inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : CharP R n inst\u271d : Fintype R h : Fintype.card R = n hn : n = 0 \u22a2 False ** rw [hn] at h ** n : \u2115 R : Type u_1 inst\u271d\u00b2 : Ring R inst\u271d\u00b9 : CharP R n inst\u271d : Fintype R h : Fintype.card R = 0 hn : n = 0 \u22a2 False ** exact (Fintype.card_eq_zero_iff.mp h).elim' 0 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.mutuallySingular_singularPart ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 singularPart s \u03bc \u27c2\u1d65 toENNRealVectorMeasure \u03bc ** rw [mutuallySingular_ennreal_iff, singularPart_totalVariation] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc + Measure.singularPart (toJordanDecomposition s).negPart \u03bc \u27c2\u2098     VectorMeasure.ennrealToMeasure (toENNRealVectorMeasure \u03bc) ** change _ \u27c2\u2098 VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun \u03bc) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc + Measure.singularPart (toJordanDecomposition s).negPart \u03bc \u27c2\u2098     Equiv.toFun VectorMeasure.equivMeasure (Equiv.invFun VectorMeasure.equivMeasure \u03bc) ** rw [VectorMeasure.equivMeasure.right_inv \u03bc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc\u271d \u03bd : Measure \u03b1 s : SignedMeasure \u03b1 \u03bc : Measure \u03b1 \u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc + Measure.singularPart (toJordanDecomposition s).negPart \u03bc \u27c2\u2098     \u03bc ** exact (mutuallySingular_singularPart _ _).add_left (mutuallySingular_singularPart _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.totalDegree_X ** R\u271d : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R\u271d e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b2 : CommSemiring R\u271d p q : MvPolynomial \u03c3 R\u271d R : Type u_3 inst\u271d\u00b9 : CommSemiring R inst\u271d : Nontrivial R s : \u03c3 \u22a2 totalDegree (X s) = 1 ** rw [totalDegree, support_X] ** R\u271d : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R\u271d e : \u2115 n m : \u03c3 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b2 : CommSemiring R\u271d p q : MvPolynomial \u03c3 R\u271d R : Type u_3 inst\u271d\u00b9 : CommSemiring R inst\u271d : Nontrivial R s : \u03c3 \u22a2 (Finset.sup {fun\u2080 | s => 1} fun s => sum s fun x e => e) = 1 ** simp only [Finset.sup, Finsupp.sum_single_index, Finset.fold_singleton, sup_bot_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Icc_eq_singleton_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 \u22a2 Icc a b = {c} \u2194 a = c \u2227 b = c ** rw [\u2190 coe_eq_singleton, coe_Icc, Set.Icc_eq_singleton_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.setToL1_smul_left' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f : { x // x \u2208 Lp E 1 } \u22a2 \u2191(setToL1 hT') f = c \u2022 \u2191(setToL1 hT) f ** suffices setToL1 hT' = c \u2022 setToL1 hT by rw [this, ContinuousLinearMap.smul_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f : { x // x \u2208 Lp E 1 } \u22a2 setToL1 hT' = c \u2022 setToL1 hT ** refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc hT') _ _ _ _ _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f : { x // x \u2208 Lp E 1 } \u22a2 ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT' ** ext1 f ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f ** suffices c \u2022 setToL1 hT f = setToL1SCLM \u03b1 E \u03bc hT' f by rw [\u2190 this]; congr ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 c \u2022 \u2191(setToL1 hT) \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f ** rw [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_left' c hT hT' h_smul] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f : { x // x \u2208 Lp E 1 } this : setToL1 hT' = c \u2022 setToL1 hT \u22a2 \u2191(setToL1 hT') f = c \u2022 \u2191(setToL1 hT) f ** rw [this, ContinuousLinearMap.smul_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } this : c \u2022 \u2191(setToL1 hT) \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f \u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f ** rw [\u2190 this] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' c : \u211d h_smul : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T' s = c \u2022 T s f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } this : c \u2022 \u2191(setToL1 hT) \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT') f \u22a2 \u2191(ContinuousLinearMap.comp (c \u2022 setToL1 hT) (coeToLp \u03b1 E \u211d)) f = c \u2022 \u2191(setToL1 hT) \u2191f ** congr ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.exists_cond_kernel ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 \u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7 ** obtain \u27e8f, hf\u27e9 := exists_measurableEmbedding_real \u03a9 ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7 ** let \u03c1' : Measure (\u03b1 \u00d7 \u211d) := \u03c1.map (Prod.map id f) ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7 ** let \u03c1_set := (toMeasurable \u03c1.fst {a | condKernelReal \u03c1' a (range f) = 1}\u1d9c)\u1d9c ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c \u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7 ** have hm : MeasurableSet \u03c1_set := (measurableSet_toMeasurable _ _).compl ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 \u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7 ** have h_prod_embed : MeasurableEmbedding (Prod.map (id : \u03b1 \u2192 \u03b1) f) :=\n  MeasurableEmbedding.id.prod_mk hf ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) \u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7 ** have h_fst : \u03c1'.fst = \u03c1.fst := by\n  ext1 u hu\n  rw [Measure.fst_apply hu, Measure.fst_apply hu,\n    Measure.map_apply h_prod_embed.measurable (measurable_fst hu)]\n  rfl ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set \u22a2 \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 ** intro a ha ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set a : \u03b1 ha : a \u2208 \u03c1_set \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 ** rw [mem_compl_iff] at ha ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set a : \u03b1 ha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 ** have h_ss := subset_toMeasurable \u03c1.fst {a : \u03b1 | condKernelReal \u03c1' a (range f) = 1}\u1d9c ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set a : \u03b1 ha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c h_ss :   {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286     toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 ** suffices ha' : a \u2209 {a : \u03b1 | condKernelReal \u03c1' a (range f) = 1}\u1d9c ** case ha' \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set a : \u03b1 ha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c h_ss :   {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286     toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u22a2 \u00aca \u2208 {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c ** exact not_mem_subset h_ss ha ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set a : \u03b1 ha : \u00aca \u2208 toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c h_ss :   {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u2286     toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c ha' : \u00aca \u2208 {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 ** rwa [not_mem_compl_iff] at ha' ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) \u22a2 Measure.fst \u03c1' = Measure.fst \u03c1 ** ext1 u hu ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) u : Set \u03b1 hu : MeasurableSet u \u22a2 \u2191\u2191(Measure.fst \u03c1') u = \u2191\u2191(Measure.fst \u03c1) u ** rw [Measure.fst_apply hu, Measure.fst_apply hu,\n  Measure.map_apply h_prod_embed.measurable (measurable_fst hu)] ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) u : Set \u03b1 hu : MeasurableSet u \u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.fst \u207b\u00b9' u)) = \u2191\u2191\u03c1 (Prod.fst \u207b\u00b9' u) ** rfl ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set ** rw [ae_iff] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u00aca \u2208 \u03c1_set} = 0 ** simp only [not_mem_compl_iff, setOf_mem_eq, measure_toMeasurable] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal (Measure.map (Prod.map id f) \u03c1)) a) (range f) = 1}\u1d9c = 0 ** change \u03c1.fst {a : \u03b1 | a \u2209 {a' : \u03b1 | condKernelReal \u03c1' a' (range f) = 1}} = 0 ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2191\u2191(Measure.fst \u03c1) {a | \u00aca \u2208 {a' | \u2191\u2191(\u2191(condKernelReal \u03c1') a') (range f) = 1}} = 0 ** rw [\u2190 ae_iff, \u2190 h_fst] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1', a \u2208 {a' | \u2191\u2191(\u2191(condKernelReal \u03c1') a') (range f) = 1} ** refine' ae_condKernelReal_eq_one \u03c1' hf.measurableSet_range _ ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2191\u2191\u03c1' {x | x.2 \u2208 (range f)\u1d9c} = 0 ** rw [Measure.map_apply h_prod_embed.measurable] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c}) = 0  \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 MeasurableSet {x | x.2 \u2208 (range f)\u1d9c} ** swap ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c}) = 0 ** convert measure_empty (\u03b1 := \u03b1 \u00d7 \u03a9) ** case h.e'_2.h.e'_3 \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c} = \u2205 ** ext1 x ** case h.e'_2.h.e'_3.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 x : \u03b1 \u00d7 \u03a9 \u22a2 x \u2208 Prod.map id f \u207b\u00b9' {x | x.2 \u2208 (range f)\u1d9c} \u2194 x \u2208 \u2205 ** simp only [mem_compl_iff, mem_range, preimage_setOf_eq, Prod_map, mem_setOf_eq,\n  mem_empty_iff_false, iff_false_iff, Classical.not_not, exists_apply_eq_apply] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 \u22a2 MeasurableSet {x | x.2 \u2208 (range f)\u1d9c} ** exact measurable_snd hf.measurableSet_range.compl ** case intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set \u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7 ** obtain \u27e8x\u2080, hx\u2080\u27e9 : \u2203 x, x \u2208 range f := range_nonempty _ ** case intro.intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7 ** let \u03b7' :=\n  kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun _ => x\u2080) measurable_const) ** case intro.intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) \u22a2 \u2203 \u03b7 _h, kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 \u03b7 ** refine' \u27e8kernel.comapRight \u03b7' hf, _, _\u27e9 ** case intro.intro.refine'_2 \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) \u22a2 kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf) ** have : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed := by\n  ext c t ht : 2\n  rw [kernel.const_apply, kernel.comapRight_apply' _ _ _ ht, kernel.const_apply,\n    Measure.map_apply h_prod_embed.measurable (h_prod_embed.measurableSet_image.mpr ht)]\n  congr with x : 1\n  rw [\u2190 @Prod.mk.eta _ _ x]\n  simp only [id.def, mem_preimage, Prod.map_mk, mem_image, Prod.mk.inj_iff, Prod.exists]\n  refine' \u27e8fun h => \u27e8x.1, x.2, h, rfl, rfl\u27e9, _\u27e9\n  rintro \u27e8a, b, h_mem, rfl, hf_eq\u27e9\n  rwa [hf.injective hf_eq] at h_mem ** case intro.intro.refine'_2 \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed \u22a2 kernel.const \u03b3 \u03c1 = kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf) ** rw [this, kernel.const_eq_compProd_real _ \u03c1'] ** case intro.intro.refine'_2 \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed \u22a2 kernel.comapRight (kernel.const \u03b3 (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) h_prod_embed =     kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf) ** ext c t ht : 2 ** case intro.intro.refine'_2.h.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t \u22a2 \u2191\u2191(\u2191(kernel.comapRight (kernel.const \u03b3 (Measure.fst \u03c1') \u2297\u2096 kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) h_prod_embed) c)       t =     \u2191\u2191(\u2191(kernel.const \u03b3 (Measure.fst \u03c1) \u2297\u2096 kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) c) t ** rw [kernel.comapRight_apply' _ _ _ ht,\n  kernel.compProd_apply _ _ _ (h_prod_embed.measurableSet_image.mpr ht), kernel.const_apply,\n  h_fst, kernel.compProd_apply _ _ _ ht, kernel.const_apply] ** case intro.intro.refine'_2.h.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t \u22a2 \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, b)) {c | (b, c) \u2208 Prod.map id f '' t} \u2202Measure.fst \u03c1 =     \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, b)) {c | (b, c) \u2208 t} \u2202Measure.fst \u03c1 ** refine' lintegral_congr_ae _ ** case intro.intro.refine'_2.h.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t \u22a2 (fun b => \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, b)) {c | (b, c) \u2208 Prod.map id f '' t}) =\u1d50[Measure.fst \u03c1]     fun b => \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, b)) {c | (b, c) \u2208 t} ** filter_upwards [h_ae] with a ha ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set \u22a2 \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (condKernelReal \u03c1')) (c, a)) {c | (a, c) \u2208 Prod.map id f '' t} =     \u2191\u2191(\u2191(kernel.prodMkLeft \u03b3 (kernel.comapRight \u03b7' hf)) (c, a)) {c | (a, c) \u2208 t} ** rw [kernel.prodMkLeft_apply', kernel.prodMkLeft_apply', kernel.comapRight_apply'] ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') (c, a).2) {c | (a, c) \u2208 Prod.map id f '' t} = \u2191\u2191(\u2191\u03b7' (c, a).2) (f '' {c | (a, c) \u2208 t})  case h.ht \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set \u22a2 MeasurableSet {c | (a, c) \u2208 t} ** swap ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set h1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t} \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') (c, a).2) {c | (a, c) \u2208 Prod.map id f '' t} = \u2191\u2191(\u2191\u03b7' (c, a).2) (f '' {c | (a, c) \u2208 t}) ** have h2 : condKernelReal \u03c1' (c, a).snd = \u03b7' (c, a).snd := by\n  rw [kernel.piecewise_apply, if_pos ha] ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set h1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t} h2 : \u2191(condKernelReal \u03c1') (c, a).2 = \u2191\u03b7' (c, a).2 \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') (c, a).2) {c | (a, c) \u2208 Prod.map id f '' t} = \u2191\u2191(\u2191\u03b7' (c, a).2) (f '' {c | (a, c) \u2208 t}) ** rw [h1, h2] ** case intro.intro.refine'_1 \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) \u22a2 IsMarkovKernel (kernel.comapRight \u03b7' hf) ** refine' kernel.IsMarkovKernel.comapRight _ _ fun a => _ ** case intro.intro.refine'_1 \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a : \u03b1 \u22a2 \u2191\u2191(\u2191\u03b7' a) (range f) = 1 ** rw [kernel.piecewise_apply'] ** case intro.intro.refine'_1 \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a : \u03b1 \u22a2 (if a \u2208 \u03c1_set then \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f)     else \u2191\u2191(\u2191(kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a) (range f)) =     1 ** split_ifs with h_mem ** case pos \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a : \u03b1 h_mem : a \u2208 \u03c1_set \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 ** exact h_eq_one_of_mem _ h_mem ** case neg \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a : \u03b1 h_mem : \u00aca \u2208 \u03c1_set \u22a2 \u2191\u2191(\u2191(kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) a) (range f) = 1 ** rw [kernel.deterministic_apply' _ _ hf.measurableSet_range, Set.indicator_apply, if_pos hx\u2080] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) \u22a2 kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed ** ext c t ht : 2 ** case h.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t \u22a2 \u2191\u2191(\u2191(kernel.const \u03b3 \u03c1) c) t = \u2191\u2191(\u2191(kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed) c) t ** rw [kernel.const_apply, kernel.comapRight_apply' _ _ _ ht, kernel.const_apply,\n  Measure.map_apply h_prod_embed.measurable (h_prod_embed.measurableSet_image.mpr ht)] ** case h.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t \u22a2 \u2191\u2191\u03c1 t = \u2191\u2191\u03c1 (Prod.map id f \u207b\u00b9' (Prod.map id f '' t)) ** congr with x : 1 ** case h.h.e_a.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t x : \u03b1 \u00d7 \u03a9 \u22a2 x \u2208 t \u2194 x \u2208 Prod.map id f \u207b\u00b9' (Prod.map id f '' t) ** rw [\u2190 @Prod.mk.eta _ _ x] ** case h.h.e_a.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t x : \u03b1 \u00d7 \u03a9 \u22a2 (x.1, x.2) \u2208 t \u2194 (x.1, x.2) \u2208 Prod.map id f \u207b\u00b9' (Prod.map id f '' t) ** simp only [id.def, mem_preimage, Prod.map_mk, mem_image, Prod.mk.inj_iff, Prod.exists] ** case h.h.e_a.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t x : \u03b1 \u00d7 \u03a9 \u22a2 (x.1, x.2) \u2208 t \u2194 \u2203 a b, (a, b) \u2208 t \u2227 a = x.1 \u2227 f b = f x.2 ** refine' \u27e8fun h => \u27e8x.1, x.2, h, rfl, rfl\u27e9, _\u27e9 ** case h.h.e_a.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t x : \u03b1 \u00d7 \u03a9 \u22a2 (\u2203 a b, (a, b) \u2208 t \u2227 a = x.1 \u2227 f b = f x.2) \u2192 (x.1, x.2) \u2208 t ** rintro \u27e8a, b, h_mem, rfl, hf_eq\u27e9 ** case h.h.e_a.h.intro.intro.intro.intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t x : \u03b1 \u00d7 \u03a9 b : \u03a9 hf_eq : f b = f x.2 h_mem : (x.1, b) \u2208 t \u22a2 (x.1, x.2) \u2208 t ** rwa [hf.injective hf_eq] at h_mem ** case h.ht \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set \u22a2 MeasurableSet {c | (a, c) \u2208 t} ** exact measurable_prod_mk_left ht ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set \u22a2 {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t} ** ext1 x ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set x : \u211d \u22a2 x \u2208 {c | (a, c) \u2208 Prod.map id f '' t} \u2194 x \u2208 f '' {c | (a, c) \u2208 t} ** simp only [Prod_map, id.def, mem_image, Prod.mk.inj_iff, Prod.exists, mem_setOf_eq] ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set x : \u211d \u22a2 (\u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x) \u2194 \u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x ** constructor ** case h.mp \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set x : \u211d \u22a2 (\u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x) \u2192 \u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x ** rintro \u27e8a', b, h_mem, rfl, hf_eq\u27e9 ** case h.mp.intro.intro.intro.intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t x : \u211d a' : \u03b1 b : \u03a9 h_mem : (a', b) \u2208 t hf_eq : f b = x ha : a' \u2208 \u03c1_set \u22a2 \u2203 x_1, (a', x_1) \u2208 t \u2227 f x_1 = x ** exact \u27e8b, h_mem, hf_eq\u27e9 ** case h.mpr \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set x : \u211d \u22a2 (\u2203 x_1, (a, x_1) \u2208 t \u2227 f x_1 = x) \u2192 \u2203 a_2 b, (a_2, b) \u2208 t \u2227 a_2 = a \u2227 f b = x ** rintro \u27e8b, h_mem, hf_eq\u27e9 ** case h.mpr.intro.intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set x : \u211d b : \u03a9 h_mem : (a, b) \u2208 t hf_eq : f b = x \u22a2 \u2203 a_1 b, (a_1, b) \u2208 t \u2227 a_1 = a \u2227 f b = x ** exact \u27e8a, b, h_mem, rfl, hf_eq\u27e9 ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : PolishSpace \u03a9 inst\u271d\u2074 : MeasurableSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d\u00b9 : IsFiniteMeasure \u03c1 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b3 f : \u03a9 \u2192 \u211d hf : MeasurableEmbedding f \u03c1' : Measure (\u03b1 \u00d7 \u211d) := Measure.map (Prod.map id f) \u03c1 \u03c1_set : Set \u03b1 := (toMeasurable (Measure.fst \u03c1) {a | \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1}\u1d9c)\u1d9c hm : MeasurableSet \u03c1_set h_eq_one_of_mem : \u2200 (a : \u03b1), a \u2208 \u03c1_set \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1') a) (range f) = 1 h_prod_embed : MeasurableEmbedding (Prod.map id f) h_fst : Measure.fst \u03c1' = Measure.fst \u03c1 h_ae : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, a \u2208 \u03c1_set x\u2080 : \u211d hx\u2080 : x\u2080 \u2208 range f \u03b7' : { x // x \u2208 kernel \u03b1 \u211d } :=   kernel.piecewise hm (condKernelReal \u03c1') (kernel.deterministic (fun x => x\u2080) (_ : Measurable fun x => x\u2080)) this : kernel.const \u03b3 \u03c1 = kernel.comapRight (kernel.const \u03b3 \u03c1') h_prod_embed c : \u03b3 t : Set (\u03b1 \u00d7 \u03a9) ht : MeasurableSet t a : \u03b1 ha : a \u2208 \u03c1_set h1 : {c | (a, c) \u2208 Prod.map id f '' t} = f '' {c | (a, c) \u2208 t} \u22a2 \u2191(condKernelReal \u03c1') (c, a).2 = \u2191\u03b7' (c, a).2 ** rw [kernel.piecewise_apply, if_pos ha] ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.le_multiplicity_of_\u03b4_of_fin ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 \u22a2 n \u2264 multiplicity E ** classical\nhave finj : Function.Injective f := by\n  intro i j hij\n  by_contra h\n  have : 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 := h' i j h\n  simp only [hij, norm_zero, sub_self] at this\n  linarith [good\u03b4_lt_one E]\nlet s := Finset.image f Finset.univ\nhave s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n\nhave hs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2 := by\n  simp only [h, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n    Finset.mem_image, imp_true_iff, true_and]\nhave h's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016 := by\n  simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n    Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]\n  intro i j hij\n  have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl\n  exact h' i j this\nhave : s.card \u2264 multiplicity E := card_le_multiplicity_of_\u03b4 hs h's\nrwa [s_card] at this ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 \u22a2 n \u2264 multiplicity E ** have finj : Function.Injective f := by\n  intro i j hij\n  by_contra h\n  have : 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 := h' i j h\n  simp only [hij, norm_zero, sub_self] at this\n  linarith [good\u03b4_lt_one E] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f \u22a2 n \u2264 multiplicity E ** let s := Finset.image f Finset.univ ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ \u22a2 n \u2264 multiplicity E ** have s_card : s.card = n := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin n ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n \u22a2 n \u2264 multiplicity E ** have hs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2 := by\n  simp only [h, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n    Finset.mem_image, imp_true_iff, true_and] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 \u22a2 n \u2264 multiplicity E ** have h's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016 := by\n  simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n    Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and]\n  intro i j hij\n  have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl\n  exact h' i j this ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016 \u22a2 n \u2264 multiplicity E ** have : s.card \u2264 multiplicity E := card_le_multiplicity_of_\u03b4 hs h's ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 h's : \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016 this : Finset.card s \u2264 multiplicity E \u22a2 n \u2264 multiplicity E ** rwa [s_card] at this ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 \u22a2 Function.Injective f ** intro i j hij ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 i j : Fin n hij : f i = f j \u22a2 i = j ** by_contra h ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 i j : Fin n hij : f i = f j h : \u00aci = j \u22a2 False ** have : 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 := h' i j h ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 i j : Fin n hij : f i = f j h : \u00aci = j this : 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 \u22a2 False ** simp only [hij, norm_zero, sub_self] at this ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 i j : Fin n hij : f i = f j h : \u00aci = j this : 1 - good\u03b4 E \u2264 0 \u22a2 False ** linarith [good\u03b4_lt_one E] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ \u22a2 Finset.card s = n ** rw [Finset.card_image_of_injective _ finj] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ \u22a2 Finset.card Finset.univ = n ** exact Finset.card_fin n ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n \u22a2 \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 ** simp only [h, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,\n  Finset.mem_image, imp_true_iff, true_and] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 \u22a2 \u2200 (c : E), c \u2208 s \u2192 \u2200 (d : E), d \u2208 s \u2192 c \u2260 d \u2192 1 - good\u03b4 E \u2264 \u2016c - d\u2016 ** simp only [forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,\n  Ne.def, exists_true_left, forall_apply_eq_imp_iff, forall_true_left, true_and] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 \u22a2 \u2200 (a a_1 : Fin n), \u00acf a = f a_1 \u2192 1 - good\u03b4 E \u2264 \u2016f a - f a_1\u2016 ** intro i j hij ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 i j : Fin n hij : \u00acf i = f j \u22a2 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 ** have : i \u2260 j := fun h => by rw [h] at hij; exact hij rfl ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 i j : Fin n hij : \u00acf i = f j this : i \u2260 j \u22a2 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 ** exact h' i j this ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 i j : Fin n hij : \u00acf i = f j h : i = j \u22a2 False ** rw [h] at hij ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : FiniteDimensional \u211d E n : \u2115 f : Fin n \u2192 E h\u271d : \u2200 (i : Fin n), \u2016f i\u2016 \u2264 2 h' : \u2200 (i j : Fin n), i \u2260 j \u2192 1 - good\u03b4 E \u2264 \u2016f i - f j\u2016 finj : Function.Injective f s : Finset E := Finset.image f Finset.univ s_card : Finset.card s = n hs : \u2200 (c : E), c \u2208 s \u2192 \u2016c\u2016 \u2264 2 i j : Fin n hij : \u00acf j = f j h : i = j \u22a2 False ** exact hij rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.simpleFunc.norm_toSimpleFunc ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedRing \ud835\udd5c inst\u271d\u00b2 : Module \ud835\udd5c E inst\u271d\u00b9 : BoundedSMul \ud835\udd5c E inst\u271d : Fact (1 \u2264 p) f : { x // x \u2208 simpleFunc E p \u03bc } \u22a2 \u2016f\u2016 = ENNReal.toReal (snorm (\u2191(toSimpleFunc f)) p \u03bc) ** simpa [toLp_toSimpleFunc] using norm_toLp (toSimpleFunc f) (simpleFunc.mem\u2112p f) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pi_univ_Ioc_update_union ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i) x y : (i : \u03b9) \u2192 \u03b1 i i\u2080 : \u03b9 m : \u03b1 i\u2080 hm : m \u2208 Icc (x i\u2080) (y i\u2080) \u22a2 ((pi univ fun i => Ioc (x i) (update y i\u2080 m i)) \u222a pi univ fun i => Ioc (update x i\u2080 m i) (y i)) =     pi univ fun i => Ioc (x i) (y i) ** simp_rw [pi_univ_Ioc_update_left hm.1, pi_univ_Ioc_update_right hm.2, \u2190 union_inter_distrib_right,\n  \u2190 setOf_or, le_or_lt, setOf_true, univ_inter] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.mem_iff_get? ** \u03b1 : Type u_1 a : \u03b1 l : List \u03b1 \u22a2 a \u2208 l \u2194 \u2203 n, get? l n = some a ** simp [get?_eq_some, Fin.exists_iff, mem_iff_get] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_mono_measure ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** by_cases hp0 : p = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : \u00acp = 0 \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** by_cases hp_top : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** simp_rw [snorm_eq_snorm' hp0 hp_top] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm' f (ENNReal.toReal p) \u03bd \u2264 snorm' f (ENNReal.toReal p) \u03bc ** exact snorm'_mono_measure f h\u03bc\u03bd ENNReal.toReal_nonneg ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : p = 0 \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** simp [hp0] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** simp [hp_top, snormEssSup_mono_measure f (Measure.absolutelyContinuous_of_le h\u03bc\u03bd)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.setToL1_congr_left' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f : { x // x \u2208 Lp E 1 } \u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f ** suffices setToL1 hT = setToL1 hT' by rw [this] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f : { x // x \u2208 Lp E 1 } \u22a2 setToL1 hT = setToL1 hT' ** refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc hT) _ _ _ _ _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f : { x // x \u2208 Lp E 1 } \u22a2 ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT ** ext1 f ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** suffices setToL1 hT' f = setToL1SCLM \u03b1 E \u03bc hT f by rw [\u2190 this]; rfl ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** rw [setToL1_eq_setToL1SCLM] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(setToL1SCLM \u03b1 E \u03bc hT') f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** exact (setToL1SCLM_congr_left' hT hT' h f).symm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f : { x // x \u2208 Lp E 1 } this : setToL1 hT = setToL1 hT' \u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f ** rw [this] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } this : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f ** rw [\u2190 this] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C\u271d C'\u271d C'' : \u211d T T' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } this : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT') \u2191f ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.trNormal_run ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 k : K s : StAct k q : Stmt\u2082 \u22a2 trNormal (stRun s q) = goto fun x x => go k s q ** cases s <;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.locallyIntegrable_finset_sum ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2074 : MeasurableSpace X inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : MeasurableSpace Y inst\u271d\u00b9 : TopologicalSpace Y inst\u271d : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc : Measure X s\u271d : Set X \u03b9 : Type u_5 s : Finset \u03b9 f : \u03b9 \u2192 X \u2192 E hf : \u2200 (i : \u03b9), i \u2208 s \u2192 LocallyIntegrable (f i) \u22a2 LocallyIntegrable fun a => \u2211 i in s, f i a ** simpa only [\u2190 Finset.sum_apply] using locallyIntegrable_finset_sum' s hf ** Qed",
        "informal": ""
    },
    {
        "formal": "generateFrom_eq_prod ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1' inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b2' inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d : NormedAddCommGroup E C : Set (Set \u03b1) D : Set (Set \u03b2) hC : generateFrom C = inst\u271d\u2075 hD : generateFrom D = inst\u271d\u00b3 h2C : IsCountablySpanning C h2D : IsCountablySpanning D \u22a2 generateFrom (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D) = Prod.instMeasurableSpace ** rw [\u2190 hC, \u2190 hD, generateFrom_prod_eq h2C h2D] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.smul_of_top_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b4 inst\u271d\u2074 : NormedAddCommGroup \u03b2 inst\u271d\u00b3 : NormedAddCommGroup \u03b3 \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NormedRing \ud835\udd5c inst\u271d\u00b9 : Module \ud835\udd5c \u03b2 inst\u271d : BoundedSMul \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 \u03c6 : \u03b1 \u2192 \ud835\udd5c hf : Integrable f h\u03c6 : Mem\u2112p \u03c6 \u22a4 \u22a2 Integrable (\u03c6 \u2022 f) ** rw [\u2190 mem\u2112p_one_iff_integrable] at hf \u22a2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b4 inst\u271d\u2074 : NormedAddCommGroup \u03b2 inst\u271d\u00b3 : NormedAddCommGroup \u03b3 \ud835\udd5c : Type u_5 inst\u271d\u00b2 : NormedRing \ud835\udd5c inst\u271d\u00b9 : Module \ud835\udd5c \u03b2 inst\u271d : BoundedSMul \ud835\udd5c \u03b2 f : \u03b1 \u2192 \u03b2 \u03c6 : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f 1 h\u03c6 : Mem\u2112p \u03c6 \u22a4 \u22a2 Mem\u2112p (\u03c6 \u2022 f) 1 ** exact Mem\u2112p.smul_of_top_right hf h\u03c6 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_sub_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc hp : 1 \u2264 p \u22a2 snorm (f - g) p \u03bc \u2264 snorm f p \u03bc + snorm g p \u03bc ** simpa [LpAddConst_of_one_le hp] using snorm_sub_le' hf hg p ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableEmbedding.measurable_rangeSplitting ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u : Set \u03b1 m\u03b1 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 \u03b3 hf : MeasurableEmbedding f s : Set \u03b1 hs : MeasurableSet s \u22a2 MeasurableSet (rangeSplitting f \u207b\u00b9' s) ** rwa [preimage_rangeSplitting hf.injective,\n  \u2190 (subtype_coe hf.measurableSet_range).measurableSet_image, \u2190 image_comp,\n  coe_comp_rangeFactorization, hf.measurableSet_image] ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.last_le_iff ** n : Nat k : Fin (n + 1) \u22a2 last n \u2264 k \u2194 k = last n ** rw [ext_iff, Nat.le_antisymm_iff, le_def, and_iff_right (by apply le_last)] ** n : Nat k : Fin (n + 1) \u22a2 \u2191k \u2264 \u2191(last n) ** apply le_last ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsFundamentalDomain.integral_eq_tsum ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E h : IsFundamentalDomain G s f : \u03b1 \u2192 E hf : Integrable f \u22a2 \u03bc \u226a \u03bc ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.const_smul\u2080 ** G\u2080 : Type u_1 \u03b1 : Type u_2 inst\u271d\u2076 : GroupWithZero G\u2080 inst\u271d\u2075 : Zero \u03b1 inst\u271d\u2074 : MulActionWithZero G\u2080 \u03b1 inst\u271d\u00b3 : MeasurableSpace G\u2080 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSMul G\u2080 \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 s : Set \u03b1 hs : MeasurableSet s a : G\u2080 \u22a2 MeasurableSet (a \u2022 s) ** rcases eq_or_ne a 0 with (rfl | ha) ** case inl G\u2080 : Type u_1 \u03b1 : Type u_2 inst\u271d\u2076 : GroupWithZero G\u2080 inst\u271d\u2075 : Zero \u03b1 inst\u271d\u2074 : MulActionWithZero G\u2080 \u03b1 inst\u271d\u00b3 : MeasurableSpace G\u2080 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSMul G\u2080 \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 MeasurableSet (0 \u2022 s)  case inr G\u2080 : Type u_1 \u03b1 : Type u_2 inst\u271d\u2076 : GroupWithZero G\u2080 inst\u271d\u2075 : Zero \u03b1 inst\u271d\u2074 : MulActionWithZero G\u2080 \u03b1 inst\u271d\u00b3 : MeasurableSpace G\u2080 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSMul G\u2080 \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 s : Set \u03b1 hs : MeasurableSet s a : G\u2080 ha : a \u2260 0 \u22a2 MeasurableSet (a \u2022 s) ** exacts [(subsingleton_zero_smul_set s).measurableSet, hs.const_smul_of_ne_zero ha] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.sub_apply_eq_zero_of_restrict_le_restrict ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 s : Set \u03b1 h_le : restrict \u03bc s \u2264 restrict \u03bd s h_meas_s : MeasurableSet s \u22a2 \u2191\u2191(\u03bc - \u03bd) s = 0 ** rw [\u2190 restrict_apply_self, restrict_sub_eq_restrict_sub_restrict, sub_eq_zero_of_le] <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.unifIntegrable_of_tendsto_Lp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), Mem\u2112p (f n) p hg : Mem\u2112p g p hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) \u22a2 UnifIntegrable f p \u03bc ** have : f = (fun _ => g) + fun n => f n - g := by ext1 n; simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), Mem\u2112p (f n) p hg : Mem\u2112p g p hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) this : f = (fun x => g) + fun n => f n - g \u22a2 UnifIntegrable f p \u03bc ** rw [this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), Mem\u2112p (f n) p hg : Mem\u2112p g p hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) this : f = (fun x => g) + fun n => f n - g \u22a2 UnifIntegrable ((fun x => g) + fun n => f n - g) p \u03bc ** refine' UnifIntegrable.add _ _ hp (fun _ => hg.aestronglyMeasurable)\n    fun n => (hf n).1.sub hg.aestronglyMeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), Mem\u2112p (f n) p hg : Mem\u2112p g p hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) \u22a2 f = (fun x => g) + fun n => f n - g ** ext1 n ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), Mem\u2112p (f n) p hg : Mem\u2112p g p hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) n : \u2115 \u22a2 f n = ((fun x => g) + fun n => f n - g) n ** simp ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), Mem\u2112p (f n) p hg : Mem\u2112p g p hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) this : f = (fun x => g) + fun n => f n - g \u22a2 UnifIntegrable (fun x => g) p \u03bc ** exact unifIntegrable_const \u03bc hp hp' hg ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 hf : \u2200 (n : \u2115), Mem\u2112p (f n) p hg : Mem\u2112p g p hfg : Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) this : f = (fun x => g) + fun n => f n - g \u22a2 UnifIntegrable (fun n => f n - g) p \u03bc ** exact unifIntegrable_of_tendsto_Lp_zero \u03bc hp hp' (fun n => (hf n).sub hg) hfg ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.prec' ** f g h : \u2115 \u2192. \u2115 hf : Partrec f hg : Partrec g hh : Partrec h a s : \u2115 \u22a2 (s \u2208       (Seq.seq (Nat.pair <$> Part.some a) fun x => f a) >>=         unpaired fun a n =>           Nat.rec (g a)             (fun y IH => do               let i \u2190 IH               h (Nat.pair a (Nat.pair y i)))             n) \u2194     s \u2208       Part.bind (f a) fun n =>         Nat.rec (g a)           (fun y IH => do             let i \u2190 IH             h (Nat.pair a (Nat.pair y i)))           n ** simp [Seq.seq] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.sections_eq_sectionsTR ** \u22a2 @sections = @sectionsTR ** funext \u03b1 L ** case h.h \u03b1 : Type u_1 L : List (List \u03b1) \u22a2 sections L = sectionsTR L ** simp [sectionsTR] ** case h.h \u03b1 : Type u_1 L : List (List \u03b1) \u22a2 sections L = bif any L isEmpty then [] else (foldr sectionsTR.go #[[]] L).data ** cases e : L.any isEmpty <;> simp [sections_eq_nil_of_isEmpty, *] ** case h.h.false \u03b1 : Type u_1 L : List (List \u03b1) e : any L isEmpty = false \u22a2 sections L = (foldr sectionsTR.go #[[]] L).data ** clear e ** case h.h.false \u03b1 : Type u_1 L : List (List \u03b1) \u22a2 sections L = (foldr sectionsTR.go #[[]] L).data ** induction L with | nil => rfl | cons l L IH => ?_ ** case h.h.false.cons \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) IH : sections L = (foldr sectionsTR.go #[[]] L).data \u22a2 sections (l :: L) = (foldr sectionsTR.go #[[]] (l :: L)).data ** simp [IH, sectionsTR.go] ** case h.h.false.cons \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) IH : sections L = (foldr sectionsTR.go #[[]] L).data \u22a2 (List.bind       (foldr           (fun l acc =>             Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0               (Array.size acc))           #[[]] L).data       fun s => map (fun a => a :: s) l) =     (Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[]         (foldr           (fun l acc =>             Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0               (Array.size acc))           #[[]] L)         0         (Array.size           (foldr             (fun l acc =>               Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0                 (Array.size acc))             #[[]] L))).data ** rw [Array.foldl_eq_foldl_data, Array.foldl_data_eq_bind] ** case h.h.false.cons \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) IH : sections L = (foldr sectionsTR.go #[[]] L).data \u22a2 (List.bind       (foldr           (fun l acc =>             Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0               (Array.size acc))           #[[]] L).data       fun s => map (fun a => a :: s) l) =     #[].data ++       List.bind         (foldr             (fun l acc =>               Array.foldl (fun acc' l' => foldl (fun acc' a => Array.push acc' (a :: l')) acc' l) #[] acc 0                 (Array.size acc))             #[[]] L).data         ?h.h.false.cons.G  case h.h.false.cons.G \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) IH : sections L = (foldr sectionsTR.go #[[]] L).data \u22a2 List \u03b1 \u2192 List (List \u03b1)  case h.h.false.cons.H \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) IH : sections L = (foldr sectionsTR.go #[[]] L).data \u22a2 \u2200 (acc : Array (List \u03b1)) (a : List \u03b1),     (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ ?h.h.false.cons.G a ** rfl ** case h.h.false.cons.H \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) IH : sections L = (foldr sectionsTR.go #[[]] L).data \u22a2 \u2200 (acc : Array (List \u03b1)) (a : List \u03b1),     (foldl (fun acc' a_1 => Array.push acc' (a_1 :: a)) acc l).data = acc.data ++ map (fun a_1 => a_1 :: a) l ** intros ** case h.h.false.cons.H \u03b1 : Type u_1 l : List \u03b1 L : List (List \u03b1) IH : sections L = (foldr sectionsTR.go #[[]] L).data acc\u271d : Array (List \u03b1) a\u271d : List \u03b1 \u22a2 (foldl (fun acc' a => Array.push acc' (a :: a\u271d)) acc\u271d l).data = acc\u271d.data ++ map (fun a => a :: a\u271d) l ** apply Array.foldl_data_eq_map ** case h.h.false.nil \u03b1 : Type u_1 \u22a2 sections [] = (foldr sectionsTR.go #[[]] []).data ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.card_Iio_finset ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 \u22a2 card (Iio s) = 2 ^ card s - 1 ** rw [Iio_eq_ssubsets, ssubsets, card_erase_of_mem (mem_powerset_self _), card_powerset] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_prod ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' f : \u03b1 \u00d7 \u03b2 \u2192 E hf : Integrable f \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc ** by_cases hE : CompleteSpace E ** case pos \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' f : \u03b1 \u00d7 \u03b2 \u2192 E hf : Integrable f hE : CompleteSpace E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc  case neg \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' f : \u03b1 \u00d7 \u03b2 \u2192 E hf : Integrable f hE : \u00acCompleteSpace E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' f : \u03b1 \u00d7 \u03b2 \u2192 E hf : Integrable f hE : CompleteSpace E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc ** revert f ** case pos \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E \u22a2 \u2200 (f : \u03b1 \u00d7 \u03b2 \u2192 E), Integrable f \u2192 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc ** apply Integrable.induction ** case neg \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' f : \u03b1 \u00d7 \u03b2 \u2192 E hf : Integrable f hE : \u00acCompleteSpace E \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc ** simp only [integral, dif_neg hE] ** case pos.h_ind \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E \u22a2 \u2200 (c : E) \u2983s : Set (\u03b1 \u00d7 \u03b2)\u2984,     MeasurableSet s \u2192       \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4 \u2192         \u222b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =           \u222b (x : \u03b1), \u222b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc ** intro c s hs h2s ** case pos.h_ind \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E c : E s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s h2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4 \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), indicator s (fun x => c) z \u2202Measure.prod \u03bc \u03bd =     \u222b (x : \u03b1), \u222b (y : \u03b2), indicator s (fun x => c) (x, y) \u2202\u03bd \u2202\u03bc ** simp_rw [integral_indicator hs, \u2190 indicator_comp_right, Function.comp,\n  integral_indicator (measurable_prod_mk_left hs), set_integral_const, integral_smul_const,\n  integral_toReal (measurable_measure_prod_mk_left hs).aemeasurable\n    (ae_measure_lt_top hs h2s.ne)] ** case pos.h_ind \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E c : E s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s h2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s < \u22a4 \u22a2 ENNReal.toReal (\u2191\u2191(Measure.prod \u03bc \u03bd) s) \u2022 c = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s) \u2202\u03bc) \u2022 c ** rw [prod_apply hs] ** case pos.h_add \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E \u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 E\u2984,     Disjoint (support f) (support g) \u2192       Integrable f \u2192         Integrable g \u2192           \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u2192             \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc \u2192               \u222b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc ** rintro f g - i_f i_g hf hg ** case pos.h_add \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E f g : \u03b1 \u00d7 \u03b2 \u2192 E i_f : Integrable f i_g : Integrable g hf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc hg : \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), (f + g) z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), (f + g) (x, y) \u2202\u03bd \u2202\u03bc ** simp_rw [integral_add' i_f i_g, integral_integral_add' i_f i_g, hf, hg] ** case pos.h_closed \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E \u22a2 IsClosed {f | \u222b (z : \u03b1 \u00d7 \u03b2), \u2191\u2191f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), \u2191\u2191f (x, y) \u2202\u03bd \u2202\u03bc} ** exact isClosed_eq continuous_integral continuous_integral_integral ** case pos.h_ae \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E \u22a2 \u2200 \u2983f g : \u03b1 \u00d7 \u03b2 \u2192 E\u2984,     f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g \u2192       Integrable f \u2192         \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u2192           \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc ** rintro f g hfg - hf ** case pos.h_ae \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E f g : \u03b1 \u00d7 \u03b2 \u2192 E hfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g hf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc ** convert hf using 1 ** case h.e'_2 \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E f g : \u03b1 \u00d7 \u03b2 \u2192 E hfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g hf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u222b (z : \u03b1 \u00d7 \u03b2), g z \u2202Measure.prod \u03bc \u03bd = \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd ** exact integral_congr_ae hfg.symm ** case h.e'_3 \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E f g : \u03b1 \u00d7 \u03b2 \u2192 E hfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g hf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u222b (x : \u03b1), \u222b (y : \u03b2), g (x, y) \u2202\u03bd \u2202\u03bc = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc ** refine' integral_congr_ae _ ** case h.e'_3 \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E f g : \u03b1 \u00d7 \u03b2 \u2192 E hfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g hf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u22a2 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) =\u1da0[ae \u03bc] fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd ** refine' (ae_ae_of_ae_prod hfg).mp _ ** case h.e'_3 \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E f g : \u03b1 \u00d7 \u03b2 \u2192 E hfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g hf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)) \u2192 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x ** apply eventually_of_forall ** case h.e'_3.hp \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E f g : \u03b1 \u00d7 \u03b2 \u2192 E hfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g hf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc \u22a2 \u2200 (x : \u03b1),     (\u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y)) \u2192 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x ** intro x hfgx ** case h.e'_3.hp \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1' inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2' inst\u271d\u2076 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SigmaFinite \u03bc E' : Type u_7 inst\u271d\u00b9 : NormedAddCommGroup E' inst\u271d : NormedSpace \u211d E' hE : CompleteSpace E f g : \u03b1 \u00d7 \u03b2 \u2192 E hfg : f =\u1da0[ae (Measure.prod \u03bc \u03bd)] g hf : \u222b (z : \u03b1 \u00d7 \u03b2), f z \u2202Measure.prod \u03bc \u03bd = \u222b (x : \u03b1), \u222b (y : \u03b2), f (x, y) \u2202\u03bd \u2202\u03bc x : \u03b1 hfgx : \u2200\u1d50 (y : \u03b2) \u2202\u03bd, f (x, y) = g (x, y) \u22a2 (fun x => \u222b (y : \u03b2), g (x, y) \u2202\u03bd) x = (fun x => \u222b (y : \u03b2), f (x, y) \u2202\u03bd) x ** exact integral_congr_ae (ae_eq_symm hfgx) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.locallyIntegrable_map_homeomorph ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X \u22a2 LocallyIntegrable f \u2194 LocallyIntegrable (f \u2218 \u2191e) ** refine' \u27e8fun h x => _, fun h x => _\u27e9 ** case refine'_1 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable f x : X \u22a2 IntegrableAtFilter (f \u2218 \u2191e) (\ud835\udcdd x) ** rcases h (e x) with \u27e8U, hU, h'U\u27e9 ** case refine'_1.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable f x : X U : Set Y hU : U \u2208 \ud835\udcdd (\u2191e x) h'U : IntegrableOn f U \u22a2 IntegrableAtFilter (f \u2218 \u2191e) (\ud835\udcdd x) ** refine' \u27e8e \u207b\u00b9' U, e.continuous.continuousAt.preimage_mem_nhds hU, _\u27e9 ** case refine'_1.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable f x : X U : Set Y hU : U \u2208 \ud835\udcdd (\u2191e x) h'U : IntegrableOn f U \u22a2 IntegrableOn (f \u2218 \u2191e) (\u2191e \u207b\u00b9' U) ** exact (integrableOn_map_equiv e.toMeasurableEquiv).1 h'U ** case refine'_2 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable (f \u2218 \u2191e) x : Y \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** rcases h (e.symm x) with \u27e8U, hU, h'U\u27e9 ** case refine'_2.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable (f \u2218 \u2191e) x : Y U : Set X hU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x) h'U : IntegrableOn (f \u2218 \u2191e) U \u22a2 IntegrableAtFilter f (\ud835\udcdd x) ** refine' \u27e8e.symm \u207b\u00b9' U, e.symm.continuous.continuousAt.preimage_mem_nhds hU, _\u27e9 ** case refine'_2.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable (f \u2218 \u2191e) x : Y U : Set X hU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x) h'U : IntegrableOn (f \u2218 \u2191e) U \u22a2 IntegrableOn f (\u2191(Homeomorph.symm e) \u207b\u00b9' U) ** apply (integrableOn_map_equiv e.toMeasurableEquiv).2 ** case refine'_2.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable (f \u2218 \u2191e) x : Y U : Set X hU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x) h'U : IntegrableOn (f \u2218 \u2191e) U \u22a2 IntegrableOn (f \u2218 \u2191(Homeomorph.toMeasurableEquiv e))     (\u2191(Homeomorph.toMeasurableEquiv e) \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U)) ** simp only [Homeomorph.toMeasurableEquiv_coe] ** case refine'_2.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable (f \u2218 \u2191e) x : Y U : Set X hU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x) h'U : IntegrableOn (f \u2218 \u2191e) U \u22a2 IntegrableOn (f \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U)) ** convert h'U ** case h.e'_6 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable (f \u2218 \u2191e) x : Y U : Set X hU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x) h'U : IntegrableOn (f \u2218 \u2191e) U \u22a2 \u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U) = U ** ext x ** case h.e'_6.h X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst\u271d\u2076 : MeasurableSpace X inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : MeasurableSpace Y inst\u271d\u00b3 : TopologicalSpace Y inst\u271d\u00b2 : NormedAddCommGroup E f\u271d g : X \u2192 E \u03bc\u271d : Measure X s : Set X inst\u271d\u00b9 : BorelSpace X inst\u271d : BorelSpace Y e : X \u2243\u209c Y f : Y \u2192 E \u03bc : Measure X h : LocallyIntegrable (f \u2218 \u2191e) x\u271d : Y U : Set X hU : U \u2208 \ud835\udcdd (\u2191(Homeomorph.symm e) x\u271d) h'U : IntegrableOn (f \u2218 \u2191e) U x : X \u22a2 x \u2208 \u2191e \u207b\u00b9' (\u2191(Homeomorph.symm e) \u207b\u00b9' U) \u2194 x \u2208 U ** simp only [mem_preimage, Homeomorph.symm_apply_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.DynkinSystem.generate_inter ** \u03b1 : Type u_1 d : DynkinSystem \u03b1 s : Set (Set \u03b1) hs : IsPiSystem s t\u2081 t\u2082 : Set \u03b1 ht\u2081 : Has (generate s) t\u2081 ht\u2082 : Has (generate s) t\u2082 s\u2081 : Set \u03b1 hs\u2081 : s\u2081 \u2208 s this\u271d\u00b9 : Has (generate s) s\u2081 this\u271d : generate s \u2264 restrictOn (generate s) this\u271d\u00b9 this : Has (generate s) (t\u2082 \u2229 s\u2081) \u22a2 Has (generate s) (s\u2081 \u2229 t\u2082) ** rwa [inter_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.filter_apply_eq_zero_of_not_mem ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 s : Set \u03b1 h : \u2203 a, a \u2208 s \u2227 a \u2208 support p a : \u03b1 ha : \u00aca \u2208 s \u22a2 \u2191(filter p s h) a = 0 ** rw [filter_apply, Set.indicator_apply_eq_zero.mpr fun ha' => absurd ha' ha, zero_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bd : ProbabilityMeasure \u03a9 s : Set \u03a9 \u22a2 \u2191((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) s) = \u2191\u2191\u2191\u03bd s ** rw [\u2190 coeFn_comp_toFiniteMeasure_eq_coeFn, FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure,\n  toMeasure_comp_toFiniteMeasure_eq_toMeasure] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.PairingHeapImp.Heap.size_tail? ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 h : NoSibling s \u22a2 tail? le s = some s' \u2192 size s = size s' + 1 ** simp only [Heap.tail?] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 h : NoSibling s \u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 size s = size s' + 1 ** intro eq ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 h : NoSibling s eq : Option.map (fun x => x.snd) (deleteMin le s) = some s' \u22a2 size s = size s' + 1 ** match eq\u2082 : s.deleteMin le, eq with\n| some (a, tl), rfl => exact size_deleteMin h eq\u2082 ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 h : NoSibling s eq : Option.map (fun x => x.snd) (deleteMin le s) = some s' a : \u03b1 tl : Heap \u03b1 eq\u2082 : deleteMin le s = some (a, tl) \u22a2 size s = size ((fun x => x.snd) (a, tl)) + 1 ** exact size_deleteMin h eq\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_insert_eq_ite ** \u03b1 : Type u_1 s t : Set \u03b1 a : \u03b1 inst\u271d : Decidable (a \u2208 s) hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard (insert a s) = if a \u2208 s then ncard s else ncard s + 1 ** by_cases h : a \u2208 s ** case pos \u03b1 : Type u_1 s t : Set \u03b1 a : \u03b1 inst\u271d : Decidable (a \u2208 s) hs : autoParam (Set.Finite s) _auto\u271d h : a \u2208 s \u22a2 ncard (insert a s) = if a \u2208 s then ncard s else ncard s + 1 ** rw [ncard_insert_of_mem h, if_pos h] ** case neg \u03b1 : Type u_1 s t : Set \u03b1 a : \u03b1 inst\u271d : Decidable (a \u2208 s) hs : autoParam (Set.Finite s) _auto\u271d h : \u00aca \u2208 s \u22a2 ncard (insert a s) = if a \u2208 s then ncard s else ncard s + 1 ** rw [ncard_insert_of_not_mem h hs, if_neg h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.div2_bit ** a b\u271d : \u2124 n\u271d : \u2115 b : Bool n : \u2124 \u22a2 div2 (bit b n) = n ** rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : \u2124) = 0), zero_add] ** a b\u271d : \u2124 n\u271d : \u2115 b : Bool n : \u2124 \u22a2 (bif b then 1 else 0) / 2 = 0  case H a b\u271d : \u2124 n\u271d : \u2115 b : Bool n : \u2124 \u22a2 2 \u2260 0 ** cases b ** case false a b : \u2124 n\u271d : \u2115 n : \u2124 \u22a2 (bif false then 1 else 0) / 2 = 0 ** simp ** case true a b : \u2124 n\u271d : \u2115 n : \u2124 \u22a2 (bif true then 1 else 0) / 2 = 0 ** show ofNat _ = _ ** case true a b : \u2124 n\u271d : \u2115 n : \u2124 \u22a2 ofNat (1 / 2) = 0 ** rw [Nat.div_eq_of_lt] <;> simp ** case H a b\u271d : \u2124 n\u271d : \u2115 b : Bool n : \u2124 \u22a2 2 \u2260 0 ** decide ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_succ ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : MetricSpace E f : \u2115 \u2192 \u03b1 \u2192 E g : \u03b1 \u2192 E hfg : TendstoInMeasure \u03bc f atTop g n : \u2115 \u22a2 seqTendstoAeSeq hfg (n + 1) = max (seqTendstoAeSeqAux hfg (n + 1)) (seqTendstoAeSeq hfg n + 1) ** rw [seqTendstoAeSeq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.pimage_inter ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : DecidableEq \u03b2 f g : \u03b1 \u2192. \u03b2 inst\u271d\u00b2 : (x : \u03b1) \u2192 Decidable (f x).Dom inst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (g x).Dom s t : Finset \u03b1 b : \u03b2 inst\u271d : DecidableEq \u03b1 \u22a2 pimage f (s \u2229 t) \u2286 pimage f s \u2229 pimage f t ** simp only [\u2190 coe_subset, coe_pimage, coe_inter, PFun.image_inter] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.of_to_int ** \u03b1 : Type u_1 n : ZNum \u22a2 \u2191\u2191n = n ** rw [\u2190 ofInt'_eq, of_to_int'] ** Qed",
        "informal": ""
    },
    {
        "formal": "upperClosure_sups ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeSup \u03b1 s t : Set \u03b1 \u22a2 upperClosure (s \u22bb t) = upperClosure s \u2294 upperClosure t ** ext a ** case a.h F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeSup \u03b1 s t : Set \u03b1 a : \u03b1 \u22a2 a \u2208 \u2191(upperClosure (s \u22bb t)) \u2194 a \u2208 \u2191(upperClosure s \u2294 upperClosure t) ** simp only [SetLike.mem_coe, mem_upperClosure, Set.mem_sups, exists_and_left, exists_prop,\n  UpperSet.coe_sup, Set.mem_inter_iff] ** case a.h F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeSup \u03b1 s t : Set \u03b1 a : \u03b1 \u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a) \u2194 (\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a ** constructor ** case a.h.mp F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeSup \u03b1 s t : Set \u03b1 a : \u03b1 \u22a2 (\u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a) \u2192 (\u2203 a_2, a_2 \u2208 s \u2227 a_2 \u2264 a) \u2227 \u2203 a_2, a_2 \u2208 t \u2227 a_2 \u2264 a ** rintro \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, ha\u27e9 ** case a.h.mp.intro.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeSup \u03b1 s t : Set \u03b1 a b : \u03b1 hb : b \u2208 s c : \u03b1 hc : c \u2208 t ha : b \u2294 c \u2264 a \u22a2 (\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a ** exact \u27e8\u27e8b, hb, le_sup_left.trans ha\u27e9, c, hc, le_sup_right.trans ha\u27e9 ** case a.h.mpr F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeSup \u03b1 s t : Set \u03b1 a : \u03b1 \u22a2 ((\u2203 a_1, a_1 \u2208 s \u2227 a_1 \u2264 a) \u2227 \u2203 a_1, a_1 \u2208 t \u2227 a_1 \u2264 a) \u2192 \u2203 a_2, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_2) \u2227 a_2 \u2264 a ** rintro \u27e8\u27e8b, hb, hab\u27e9, c, hc, hac\u27e9 ** case a.h.mpr.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : SemilatticeSup \u03b1 s t : Set \u03b1 a b : \u03b1 hb : b \u2208 s hab : b \u2264 a c : \u03b1 hc : c \u2208 t hac : c \u2264 a \u22a2 \u2203 a_1, (\u2203 a, a \u2208 s \u2227 \u2203 b, b \u2208 t \u2227 a \u2294 b = a_1) \u2227 a_1 \u2264 a ** exact \u27e8_, \u27e8b, hb, c, hc, rfl\u27e9, sup_le hab hac\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.expand_bind\u2081 ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S p : \u2115 f : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 \u2191(expand p) (\u2191(bind\u2081 f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(expand p) (f i)) \u03c6 ** rw [\u2190 AlgHom.comp_apply, expand_comp_bind\u2081] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hausdorffMeasure_prod_real ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u2075 : EMetricSpace X inst\u271d\u2074 : EMetricSpace Y inst\u271d\u00b3 : MeasurableSpace X inst\u271d\u00b2 : BorelSpace X inst\u271d\u00b9 : MeasurableSpace Y inst\u271d : BorelSpace Y \u22a2 \u03bcH[2] = volume ** rw [\u2190 (volume_preserving_piFinTwo fun _ => \u211d).map_eq,\n  \u2190 (hausdorffMeasure_measurePreserving_piFinTwo (fun _ => \u211d) _).map_eq,\n  \u2190 hausdorffMeasure_pi_real, Fintype.card_fin, Nat.cast_two] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableEmbedding.stronglyMeasurable_extend ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 f\u271d g\u271d f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 g' : \u03b3 \u2192 \u03b2 m\u03b1 : MeasurableSpace \u03b1 m\u03b3 : MeasurableSpace \u03b3 inst\u271d : TopologicalSpace \u03b2 hg : MeasurableEmbedding g hf : StronglyMeasurable f hg' : StronglyMeasurable g' \u22a2 StronglyMeasurable (Function.extend g f g') ** refine' \u27e8fun n => SimpleFunc.extend (hf.approx n) g hg (hg'.approx n), _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 f\u271d g\u271d f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 g' : \u03b3 \u2192 \u03b2 m\u03b1 : MeasurableSpace \u03b1 m\u03b3 : MeasurableSpace \u03b3 inst\u271d : TopologicalSpace \u03b2 hg : MeasurableEmbedding g hf : StronglyMeasurable f hg' : StronglyMeasurable g' \u22a2 \u2200 (x : \u03b3),     Tendsto       (fun n =>         \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)       atTop (\ud835\udcdd (Function.extend g f g' x)) ** intro x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 f\u271d g\u271d f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 g' : \u03b3 \u2192 \u03b2 m\u03b1 : MeasurableSpace \u03b1 m\u03b3 : MeasurableSpace \u03b3 inst\u271d : TopologicalSpace \u03b2 hg : MeasurableEmbedding g hf : StronglyMeasurable f hg' : StronglyMeasurable g' x : \u03b3 \u22a2 Tendsto     (fun n =>       \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)     atTop (\ud835\udcdd (Function.extend g f g' x)) ** by_cases hx : \u2203 y, g y = x ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 f\u271d g\u271d f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 g' : \u03b3 \u2192 \u03b2 m\u03b1 : MeasurableSpace \u03b1 m\u03b3 : MeasurableSpace \u03b3 inst\u271d : TopologicalSpace \u03b2 hg : MeasurableEmbedding g hf : StronglyMeasurable f hg' : StronglyMeasurable g' x : \u03b3 hx : \u2203 y, g y = x \u22a2 Tendsto     (fun n =>       \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)     atTop (\ud835\udcdd (Function.extend g f g' x)) ** rcases hx with \u27e8y, rfl\u27e9 ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 f\u271d g\u271d f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 g' : \u03b3 \u2192 \u03b2 m\u03b1 : MeasurableSpace \u03b1 m\u03b3 : MeasurableSpace \u03b3 inst\u271d : TopologicalSpace \u03b2 hg : MeasurableEmbedding g hf : StronglyMeasurable f hg' : StronglyMeasurable g' y : \u03b1 \u22a2 Tendsto     (fun n =>       \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) (g y))     atTop (\ud835\udcdd (Function.extend g f g' (g y))) ** simpa only [SimpleFunc.extend_apply, hg.injective, Injective.extend_apply] using\n  hf.tendsto_approx y ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b9 : Countable \u03b9 f\u271d g\u271d f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 g' : \u03b3 \u2192 \u03b2 m\u03b1 : MeasurableSpace \u03b1 m\u03b3 : MeasurableSpace \u03b3 inst\u271d : TopologicalSpace \u03b2 hg : MeasurableEmbedding g hf : StronglyMeasurable f hg' : StronglyMeasurable g' x : \u03b3 hx : \u00ac\u2203 y, g y = x \u22a2 Tendsto     (fun n =>       \u2191((fun n => SimpleFunc.extend (StronglyMeasurable.approx hf n) g hg (StronglyMeasurable.approx hg' n)) n) x)     atTop (\ud835\udcdd (Function.extend g f g' x)) ** simpa only [hx, SimpleFunc.extend_apply', not_false_iff, extend_apply'] using\n  hg'.tendsto_approx x ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.diag_mem_sym2_iff ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 a b : \u03b1 m : Sym2 \u03b1 \u22a2 Sym2.diag a \u2208 Finset.sym2 s \u2194 a \u2208 s ** simp [diag_mem_sym2_mem_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.comp_snd_map_prod_mk ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9\u271d inst\u271d\u2075 : MeasurableSpace \u03a9\u271d inst\u271d\u2074 : PolishSpace \u03a9\u271d inst\u271d\u00b3 : BorelSpace \u03a9\u271d inst\u271d\u00b2 : Nonempty \u03a9\u271d inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F \u03a9 : Type u_5 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F hf_int : Integrable f \u22a2 Integrable fun x => f x.2 ** by_cases hX : AEMeasurable X \u03bc ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9\u271d inst\u271d\u2075 : MeasurableSpace \u03a9\u271d inst\u271d\u2074 : PolishSpace \u03a9\u271d inst\u271d\u00b3 : BorelSpace \u03a9\u271d inst\u271d\u00b2 : Nonempty \u03a9\u271d inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F \u03a9 : Type u_5 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F hf_int : Integrable f hX : AEMeasurable X \u22a2 Integrable fun x => f x.2 ** have hf := hf_int.1.comp_snd_map_prod_mk X (m\u03a9 := m\u03a9) (m\u03b2 := m\u03b2) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9\u271d inst\u271d\u2075 : MeasurableSpace \u03a9\u271d inst\u271d\u2074 : PolishSpace \u03a9\u271d inst\u271d\u00b3 : BorelSpace \u03a9\u271d inst\u271d\u00b2 : Nonempty \u03a9\u271d inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F \u03a9 : Type u_5 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F hf_int : Integrable f hX : AEMeasurable X hf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) \u22a2 Integrable fun x => f x.2 ** refine' \u27e8hf, _\u27e9 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9\u271d inst\u271d\u2075 : MeasurableSpace \u03a9\u271d inst\u271d\u2074 : PolishSpace \u03a9\u271d inst\u271d\u00b3 : BorelSpace \u03a9\u271d inst\u271d\u00b2 : Nonempty \u03a9\u271d inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F \u03a9 : Type u_5 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F hf_int : Integrable f hX : AEMeasurable X hf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) \u22a2 HasFiniteIntegral fun x => f x.2 ** rw [HasFiniteIntegral, lintegral_map' hf.ennnorm (hX.prod_mk aemeasurable_id)] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9\u271d inst\u271d\u2075 : MeasurableSpace \u03a9\u271d inst\u271d\u2074 : PolishSpace \u03a9\u271d inst\u271d\u00b3 : BorelSpace \u03a9\u271d inst\u271d\u00b2 : Nonempty \u03a9\u271d inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F \u03a9 : Type u_5 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F hf_int : Integrable f hX : AEMeasurable X hf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun \u03c9 => (X \u03c9, \u03c9)) \u03bc) \u22a2 \u222b\u207b (a : \u03a9), \u2191\u2016f (X a, a).2\u2016\u208a \u2202\u03bc < \u22a4 ** exact hf_int.2 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9\u271d inst\u271d\u2075 : MeasurableSpace \u03a9\u271d inst\u271d\u2074 : PolishSpace \u03a9\u271d inst\u271d\u00b3 : BorelSpace \u03a9\u271d inst\u271d\u00b2 : Nonempty \u03a9\u271d inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F \u03a9 : Type u_5 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F hf_int : Integrable f hX : \u00acAEMeasurable X \u22a2 Integrable fun x => f x.2 ** rw [Measure.map_of_not_aemeasurable] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9\u271d inst\u271d\u2075 : MeasurableSpace \u03a9\u271d inst\u271d\u2074 : PolishSpace \u03a9\u271d inst\u271d\u00b3 : BorelSpace \u03a9\u271d inst\u271d\u00b2 : Nonempty \u03a9\u271d inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F \u03a9 : Type u_5 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F hf_int : Integrable f hX : \u00acAEMeasurable X \u22a2 Integrable fun x => f x.2 ** simp ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9\u271d inst\u271d\u2075 : MeasurableSpace \u03a9\u271d inst\u271d\u2074 : PolishSpace \u03a9\u271d inst\u271d\u00b3 : BorelSpace \u03a9\u271d inst\u271d\u00b2 : Nonempty \u03a9\u271d inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F \u03a9 : Type u_5 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F hf_int : Integrable f hX : \u00acAEMeasurable X \u22a2 \u00acAEMeasurable fun \u03c9 => (X \u03c9, \u03c9) ** contrapose! hX ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9\u271d : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9\u271d inst\u271d\u2075 : MeasurableSpace \u03a9\u271d inst\u271d\u2074 : PolishSpace \u03a9\u271d inst\u271d\u00b3 : BorelSpace \u03a9\u271d inst\u271d\u00b2 : Nonempty \u03a9\u271d inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc\u271d X\u271d : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9\u271d m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9\u271d t : Set \u03b2 f\u271d : \u03b2 \u00d7 \u03a9\u271d \u2192 F \u03a9 : Type u_5 m\u03a9 : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u03b2 \u03bc : Measure \u03a9 f : \u03a9 \u2192 F hf_int : Integrable f hX : AEMeasurable fun \u03c9 => (X \u03c9, \u03c9) \u22a2 AEMeasurable X ** exact measurable_fst.comp_aemeasurable hX ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.rnDeriv_withDensity ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d \u03bd : Measure \u03b1 inst\u271d : SigmaFinite \u03bd f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 withDensity \u03bd f = 0 + withDensity \u03bd f ** rw [zero_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.const_eq_compProd_real ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1\u271d : Measure (\u03b1 \u00d7 \u211d) inst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d \u03b3 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 \u22a2 const \u03b3 \u03c1 = const \u03b3 (Measure.fst \u03c1) \u2297\u2096 prodMkLeft \u03b3 (condKernelReal \u03c1) ** ext a s hs : 2 ** case h.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1\u271d : Measure (\u03b1 \u00d7 \u211d) inst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d \u03b3 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 a : \u03b3 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(const \u03b3 \u03c1) a) s = \u2191\u2191(\u2191(const \u03b3 (Measure.fst \u03c1) \u2297\u2096 prodMkLeft \u03b3 (condKernelReal \u03c1)) a) s ** rw [kernel.compProd_apply _ _ _ hs, kernel.const_apply, kernel.const_apply] ** case h.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1\u271d : Measure (\u03b1 \u00d7 \u211d) inst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d \u03b3 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 a : \u03b3 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2191\u2191\u03c1 s = \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(prodMkLeft \u03b3 (condKernelReal \u03c1)) (a, b)) {c | (b, c) \u2208 s} \u2202Measure.fst \u03c1 ** simp_rw [kernel.prodMkLeft_apply] ** case h.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1\u271d : Measure (\u03b1 \u00d7 \u211d) inst\u271d\u00b2 : IsFiniteMeasure \u03c1\u271d \u03b3 : Type u_2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 a : \u03b3 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2191\u2191\u03c1 s = \u222b\u207b (b : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) b) {c | (b, c) \u2208 s} \u2202Measure.fst \u03c1 ** rw [lintegral_condKernelReal_mem \u03c1 hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_bdd_liminf_atTop_of_snorm_bdd ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** by_cases hp' : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R hp' : \u00acp = \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** filter_upwards [ae_bdd_liminf_atTop_rpow_of_snorm_bdd hfmeas hbdd] with x hx ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R hp' : \u00acp = \u22a4 x : \u03b1 hx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4 \u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** have hppos : 0 < p.toReal := ENNReal.toReal_pos hp hp' ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R hp' : \u00acp = \u22a4 x : \u03b1 hx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4 hppos : 0 < ENNReal.toReal p \u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** have :\n  liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) atTop =\n    liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop ^ p.toReal := by\n  change\n    liminf (fun n => ENNReal.orderIsoRpow p.toReal hppos (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop =\n      ENNReal.orderIsoRpow p.toReal hppos (liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop)\n  refine' (OrderIso.liminf_apply (ENNReal.orderIsoRpow p.toReal _) _ _ _ _).symm <;>\n    isBoundedDefault ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R hp' : \u00acp = \u22a4 x : \u03b1 hx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4 hppos : 0 < ENNReal.toReal p this : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p \u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** rw [this] at hx ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R hp' : \u00acp = \u22a4 x : \u03b1 hx : liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p < \u22a4 hppos : 0 < ENNReal.toReal p this : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p \u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** rw [\u2190 ENNReal.rpow_one (liminf (fun n => \u2016f n x\u2016\u208a) atTop), \u2190 mul_inv_cancel hppos.ne.symm,\n  ENNReal.rpow_mul] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R hp' : \u00acp = \u22a4 x : \u03b1 hx : liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p < \u22a4 hppos : 0 < ENNReal.toReal p this : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p \u22a2 (liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p) ^ (ENNReal.toReal p)\u207b\u00b9 < \u22a4 ** exact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.2 hppos.le) hx.ne ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R hp' : p = \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** subst hp' ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hp : \u22a4 \u2260 0 hbdd : \u2200 (n : \u2115), snorm (f n) \u22a4 \u03bc \u2264 \u2191R \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** simp_rw [snorm_exponent_top] at hbdd ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hp : \u22a4 \u2260 0 hbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** have : \u2200 n, \u2200\u1d50 x \u2202\u03bc, (\u2016f n x\u2016\u208a : \u211d\u22650\u221e) < R + 1 := fun n =>\n  ae_lt_of_essSup_lt\n    (lt_of_le_of_lt (hbdd n) <| ENNReal.lt_add_right ENNReal.coe_ne_top one_ne_zero) ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hp : \u22a4 \u2260 0 hbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R this : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2016f n x\u2016\u208a < \u2191R + 1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** rw [\u2190 ae_all_iff] at this ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hp : \u22a4 \u2260 0 hbdd : \u2200 (n : \u2115), snormEssSup (f n) \u03bc \u2264 \u2191R this : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2016f i a\u2016\u208a < \u2191R + 1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop < \u22a4 ** filter_upwards [this] with x hx using lt_of_le_of_lt\n    (liminf_le_of_frequently_le' <| frequently_of_forall fun n => (hx n).le)\n    (ENNReal.add_lt_top.2 \u27e8ENNReal.coe_lt_top, ENNReal.one_lt_top\u27e9) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R hp' : \u00acp = \u22a4 x : \u03b1 hx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4 hppos : 0 < ENNReal.toReal p \u22a2 liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop = liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop ^ ENNReal.toReal p ** change\n  liminf (fun n => ENNReal.orderIsoRpow p.toReal hppos (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop =\n    ENNReal.orderIsoRpow p.toReal hppos (liminf (fun n => (\u2016f n x\u2016\u208a : \u211d\u22650\u221e)) atTop) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p\u271d : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedAddCommGroup G inst\u271d\u00b9 : MeasurableSpace E inst\u271d : OpensMeasurableSpace E R : \u211d\u22650 p : \u211d\u22650\u221e hp : p \u2260 0 f : \u2115 \u2192 \u03b1 \u2192 E hfmeas : \u2200 (n : \u2115), Measurable (f n) hbdd : \u2200 (n : \u2115), snorm (f n) p \u03bc \u2264 \u2191R hp' : \u00acp = \u22a4 x : \u03b1 hx : liminf (fun n => \u2191\u2016f n x\u2016\u208a ^ ENNReal.toReal p) atTop < \u22a4 hppos : 0 < ENNReal.toReal p \u22a2 liminf (fun n => \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) \u2191\u2016f n x\u2016\u208a) atTop =     \u2191(ENNReal.orderIsoRpow (ENNReal.toReal p) hppos) (liminf (fun n => \u2191\u2016f n x\u2016\u208a) atTop) ** refine' (OrderIso.liminf_apply (ENNReal.orderIsoRpow p.toReal _) _ _ _ _).symm <;>\n  isBoundedDefault ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.integral_eq_sum ** \u03b1 : Type u_1 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSingletonClass \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E inst\u271d : Fintype \u03b1 p : PMF \u03b1 f : \u03b1 \u2192 E \u22a2 \u222b (a : \u03b1), f a \u2202toMeasure p = \u2211 a : \u03b1, ENNReal.toReal (\u2191p a) \u2022 f a ** rw [integral_fintype _ (integrable_of_fintype _ f)] ** \u03b1 : Type u_1 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSingletonClass \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E inst\u271d : Fintype \u03b1 p : PMF \u03b1 f : \u03b1 \u2192 E \u22a2 \u2211 x : \u03b1, ENNReal.toReal (\u2191\u2191(toMeasure p) {x}) \u2022 f x = \u2211 a : \u03b1, ENNReal.toReal (\u2191p a) \u2022 f a ** congr with x ** case e_f.h \u03b1 : Type u_1 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSingletonClass \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E inst\u271d : Fintype \u03b1 p : PMF \u03b1 f : \u03b1 \u2192 E x : \u03b1 \u22a2 ENNReal.toReal (\u2191\u2191(toMeasure p) {x}) \u2022 f x = ENNReal.toReal (\u2191p x) \u2022 f x ** congr ** case e_f.h.e_a.e_a \u03b1 : Type u_1 inst\u271d\u2075 : MeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSingletonClass \u03b1 E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E inst\u271d : Fintype \u03b1 p : PMF \u03b1 f : \u03b1 \u2192 E x : \u03b1 \u22a2 \u2191\u2191(toMeasure p) {x} = \u2191p x ** exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _) ** Qed",
        "informal": ""
    },
    {
        "formal": "BoundedContinuousFunction.range_toLpHom ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : Fact (1 \u2264 p) \u22a2 NormedAddGroupHom.range (toLpHom p \u03bc) = Lp.boundedContinuousFunction E p \u03bc ** symm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : Fact (1 \u2264 p) \u22a2 Lp.boundedContinuousFunction E p \u03bc = NormedAddGroupHom.range (toLpHom p \u03bc) ** convert AddMonoidHom.addSubgroupOf_range_eq_of_le\n    ((ContinuousMap.toAEEqFunAddHom \u03bc).comp (toContinuousMapAddHom \u03b1 E))\n    (by rintro - \u27e8f, rfl\u27e9; exact mem_Lp f : _ \u2264 Lp E p \u03bc) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : Fact (1 \u2264 p) \u22a2 AddMonoidHom.range (AddMonoidHom.comp (ContinuousMap.toAEEqFunAddHom \u03bc) (toContinuousMapAddHom \u03b1 E)) \u2264 Lp E p ** rintro - \u27e8f, rfl\u27e9 ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 inst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : Fact (1 \u2264 p) f : \u03b1 \u2192\u1d47 E \u22a2 \u2191(AddMonoidHom.comp (ContinuousMap.toAEEqFunAddHom \u03bc) (toContinuousMapAddHom \u03b1 E)) f \u2208 Lp E p ** exact mem_Lp f ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integrableOn_deriv_right_of_nonneg ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x \u22a2 IntegrableOn g' (Ioc a b) ** by_cases hab : a < b ** case pos \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b \u22a2 IntegrableOn g' (Ioc a b)  case neg \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : \u00aca < b \u22a2 IntegrableOn g' (Ioc a b) ** swap ** case pos \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b \u22a2 IntegrableOn g' (Ioc a b) ** rw [integrableOn_Ioc_iff_integrableOn_Ioo] ** case pos \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b \u22a2 IntegrableOn g' (Ioo a b) ** have meas_g' : AEMeasurable g' (volume.restrict (Ioo a b)) := by\n  apply (aemeasurable_derivWithin_Ioi g _).congr\n  refine' (ae_restrict_mem measurableSet_Ioo).mono fun x hx => _\n  exact (hderiv x hx).derivWithin (uniqueDiffWithinAt_Ioi _) ** case pos \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' \u22a2 IntegrableOn g' (Ioo a b) ** suffices H : (\u222b\u207b x in Ioo a b, \u2016g' x\u2016\u208a) \u2264 ENNReal.ofReal (g b - g a) from\n  \u27e8meas_g'.aestronglyMeasurable, H.trans_lt ENNReal.ofReal_lt_top\u27e9 ** case pos \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' \u22a2 \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a \u2264 ENNReal.ofReal (g b - g a) ** by_contra' H ** case pos \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a \u22a2 False ** obtain \u27e8f, fle, fint, hf\u27e9 :\n  \u2203 f : SimpleFunc \u211d \u211d\u22650,\n    (\u2200 x, f x \u2264 \u2016g' x\u2016\u208a) \u2227\n      (\u222b\u207b x : \u211d in Ioo a b, f x) < \u221e \u2227 ENNReal.ofReal (g b - g a) < \u222b\u207b x : \u211d in Ioo a b, f x :=\n  exists_lt_lintegral_simpleFunc_of_lt_lintegral H ** case pos.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 hf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) \u22a2 False ** let F : \u211d \u2192 \u211d := (\u2191) \u2218 f ** case pos.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 hf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f \u22a2 False ** have intF : IntegrableOn F (Ioo a b) := by\n  refine' \u27e8f.measurable.coe_nnreal_real.aestronglyMeasurable, _\u27e9\n  simpa only [HasFiniteIntegral, comp_apply, NNReal.nnnorm_eq] using fint ** case pos.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 hf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f intF : IntegrableOn F (Ioo a b) \u22a2 False ** have A : \u222b\u207b x : \u211d in Ioo a b, f x = ENNReal.ofReal (\u222b x in Ioo a b, F x) :=\n  lintegral_coe_eq_integral _ intF ** case pos.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 hf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f intF : IntegrableOn F (Ioo a b) A : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) \u22a2 False ** rw [A] at hf ** case pos.intro.intro.intro \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) intF : IntegrableOn F (Ioo a b) A : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) B : \u222b (x : \u211d) in Ioo a b, F x \u2264 g b - g a \u22a2 False ** exact lt_irrefl _ (hf.trans_le (ENNReal.ofReal_le_ofReal B)) ** case neg \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : \u00aca < b \u22a2 IntegrableOn g' (Ioc a b) ** simp [Ioc_eq_empty hab] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b \u22a2 AEMeasurable g' ** apply (aemeasurable_derivWithin_Ioi g _).congr ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b \u22a2 (fun x => derivWithin g (Ioi x) x) =\u1d50[Measure.restrict volume (Ioo a b)] g' ** refine' (ae_restrict_mem measurableSet_Ioo).mono fun x hx => _ ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b x : \u211d hx : x \u2208 Ioo a b \u22a2 (fun x => derivWithin g (Ioi x) x) x = g' x ** exact (hderiv x hx).derivWithin (uniqueDiffWithinAt_Ioi _) ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 hf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f \u22a2 IntegrableOn F (Ioo a b) ** refine' \u27e8f.measurable.coe_nnreal_real.aestronglyMeasurable, _\u27e9 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 hf : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f \u22a2 HasFiniteIntegral F ** simpa only [HasFiniteIntegral, comp_apply, NNReal.nnnorm_eq] using fint ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) intF : IntegrableOn F (Ioo a b) A : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) \u22a2 \u222b (x : \u211d) in Ioo a b, F x \u2264 g b - g a ** rw [\u2190 integral_Ioc_eq_integral_Ioo, \u2190 intervalIntegral.integral_of_le hab.le] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) intF : IntegrableOn F (Ioo a b) A : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) \u22a2 \u222b (x : \u211d) in a..b, F x \u2264 g b - g a ** refine integral_le_sub_of_hasDeriv_right_of_le hab.le hcont hderiv ?_ fun x hx => ?_ ** case refine_1 \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) intF : IntegrableOn F (Ioo a b) A : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) \u22a2 IntegrableOn (fun x => F x) (Icc a b) ** rwa [integrableOn_Icc_iff_integrableOn_Ioo] ** case refine_2 \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) intF : IntegrableOn F (Ioo a b) A : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) x : \u211d hx : x \u2208 Ioo a b \u22a2 F x \u2264 g' x ** convert NNReal.coe_le_coe.2 (fle x) ** case h.e'_4 \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F\u271d : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f\u271d f' : \u211d \u2192 E a b : \u211d hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x g'pos : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 0 \u2264 g' x hab : a < b meas_g' : AEMeasurable g' H : ENNReal.ofReal (g b - g a) < \u222b\u207b (x : \u211d) in Ioo a b, \u2191\u2016g' x\u2016\u208a f : SimpleFunc \u211d \u211d\u22650 fle : \u2200 (x : \u211d), \u2191f x \u2264 \u2016g' x\u2016\u208a fint : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) < \u22a4 F : \u211d \u2192 \u211d := NNReal.toReal \u2218 \u2191f hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) intF : IntegrableOn F (Ioo a b) A : \u222b\u207b (x : \u211d) in Ioo a b, \u2191(\u2191f x) = ENNReal.ofReal (\u222b (x : \u211d) in Ioo a b, F x) x : \u211d hx : x \u2208 Ioo a b \u22a2 g' x = \u2191\u2016g' x\u2016\u208a ** simp only [Real.norm_of_nonneg (g'pos x hx), coe_nnnorm] ** Qed",
        "informal": ""
    },
    {
        "formal": "pi_le_borel_pi ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 \u03b9 : Type u_6 \u03c0 : \u03b9 \u2192 Type u_7 inst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03c0 i) inst\u271d : \u2200 (i : \u03b9), BorelSpace (\u03c0 i) \u22a2 MeasurableSpace.pi \u2264 borel ((i : \u03b9) \u2192 \u03c0 i) ** have : \u2039\u2200 i, MeasurableSpace (\u03c0 i)\u203a = fun i => borel (\u03c0 i) :=\n  funext fun i => BorelSpace.measurable_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 \u03b9 : Type u_6 \u03c0 : \u03b9 \u2192 Type u_7 inst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03c0 i) inst\u271d : \u2200 (i : \u03b9), BorelSpace (\u03c0 i) this : inst\u271d\u00b9 = fun i => borel (\u03c0 i) \u22a2 MeasurableSpace.pi \u2264 borel ((i : \u03b9) \u2192 \u03c0 i) ** exact iSup_le fun i => comap_le_iff_le_map.2 <| (continuous_apply i).borel_measurable ** Qed",
        "informal": ""
    },
    {
        "formal": "List.TProd.ext ** \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i\u271d j : \u03b9 l : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 i : \u03b9 is : List \u03b9 hl : Nodup (i :: is) v w : TProd \u03b1 (i :: is) hvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi \u22a2 v = w ** apply Prod.ext ** case h\u2081 \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i\u271d j : \u03b9 l : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 i : \u03b9 is : List \u03b9 hl : Nodup (i :: is) v w : TProd \u03b1 (i :: is) hvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi \u22a2 v.1 = w.1  case h\u2082 \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i\u271d j : \u03b9 l : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 i : \u03b9 is : List \u03b9 hl : Nodup (i :: is) v w : TProd \u03b1 (i :: is) hvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi \u22a2 v.2 = w.2 ** rw [\u2190 elim_self v, hvw, elim_self] ** case h\u2082 \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i\u271d j : \u03b9 l : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 i : \u03b9 is : List \u03b9 hl : Nodup (i :: is) v w : TProd \u03b1 (i :: is) hvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi \u22a2 v.2 = w.2 ** refine' ext (nodup_cons.mp hl).2 fun j hj => _ ** case h\u2082 \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i\u271d j\u271d : \u03b9 l : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 i : \u03b9 is : List \u03b9 hl : Nodup (i :: is) v w : TProd \u03b1 (i :: is) hvw : \u2200 (i_1 : \u03b9) (hi : i_1 \u2208 i :: is), TProd.elim v hi = TProd.elim w hi j : \u03b9 hj : j \u2208 is \u22a2 TProd.elim v.2 hj = TProd.elim w.2 hj ** rw [\u2190 elim_of_mem hl, hvw, elim_of_mem hl] ** Qed",
        "informal": ""
    },
    {
        "formal": "WithTop.image_coe_Iic ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Iic a = Iic \u2191a ** rw [\u2190 preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Quot.subsingleton_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop \u22a2 Subsingleton (Quot r) \u2194 EqvGen r = \u22a4 ** simp only [_root_.subsingleton_iff, _root_.eq_top_iff, Pi.le_def, Pi.top_apply, forall_const] ** \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop \u22a2 (\u2200 (x y : Quot r), x = y) \u2194 \u2200 (i i_1 : \u03b1), \u22a4 \u2264 EqvGen r i i_1 ** refine' (surjective_quot_mk _).forall.trans (forall_congr' fun a => _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop a : \u03b1 \u22a2 (\u2200 (y : Quot r), mk r a = y) \u2194 \u2200 (i : \u03b1), \u22a4 \u2264 EqvGen r a i ** refine' (surjective_quot_mk _).forall.trans (forall_congr' fun b => _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop a b : \u03b1 \u22a2 mk r a = mk r b \u2194 \u22a4 \u2264 EqvGen r a b ** rw [Quot.eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop a b : \u03b1 \u22a2 EqvGen r a b \u2194 \u22a4 \u2264 EqvGen r a b ** simp only [forall_const, le_Prop_eq, OrderTop.toTop, Pi.orderTop, Pi.top_apply,\n           Prop.top_eq_true, true_implies] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEMeasurable.ae_eq_of_forall_set_lintegral_eq ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u22a2 f =\u1d50[\u03bc] g ** refine'\n  ENNReal.eventuallyEq_of_toReal_eventuallyEq (ae_lt_top' hf hfi).ne_of_lt\n    (ae_lt_top' hg hgi).ne_of_lt\n    (Integrable.ae_eq_of_forall_set_integral_eq _ _\n      (integrable_toReal_of_lintegral_ne_top hf hfi)\n      (integrable_toReal_of_lintegral_ne_top hg hgi) fun s hs hs' => _) ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, ENNReal.toReal (f x) \u2202\u03bc = \u222b (x : \u03b1) in s, ENNReal.toReal (g x) \u2202\u03bc ** rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae] ** case hf \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 0 \u2264\u1d50[Measure.restrict \u03bc s] fun x => ENNReal.toReal (g x)  case hfm \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) (Measure.restrict \u03bc s)  case hf \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 0 \u2264\u1d50[Measure.restrict \u03bc s] fun x => ENNReal.toReal (f x)  case hfm \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (f x)) (Measure.restrict \u03bc s) ** exacts [ae_of_all _ fun x => ENNReal.toReal_nonneg,\n  hg.ennreal_toReal.restrict.aestronglyMeasurable, ae_of_all _ fun x => ENNReal.toReal_nonneg,\n  hf.ennreal_toReal.restrict.aestronglyMeasurable] ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (f a)) \u2202\u03bc) =     ENNReal.toReal (\u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) ** congr 1 ** case e_a \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (f a)) \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc ** rw [lintegral_congr_ae (ofReal_toReal_ae_eq _), lintegral_congr_ae (ofReal_toReal_ae_eq _)] ** case e_a \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b\u207b (a : \u03b1) in s, f a \u2202\u03bc = \u222b\u207b (a : \u03b1) in s, g a \u2202\u03bc ** exact hfg hs hs' ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, g x < \u22a4 ** refine' ae_lt_top' hg.restrict (ne_of_lt (lt_of_le_of_lt _ hgi.lt_top)) ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x \u2202\u03bc ** exact @set_lintegral_univ \u03b1 _ \u03bc g \u25b8 lintegral_mono_set (Set.subset_univ _) ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x < \u22a4 ** refine' ae_lt_top' hf.restrict (ne_of_lt (lt_of_le_of_lt _ hfi.lt_top)) ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hg : AEMeasurable g hfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 hgi : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4 hfg : \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc s : Set \u03b1 hs : MeasurableSet s hs' : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), f x \u2202\u03bc ** exact @set_lintegral_univ \u03b1 _ \u03bc f \u25b8 lintegral_mono_set (Set.subset_univ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "List.lookup_cons_self ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 b : \u03b2\u271d es : List (\u03b1 \u00d7 \u03b2\u271d) inst\u271d\u00b9 : BEq \u03b1 inst\u271d : LawfulBEq \u03b1 k : \u03b1 \u22a2 lookup k ((k, b) :: es) = some b ** simp [lookup_cons] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.sub.aux ** a b : Rat g ad bd : Nat hg : g = Nat.gcd a.den b.den had : ad = a.den / g hbd : bd = b.den / g \u22a2 let den := ad * b.den;   let num := a.num * \u2191bd - b.num * \u2191ad;   Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den ** have := add.aux a (-b) hg had hbd ** a b : Rat g ad bd : Nat hg : g = Nat.gcd a.den b.den had : ad = a.den / g hbd : bd = b.den / g this :   let den := ad * (-b).den;   let num := a.num * \u2191bd + (-b).num * \u2191ad;   Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den \u22a2 let den := ad * b.den;   let num := a.num * \u2191bd - b.num * \u2191ad;   Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den ** simp only [show (-b).num = -b.num from rfl, Int.neg_mul] at this ** a b : Rat g ad bd : Nat hg : g = Nat.gcd a.den b.den had : ad = a.den / g hbd : bd = b.den / g this :   Nat.gcd (Int.natAbs (a.num * \u2191bd + -(b.num * \u2191ad))) g =     Nat.gcd (Int.natAbs (a.num * \u2191bd + -(b.num * \u2191ad))) (ad * (-b).den) \u22a2 let den := ad * b.den;   let num := a.num * \u2191bd - b.num * \u2191ad;   Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den ** exact this ** Qed",
        "informal": ""
    },
    {
        "formal": "ManyOneDegree.le_antisymm ** \u03b1 : Type u inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Inhabited \u03b1 \u03b2 : Type v inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Inhabited \u03b2 \u03b3 : Type w inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Inhabited \u03b3 d\u2081 d\u2082 : ManyOneDegree \u22a2 d\u2081 \u2264 d\u2082 \u2192 d\u2082 \u2264 d\u2081 \u2192 d\u2081 = d\u2082 ** induction d\u2081 using ManyOneDegree.ind_on ** case h \u03b1 : Type u inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Inhabited \u03b1 \u03b2 : Type v inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Inhabited \u03b2 \u03b3 : Type w inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Inhabited \u03b3 d\u2082 : ManyOneDegree p\u271d : Set \u2115 \u22a2 of p\u271d \u2264 d\u2082 \u2192 d\u2082 \u2264 of p\u271d \u2192 of p\u271d = d\u2082 ** induction d\u2082 using ManyOneDegree.ind_on ** case h.h \u03b1 : Type u inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Inhabited \u03b1 \u03b2 : Type v inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Inhabited \u03b2 \u03b3 : Type w inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Inhabited \u03b3 p\u271d\u00b9 p\u271d : Set \u2115 \u22a2 of p\u271d\u00b9 \u2264 of p\u271d \u2192 of p\u271d \u2264 of p\u271d\u00b9 \u2192 of p\u271d\u00b9 = of p\u271d ** intro hp hq ** case h.h \u03b1 : Type u inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Inhabited \u03b1 \u03b2 : Type v inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Inhabited \u03b2 \u03b3 : Type w inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Inhabited \u03b3 p\u271d\u00b9 p\u271d : Set \u2115 hp : of p\u271d\u00b9 \u2264 of p\u271d hq : of p\u271d \u2264 of p\u271d\u00b9 \u22a2 of p\u271d\u00b9 = of p\u271d ** simp_all only [ManyOneEquiv, of_le_of, of_eq_of, true_and_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.fold_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 op : \u03b2 \u2192 \u03b2 \u2192 \u03b2 hc : IsCommutative \u03b2 op ha : IsAssociative \u03b2 op f : \u03b1 \u2192 \u03b2 b : \u03b2 s : Finset \u03b1 a : \u03b1 hd : Decidable (s = \u2205) c : \u03b2 h : op c (op b c) = op b c \u22a2 fold op b (fun x => c) s = if s = \u2205 then b else op b c ** induction' s using Finset.induction_on with x s hx IH generalizing hd ** case empty \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 op : \u03b2 \u2192 \u03b2 \u2192 \u03b2 hc : IsCommutative \u03b2 op ha : IsAssociative \u03b2 op f : \u03b1 \u2192 \u03b2 b : \u03b2 s : Finset \u03b1 a : \u03b1 hd\u271d : Decidable (s = \u2205) c : \u03b2 h : op c (op b c) = op b c hd : Decidable (\u2205 = \u2205) \u22a2 fold op b (fun x => c) \u2205 = if \u2205 = \u2205 then b else op b c ** simp ** case insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 op : \u03b2 \u2192 \u03b2 \u2192 \u03b2 hc : IsCommutative \u03b2 op ha : IsAssociative \u03b2 op f : \u03b1 \u2192 \u03b2 b : \u03b2 s\u271d : Finset \u03b1 a : \u03b1 hd\u271d : Decidable (s\u271d = \u2205) c : \u03b2 h : op c (op b c) = op b c x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c hd : Decidable (insert x s = \u2205) \u22a2 fold op b (fun x => c) (insert x s) = if insert x s = \u2205 then b else op b c ** simp only [Finset.fold_insert hx, IH, if_false, Finset.insert_ne_empty] ** case insert \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 op : \u03b2 \u2192 \u03b2 \u2192 \u03b2 hc : IsCommutative \u03b2 op ha : IsAssociative \u03b2 op f : \u03b1 \u2192 \u03b2 b : \u03b2 s\u271d : Finset \u03b1 a : \u03b1 hd\u271d : Decidable (s\u271d = \u2205) c : \u03b2 h : op c (op b c) = op b c x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c hd : Decidable (insert x s = \u2205) \u22a2 op c (if s = \u2205 then b else op b c) = op b c ** split_ifs ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 op : \u03b2 \u2192 \u03b2 \u2192 \u03b2 hc : IsCommutative \u03b2 op ha : IsAssociative \u03b2 op f : \u03b1 \u2192 \u03b2 b : \u03b2 s\u271d : Finset \u03b1 a : \u03b1 hd\u271d : Decidable (s\u271d = \u2205) c : \u03b2 h : op c (op b c) = op b c x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c hd : Decidable (insert x s = \u2205) h\u271d : s = \u2205 \u22a2 op c b = op b c ** rw [hc.comm] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 op : \u03b2 \u2192 \u03b2 \u2192 \u03b2 hc : IsCommutative \u03b2 op ha : IsAssociative \u03b2 op f : \u03b1 \u2192 \u03b2 b : \u03b2 s\u271d : Finset \u03b1 a : \u03b1 hd\u271d : Decidable (s\u271d = \u2205) c : \u03b2 h : op c (op b c) = op b c x : \u03b1 s : Finset \u03b1 hx : \u00acx \u2208 s IH : \u2200 [hd : Decidable (s = \u2205)], fold op b (fun x => c) s = if s = \u2205 then b else op b c hd : Decidable (insert x s = \u2205) h\u271d : \u00acs = \u2205 \u22a2 op c (op b c) = op b c ** exact h ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.lintegral_mul_le_one_of_lintegral_rpow_eq_one ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1 hg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1 \u22a2 \u222b\u207b (a : \u03b1), f a ^ p / ENNReal.ofReal p + g a ^ q / ENNReal.ofReal q \u2202\u03bc = 1 ** simp only [div_eq_mul_inv] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1 hg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1 \u22a2 \u222b\u207b (a : \u03b1), f a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + g a ^ q * (ENNReal.ofReal q)\u207b\u00b9 \u2202\u03bc = 1 ** rw [lintegral_add_left'] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1 hg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1 \u22a2 \u222b\u207b (a : \u03b1), f a ^ p * (ENNReal.ofReal p)\u207b\u00b9 \u2202\u03bc + \u222b\u207b (a : \u03b1), g a ^ q * (ENNReal.ofReal q)\u207b\u00b9 \u2202\u03bc = 1 ** rw [lintegral_mul_const'' _ (hf.pow_const p), lintegral_mul_const', hf_norm, hg_norm, \u2190\n  div_eq_mul_inv, \u2190 div_eq_mul_inv, hpq.inv_add_inv_conj_ennreal] ** case hr \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1 hg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1 \u22a2 (ENNReal.ofReal q)\u207b\u00b9 \u2260 \u22a4 ** simp [hpq.symm.pos] ** case hf \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_norm : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 1 hg_norm : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 1 \u22a2 AEMeasurable fun a => f a ^ p * (ENNReal.ofReal p)\u207b\u00b9 ** exact (hf.pow_const _).mul_const _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.insert_singleton_eq ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b b' : \u03b2 a \u22a2 insert a b (singleton a b') = singleton a b ** simp only [singleton, Finmap.insert_toFinmap, AList.insert_singleton_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image_add_left_Ioo ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1 inst\u271d\u00b2 : ExistsAddOfLE \u03b1 inst\u271d\u00b9 : LocallyFiniteOrder \u03b1 inst\u271d : DecidableEq \u03b1 a b c : \u03b1 \u22a2 image ((fun x x_1 => x + x_1) c) (Ioo a b) = Ioo (c + a) (c + b) ** rw [\u2190 map_add_left_Ioo, map_eq_image, addLeftEmbedding, Embedding.coeFn_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.PairingHeapImp.Heap.noSibling_tail? ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 \u22a2 tail? le s = some s' \u2192 NoSibling s' ** simp only [Heap.tail?] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 \u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 NoSibling s' ** intro eq ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 eq : Option.map (fun x => x.snd) (deleteMin le s) = some s' \u22a2 NoSibling s' ** match eq\u2082 : s.deleteMin le, eq with\n| some (a, tl), rfl => exact noSibling_deleteMin eq\u2082 ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 eq : Option.map (fun x => x.snd) (deleteMin le s) = some s' a : \u03b1 tl : Heap \u03b1 eq\u2082 : deleteMin le s = some (a, tl) \u22a2 NoSibling ((fun x => x.snd) (a, tl)) ** exact noSibling_deleteMin eq\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "Ctop.Realizer.is_basis ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 \u03c4 : Type u_4 T : TopologicalSpace \u03b1 F : Realizer \u03b1 \u22a2 TopologicalSpace.IsTopologicalBasis (range F.F.f) ** have := toTopsp_isTopologicalBasis F.F ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 \u03c4 : Type u_4 T : TopologicalSpace \u03b1 F : Realizer \u03b1 this : TopologicalSpace.IsTopologicalBasis (range F.F.f) \u22a2 TopologicalSpace.IsTopologicalBasis (range F.F.f) ** rwa [F.eq] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "PFun.mem_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 f : \u03b1 \u2192. \u03b2 s : Set \u03b1 h : s \u2286 Dom f a : \u03b1 b : \u03b2 \u22a2 b \u2208 restrict f h a \u2194 a \u2208 s \u2227 b \u2208 f a ** simp [restrict] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.simpleFunc.coeFn_nonneg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 G : Type u_7 inst\u271d : NormedLatticeAddCommGroup G f : { x // x \u2208 simpleFunc G p \u03bc } \u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191f \u2194 0 \u2264 f ** rw [\u2190 Subtype.coe_le_coe, Lp.coeFn_nonneg, AddSubmonoid.coe_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZNum.gcd_to_nat ** a b : ZNum \u22a2 Nat.gcd \u2191(abs a) \u2191(abs b) = Int.gcd \u2191a \u2191b ** simp ** a b : ZNum \u22a2 Nat.gcd (natAbs \u2191a) (natAbs \u2191b) = Int.gcd \u2191a \u2191b ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.mk_eq_divInt ** num : Int den : Nat nz : den \u2260 0 c : Nat.Coprime (Int.natAbs num) den \u22a2 mk' num den = num /. \u2191den ** simp [mk_eq_mkRat] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.CountablyGenerated.sup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s t u : Set \u03b1 m\u2081 m\u2082 : MeasurableSpace \u03b2 h\u2081 : CountablyGenerated \u03b2 h\u2082 : CountablyGenerated \u03b2 \u22a2 CountablyGenerated \u03b2 ** rcases h\u2081 with \u27e8\u27e8b\u2081, hb\u2081c, rfl\u27e9\u27e9 ** case mk.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s t u : Set \u03b1 m\u2082 : MeasurableSpace \u03b2 h\u2082 : CountablyGenerated \u03b2 b\u2081 : Set (Set \u03b2) hb\u2081c : Set.Countable b\u2081 \u22a2 CountablyGenerated \u03b2 ** rcases h\u2082 with \u27e8\u27e8b\u2082, hb\u2082c, rfl\u27e9\u27e9 ** case mk.intro.intro.mk.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s t u : Set \u03b1 b\u2081 : Set (Set \u03b2) hb\u2081c : Set.Countable b\u2081 b\u2082 : Set (Set \u03b2) hb\u2082c : Set.Countable b\u2082 \u22a2 CountablyGenerated \u03b2 ** exact @mk _ (_ \u2294 _) \u27e8_, hb\u2081c.union hb\u2082c, generateFrom_sup_generateFrom\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "List.foldr_inf_eq_inf_toFinset ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : OrderTop \u03b1 s s\u2081 s\u2082 : Finset \u03b2 f g : \u03b2 \u2192 \u03b1 a : \u03b1 inst\u271d : DecidableEq \u03b1 l : List \u03b1 \u22a2 List.foldr (fun x x_1 => x \u2293 x_1) \u22a4 l = inf (List.toFinset l) id ** rw [\u2190 coe_fold_r, \u2190 Multiset.fold_dedup_idem, inf_def, \u2190 List.toFinset_coe, toFinset_val,\n  Multiset.map_id] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : SemilatticeInf \u03b1 inst\u271d\u00b9 : OrderTop \u03b1 s s\u2081 s\u2082 : Finset \u03b2 f g : \u03b2 \u2192 \u03b1 a : \u03b1 inst\u271d : DecidableEq \u03b1 l : List \u03b1 \u22a2 Multiset.fold (fun x x_1 => x \u2293 x_1) \u22a4 (dedup \u2191l) = Multiset.inf (dedup \u2191l) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.compProd_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t \u22a2 kernel.restrict \u03ba hs \u2297\u2096 kernel.restrict \u03b7 ht = kernel.restrict (\u03ba \u2297\u2096 \u03b7) (_ : MeasurableSet (s \u00d7\u02e2 t)) ** ext a u hu ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u \u22a2 \u2191\u2191(\u2191(kernel.restrict \u03ba hs \u2297\u2096 kernel.restrict \u03b7 ht) a) u =     \u2191\u2191(\u2191(kernel.restrict (\u03ba \u2297\u2096 \u03b7) (_ : MeasurableSet (s \u00d7\u02e2 t))) a) u ** rw [compProd_apply _ _ _ hu, restrict_apply' _ _ _ hu,\n  compProd_apply _ _ _ (hu.inter (hs.prod ht))] ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u \u22a2 \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191(kernel.restrict \u03b7 ht) (a, b)) {c | (b, c) \u2208 u} \u2202\u2191(kernel.restrict \u03ba hs) a =     \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2229 s \u00d7\u02e2 t} \u2202\u2191\u03ba a ** simp only [kernel.restrict_apply, Measure.restrict_apply' ht, Set.mem_inter_iff,\n  Set.prod_mk_mem_set_prod_eq] ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u this :   \u2200 (b : \u03b2),     \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b \u22a2 \u222b\u207b (b : \u03b2) in s, \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) \u2202\u2191\u03ba a =     \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} \u2202\u2191\u03ba a ** simp_rw [this] ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u this :   \u2200 (b : \u03b2),     \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b \u22a2 \u222b\u207b (b : \u03b2) in s, \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) \u2202\u2191\u03ba a =     \u222b\u207b (b : \u03b2), Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b \u2202\u2191\u03ba a ** rw [lintegral_indicator _ hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u \u22a2 \u2200 (b : \u03b2),     \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b ** intro b ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u b : \u03b2 \u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = Set.indicator s (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t)) b ** rw [Set.indicator_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u b : \u03b2 \u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = if b \u2208 s then \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) else 0 ** split_ifs with h ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u b : \u03b2 h : b \u2208 s \u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) ** simp only [h, true_and_iff] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u b : \u03b2 h : b \u2208 s \u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 c \u2208 t} = \u2191\u2191(\u2191\u03b7 (a, b)) ({c | (b, c) \u2208 u} \u2229 t) ** rfl ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s\u271d : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d : \u03b1 s : Set \u03b2 t : Set \u03b3 hs : MeasurableSet s ht : MeasurableSet t a : \u03b1 u : Set (\u03b2 \u00d7 \u03b3) hu : MeasurableSet u b : \u03b2 h : \u00acb \u2208 s \u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 u \u2227 b \u2208 s \u2227 c \u2208 t} = 0 ** simp only [h, false_and_iff, and_false_iff, Set.setOf_false, measure_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ofReal_setAverage ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f : \u03b1 \u2192 \u211d hf : IntegrableOn f s hf\u2080 : 0 \u2264\u1da0[ae (Measure.restrict \u03bc s)] f \u22a2 ENNReal.ofReal (\u2a0d (x : \u03b1) in s, f x \u2202\u03bc) = (\u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (f x) \u2202\u03bc) / \u2191\u2191\u03bc s ** simpa using ofReal_average hf hf\u2080 ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.mul_inv_cancel_aux ** n a\u271d p : \u2115 inst\u271d : Fact (Nat.Prime p) a : ZMod p h : a \u2260 0 \u22a2 a * a\u207b\u00b9 = 1 ** obtain \u27e8k, rfl\u27e9 := nat_cast_zmod_surjective a ** case intro n a p : \u2115 inst\u271d : Fact (Nat.Prime p) k : \u2115 h : \u2191k \u2260 0 \u22a2 \u2191k * (\u2191k)\u207b\u00b9 = 1 ** apply coe_mul_inv_eq_one ** case intro.h n a p : \u2115 inst\u271d : Fact (Nat.Prime p) k : \u2115 h : \u2191k \u2260 0 \u22a2 Nat.Coprime k p ** apply Nat.Coprime.symm ** case intro.h.a n a p : \u2115 inst\u271d : Fact (Nat.Prime p) k : \u2115 h : \u2191k \u2260 0 \u22a2 Nat.Coprime p k ** rwa [Nat.Prime.coprime_iff_not_dvd Fact.out, \u2190 CharP.cast_eq_zero_iff (ZMod p)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.swap_def ** \u03b1 : Type ?u.17701 a : Array \u03b1 i j : Fin (size a) \u22a2 j.val < size (set a i (get a j)) ** simp [j.2] ** \u03b1 : Type u_1 a : Array \u03b1 i j : Fin (size a) \u22a2 swap a i j = set (set a i (get a j)) { val := j.val, isLt := (_ : j.val < size (set a i a[j.val])) } (get a i) ** simp [swap, fin_cast_val] ** Qed",
        "informal": ""
    },
    {
        "formal": "parallelepiped_eq_convexHull ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : Fintype \u03b9' inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : AddCommGroup F inst\u271d : Module \u211d F v : \u03b9 \u2192 E \u22a2 parallelepiped v = \u2191(convexHull \u211d) (\u2211 i : \u03b9, {0, v i}) ** simp_rw [convexHull_sum, convexHull_pair, parallelepiped_eq_sum_segment] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.univ_pow ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : Monoid \u03b1 s t : Set \u03b1 a : \u03b1 m n\u271d n : \u2115 x\u271d : n + 2 \u2260 0 \u22a2 univ ^ (n + 2) = univ ** rw [pow_succ, univ_pow n.succ_ne_zero, univ_mul_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_image_eq_lintegral_abs_det_fderiv_mul ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (x : E) in f '' s, g x \u2202\u03bc = \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc ** rw [\u2190 restrict_map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs hf' hf,\n  (measurableEmbedding_of_fderivWithin hs hf' hf).lintegral_map] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (a : \u2191s),       g         (Set.restrict s f           a) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =     \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc ** have : \u2200 x : s, g (s.restrict f x) = (g \u2218 f) x := fun x => rfl ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 \u211d\u22650\u221e this : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x \u22a2 \u222b\u207b (a : \u2191s),       g         (Set.restrict s f           a) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =     \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc ** simp only [this] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 \u211d\u22650\u221e this : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x \u22a2 \u222b\u207b (a : \u2191s),       (g \u2218 f) \u2191a \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =     \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc ** rw [\u2190 (MeasurableEmbedding.subtype_coe hs).lintegral_map, map_comap_subtype_coe hs,\n  set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable\u2080 _ _ _ hs] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 \u211d\u22650\u221e this : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x \u22a2 \u222b\u207b (a : E) in s, ((fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) * fun a => (g \u2218 f) a) a \u2202\u03bc =     \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc ** rfl ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 \u211d\u22650\u221e this : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x \u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| < \u22a4 ** simp only [eventually_true, ENNReal.ofReal_lt_top] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E \u2192 \u211d\u22650\u221e this : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x \u22a2 AEMeasurable fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)| ** exact aemeasurable_ofReal_abs_det_fderivWithin \u03bc hs hf' ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.MulAntidiagonal.finite_of_isPwo ** \u03b1 : Type u_1 inst\u271d : OrderedCancelCommMonoid \u03b1 s t : Set \u03b1 a\u271d : \u03b1 x y : \u2191(mulAntidiagonal s t a\u271d) hs : IsPwo s ht : IsPwo t a : \u03b1 \u22a2 Set.Finite (mulAntidiagonal s t a) ** refine' not_infinite.1 fun h => _ ** \u03b1 : Type u_1 inst\u271d : OrderedCancelCommMonoid \u03b1 s t : Set \u03b1 a\u271d : \u03b1 x y : \u2191(mulAntidiagonal s t a\u271d) hs : IsPwo s ht : IsPwo t a : \u03b1 h : Set.Infinite (mulAntidiagonal s t a) h1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) h2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u22a2 False ** obtain \u27e8g, hg\u27e9 :=\n  h1.exists_monotone_subseq (fun n => h.natEmbedding _ n) fun n => (h.natEmbedding _ n).2 ** case intro \u03b1 : Type u_1 inst\u271d : OrderedCancelCommMonoid \u03b1 s t : Set \u03b1 a\u271d : \u03b1 x y : \u2191(mulAntidiagonal s t a\u271d) hs : IsPwo s ht : IsPwo t a : \u03b1 h : Set.Infinite (mulAntidiagonal s t a) h1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) h2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) g : \u2115 \u21aao \u2115 hg :   \u2200 (m n : \u2115),     m \u2264 n \u2192       (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))         \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n)) \u22a2 False ** obtain \u27e8m, n, mn, h2'\u27e9 := h2 (fun x => (h.natEmbedding _) (g x)) fun n => (h.natEmbedding _ _).2 ** case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : OrderedCancelCommMonoid \u03b1 s t : Set \u03b1 a\u271d : \u03b1 x y : \u2191(mulAntidiagonal s t a\u271d) hs : IsPwo s ht : IsPwo t a : \u03b1 h : Set.Infinite (mulAntidiagonal s t a) h1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) h2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) g : \u2115 \u21aao \u2115 hg :   \u2200 (m n : \u2115),     m \u2264 n \u2192       (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))         \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n)) m n : \u2115 mn : m < n h2' :   (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))     \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n)) \u22a2 False ** refine' mn.ne (g.injective <| (h.natEmbedding _).injective _) ** case intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : OrderedCancelCommMonoid \u03b1 s t : Set \u03b1 a\u271d : \u03b1 x y : \u2191(mulAntidiagonal s t a\u271d) hs : IsPwo s ht : IsPwo t a : \u03b1 h : Set.Infinite (mulAntidiagonal s t a) h1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) h2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) g : \u2115 \u21aao \u2115 hg :   \u2200 (m n : \u2115),     m \u2264 n \u2192       (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))         \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n)) m n : \u2115 mn : m < n h2' :   (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))     \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n)) \u22a2 \u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m) = \u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n) ** exact eq_of_fst_le_fst_of_snd_le_snd _ _ _ (hg _ _ mn.le) h2' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.integrable_iff_mem_L1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 Integrable f \u2194 f \u2208 Lp \u03b2 1 ** rw [\u2190 integrable_coeFn, \u2190 mem\u2112p_one_iff_integrable, Lp.mem_Lp_iff_mem\u2112p] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.setToL1S_nonneg ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \u211d F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E G'' : Type u_7 G' : Type u_8 inst\u271d\u00b3 : NormedLatticeAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : NormedLatticeAddCommGroup G'' inst\u271d : NormedSpace \u211d G'' T : Set \u03b1 \u2192 G'' \u2192L[\u211d] G' h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x f : { x // x \u2208 simpleFunc G'' 1 \u03bc } hf : 0 \u2264 f \u22a2 0 \u2264 setToL1S T f ** simp_rw [setToL1S] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \u211d F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E G'' : Type u_7 G' : Type u_8 inst\u271d\u00b3 : NormedLatticeAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : NormedLatticeAddCommGroup G'' inst\u271d : NormedSpace \u211d G'' T : Set \u03b1 \u2192 G'' \u2192L[\u211d] G' h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x f : { x // x \u2208 simpleFunc G'' 1 \u03bc } hf : 0 \u2264 f \u22a2 0 \u2264 SimpleFunc.setToSimpleFunc T (toSimpleFunc f) ** obtain \u27e8f', hf', hff'\u27e9 : \u2203 f' : \u03b1 \u2192\u209b G'', 0 \u2264 f' \u2227 simpleFunc.toSimpleFunc f =\u1d50[\u03bc] f' := by\n  obtain \u27e8f'', hf'', hff''\u27e9 := exists_simpleFunc_nonneg_ae_eq hf\n  exact \u27e8f'', hf'', (Lp.simpleFunc.toSimpleFunc_eq_toFun f).trans hff''\u27e9 ** case intro.intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \u211d F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E G'' : Type u_7 G' : Type u_8 inst\u271d\u00b3 : NormedLatticeAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : NormedLatticeAddCommGroup G'' inst\u271d : NormedSpace \u211d G'' T : Set \u03b1 \u2192 G'' \u2192L[\u211d] G' h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x f : { x // x \u2208 simpleFunc G'' 1 \u03bc } hf : 0 \u2264 f f' : \u03b1 \u2192\u209b G'' hf' : 0 \u2264 f' hff' : \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f' \u22a2 0 \u2264 SimpleFunc.setToSimpleFunc T (toSimpleFunc f) ** rw [SimpleFunc.setToSimpleFunc_congr _ h_zero h_add (SimpleFunc.integrable _) hff'] ** case intro.intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \u211d F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E G'' : Type u_7 G' : Type u_8 inst\u271d\u00b3 : NormedLatticeAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : NormedLatticeAddCommGroup G'' inst\u271d : NormedSpace \u211d G'' T : Set \u03b1 \u2192 G'' \u2192L[\u211d] G' h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x f : { x // x \u2208 simpleFunc G'' 1 \u03bc } hf : 0 \u2264 f f' : \u03b1 \u2192\u209b G'' hf' : 0 \u2264 f' hff' : \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f' \u22a2 0 \u2264 SimpleFunc.setToSimpleFunc (fun s => T s) f' ** exact\n  SimpleFunc.setToSimpleFunc_nonneg' T hT_nonneg _ hf' ((SimpleFunc.integrable f).congr hff') ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \u211d F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E G'' : Type u_7 G' : Type u_8 inst\u271d\u00b3 : NormedLatticeAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : NormedLatticeAddCommGroup G'' inst\u271d : NormedSpace \u211d G'' T : Set \u03b1 \u2192 G'' \u2192L[\u211d] G' h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x f : { x // x \u2208 simpleFunc G'' 1 \u03bc } hf : 0 \u2264 f \u22a2 \u2203 f', 0 \u2264 f' \u2227 \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f' ** obtain \u27e8f'', hf'', hff''\u27e9 := exists_simpleFunc_nonneg_ae_eq hf ** case intro.intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \u211d F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E G'' : Type u_7 G' : Type u_8 inst\u271d\u00b3 : NormedLatticeAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : NormedLatticeAddCommGroup G'' inst\u271d : NormedSpace \u211d G'' T : Set \u03b1 \u2192 G'' \u2192L[\u211d] G' h_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s = 0 \u2192 T s = 0 h_add : FinMeasAdditive \u03bc T hT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G''), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x f : { x // x \u2208 simpleFunc G'' 1 \u03bc } hf : 0 \u2264 f f'' : \u03b1 \u2192\u209b G'' hf'' : 0 \u2264 f'' hff'' : \u2191\u2191\u2191f =\u1d50[\u03bc] \u2191f'' \u22a2 \u2203 f', 0 \u2264 f' \u2227 \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191f' ** exact \u27e8f'', hf'', (Lp.simpleFunc.toSimpleFunc_eq_toFun f).trans hff''\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Measurable.measurableSet_preimage_iff_of_surjective ** \u03b1 : Type u_1 \u03b9 : Type u_2 X : Type u_3 Y : Type u_4 \u03b2 : Type u_5 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : StandardBorelSpace X inst\u271d\u2075 : TopologicalSpace Y inst\u271d\u2074 : T2Space Y inst\u271d\u00b3 : MeasurableSpace Y inst\u271d\u00b2 : OpensMeasurableSpace Y inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : SecondCountableTopology Y f : X \u2192 Y hf : Measurable f hsurj : Surjective f s : Set Y \u22a2 MeasurableSet (f \u207b\u00b9' s) \u2194 MeasurableSet s ** refine \u27e8fun h => ?_, fun h => hf h\u27e9 ** \u03b1 : Type u_1 \u03b9 : Type u_2 X : Type u_3 Y : Type u_4 \u03b2 : Type u_5 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : StandardBorelSpace X inst\u271d\u2075 : TopologicalSpace Y inst\u271d\u2074 : T2Space Y inst\u271d\u00b3 : MeasurableSpace Y inst\u271d\u00b2 : OpensMeasurableSpace Y inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : SecondCountableTopology Y f : X \u2192 Y hf : Measurable f hsurj : Surjective f s : Set Y h : MeasurableSet (f \u207b\u00b9' s) \u22a2 MeasurableSet s ** apply AnalyticSet.measurableSet_of_compl ** case hs \u03b1 : Type u_1 \u03b9 : Type u_2 X : Type u_3 Y : Type u_4 \u03b2 : Type u_5 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : StandardBorelSpace X inst\u271d\u2075 : TopologicalSpace Y inst\u271d\u2074 : T2Space Y inst\u271d\u00b3 : MeasurableSpace Y inst\u271d\u00b2 : OpensMeasurableSpace Y inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : SecondCountableTopology Y f : X \u2192 Y hf : Measurable f hsurj : Surjective f s : Set Y h : MeasurableSet (f \u207b\u00b9' s) \u22a2 AnalyticSet s ** rw [\u2190 image_preimage_eq s hsurj] ** case hs \u03b1 : Type u_1 \u03b9 : Type u_2 X : Type u_3 Y : Type u_4 \u03b2 : Type u_5 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : StandardBorelSpace X inst\u271d\u2075 : TopologicalSpace Y inst\u271d\u2074 : T2Space Y inst\u271d\u00b3 : MeasurableSpace Y inst\u271d\u00b2 : OpensMeasurableSpace Y inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : SecondCountableTopology Y f : X \u2192 Y hf : Measurable f hsurj : Surjective f s : Set Y h : MeasurableSet (f \u207b\u00b9' s) \u22a2 AnalyticSet (f '' (f \u207b\u00b9' s)) ** exact h.analyticSet_image hf ** case hsc \u03b1 : Type u_1 \u03b9 : Type u_2 X : Type u_3 Y : Type u_4 \u03b2 : Type u_5 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : StandardBorelSpace X inst\u271d\u2075 : TopologicalSpace Y inst\u271d\u2074 : T2Space Y inst\u271d\u00b3 : MeasurableSpace Y inst\u271d\u00b2 : OpensMeasurableSpace Y inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : SecondCountableTopology Y f : X \u2192 Y hf : Measurable f hsurj : Surjective f s : Set Y h : MeasurableSet (f \u207b\u00b9' s) \u22a2 AnalyticSet s\u1d9c ** rw [\u2190 image_preimage_eq s\u1d9c hsurj] ** case hsc \u03b1 : Type u_1 \u03b9 : Type u_2 X : Type u_3 Y : Type u_4 \u03b2 : Type u_5 inst\u271d\u2077 : MeasurableSpace X inst\u271d\u2076 : StandardBorelSpace X inst\u271d\u2075 : TopologicalSpace Y inst\u271d\u2074 : T2Space Y inst\u271d\u00b3 : MeasurableSpace Y inst\u271d\u00b2 : OpensMeasurableSpace Y inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : SecondCountableTopology Y f : X \u2192 Y hf : Measurable f hsurj : Surjective f s : Set Y h : MeasurableSet (f \u207b\u00b9' s) \u22a2 AnalyticSet (f '' (f \u207b\u00b9' s\u1d9c)) ** exact h.compl.analyticSet_image hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_congr_smul_measure ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E c : \u211d\u22650\u221e hc_ne_top : c \u2260 \u22a4 hT : DominatedFinMeasAdditive \u03bc T C hT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C' f : \u03b1 \u2192 E \u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul f ** by_cases hc0 : c = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E c : \u211d\u22650\u221e hc_ne_top : c \u2260 \u22a4 hT : DominatedFinMeasAdditive \u03bc T C hT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C' f : \u03b1 \u2192 E hc0 : \u00acc = 0 \u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul f ** refine' setToFun_congr_measure c\u207b\u00b9 c _ hc_ne_top (le_of_eq _) le_rfl hT hT_smul f ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E c : \u211d\u22650\u221e hc_ne_top : c \u2260 \u22a4 hT : DominatedFinMeasAdditive \u03bc T C hT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C' f : \u03b1 \u2192 E hc0 : c = 0 \u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul f ** simp [hc0] at hT_smul ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E c : \u211d\u22650\u221e hc_ne_top : c \u2260 \u22a4 hT : DominatedFinMeasAdditive \u03bc T C hT_smul\u271d : DominatedFinMeasAdditive (c \u2022 \u03bc) T C' f : \u03b1 \u2192 E hc0 : c = 0 hT_smul : DominatedFinMeasAdditive 0 T C' \u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul\u271d f ** have h : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 T s = 0 := fun s hs _ => hT_smul.eq_zero hs ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E c : \u211d\u22650\u221e hc_ne_top : c \u2260 \u22a4 hT : DominatedFinMeasAdditive \u03bc T C hT_smul\u271d : DominatedFinMeasAdditive (c \u2022 \u03bc) T C' f : \u03b1 \u2192 E hc0 : c = 0 hT_smul : DominatedFinMeasAdditive 0 T C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 \u22a2 setToFun \u03bc T hT f = setToFun (c \u2022 \u03bc) T hT_smul\u271d f ** rw [setToFun_zero_left' _ h, setToFun_measure_zero] ** case pos.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E c : \u211d\u22650\u221e hc_ne_top : c \u2260 \u22a4 hT : DominatedFinMeasAdditive \u03bc T C hT_smul\u271d : DominatedFinMeasAdditive (c \u2022 \u03bc) T C' f : \u03b1 \u2192 E hc0 : c = 0 hT_smul : DominatedFinMeasAdditive 0 T C' h : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = 0 \u22a2 c \u2022 \u03bc = 0 ** simp [hc0] ** case neg.refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E c : \u211d\u22650\u221e hc_ne_top : c \u2260 \u22a4 hT : DominatedFinMeasAdditive \u03bc T C hT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C' f : \u03b1 \u2192 E hc0 : \u00acc = 0 \u22a2 c\u207b\u00b9 \u2260 \u22a4 ** simp [hc0] ** case neg.refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E c : \u211d\u22650\u221e hc_ne_top : c \u2260 \u22a4 hT : DominatedFinMeasAdditive \u03bc T C hT_smul : DominatedFinMeasAdditive (c \u2022 \u03bc) T C' f : \u03b1 \u2192 E hc0 : \u00acc = 0 \u22a2 \u03bc = c\u207b\u00b9 \u2022 c \u2022 \u03bc ** rw [smul_smul, ENNReal.inv_mul_cancel hc0 hc_ne_top, one_smul] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snormEssSup_indicator_eq_snormEssSup_restrict ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s \u22a2 snormEssSup (Set.indicator s f) \u03bc = snormEssSup f (Measure.restrict \u03bc s) ** simp_rw [snormEssSup, nnnorm_indicator_eq_indicator_nnnorm, ENNReal.coe_indicator] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s \u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) (Measure.restrict \u03bc s) ** by_cases hs_null : \u03bc s = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) (Measure.restrict \u03bc s) ** rw [essSup_indicator_eq_essSup_restrict (eventually_of_forall fun x => ?_) hs hs_null] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u00ac\u2191\u2191\u03bc s = 0 x : \u03b1 \u22a2 OfNat.ofNat 0 x \u2264 \u2191\u2016f x\u2016\u208a ** rw [Pi.zero_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u00ac\u2191\u2191\u03bc s = 0 x : \u03b1 \u22a2 0 \u2264 \u2191\u2016f x\u2016\u208a ** exact zero_le _ ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u2191\u2191\u03bc s = 0 \u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) (Measure.restrict \u03bc s) ** rw [Measure.restrict_zero_set hs_null] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u2191\u2191\u03bc s = 0 \u22a2 essSup (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) \u03bc = essSup (fun x => \u2191\u2016f x\u2016\u208a) 0 ** simp only [essSup_measure_zero, ENNReal.essSup_eq_zero_iff, ENNReal.bot_eq_zero] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u2191\u2191\u03bc s = 0 \u22a2 (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) =\u1d50[\u03bc] 0 ** have hs_empty : s =\u1d50[\u03bc] (\u2205 : Set \u03b1) := by rw [ae_eq_set]; simpa using hs_null ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u2191\u2191\u03bc s = 0 hs_empty : s =\u1d50[\u03bc] \u2205 \u22a2 (fun x => Set.indicator s (fun x => \u2191\u2016f x\u2016\u208a) x) =\u1d50[\u03bc] 0 ** refine' (indicator_ae_eq_of_ae_eq_set hs_empty).trans _ ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u2191\u2191\u03bc s = 0 hs_empty : s =\u1d50[\u03bc] \u2205 \u22a2 (Set.indicator \u2205 fun x => \u2191\u2016f x\u2016\u208a) =\u1d50[\u03bc] 0 ** rw [Set.indicator_empty] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u2191\u2191\u03bc s = 0 hs_empty : s =\u1d50[\u03bc] \u2205 \u22a2 (fun x => 0) =\u1d50[\u03bc] 0 ** rfl ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u2191\u2191\u03bc s = 0 \u22a2 s =\u1d50[\u03bc] \u2205 ** rw [ae_eq_set] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc s : Set \u03b1 f : \u03b1 \u2192 F hs : MeasurableSet s hs_null : \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191\u03bc (s \\ \u2205) = 0 \u2227 \u2191\u2191\u03bc (\u2205 \\ s) = 0 ** simpa using hs_null ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.evariance_lt_top_iff_mem\u2112p ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : AEStronglyMeasurable X \u03bc \u22a2 evariance X \u03bc < \u22a4 \u2194 Mem\u2112p X 2 ** refine' \u27e8_, MeasureTheory.Mem\u2112p.evariance_lt_top\u27e9 ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : AEStronglyMeasurable X \u03bc \u22a2 evariance X \u03bc < \u22a4 \u2192 Mem\u2112p X 2 ** contrapose ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : AEStronglyMeasurable X \u03bc \u22a2 \u00acMem\u2112p X 2 \u2192 \u00acevariance X \u03bc < \u22a4 ** rw [not_lt, top_le_iff] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : AEStronglyMeasurable X \u03bc \u22a2 \u00acMem\u2112p X 2 \u2192 evariance X \u03bc = \u22a4 ** exact evariance_eq_top hX ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpL2_indicator_ae_eq_smul ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E' \u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) =\u1d50[\u03bc] fun a =>     \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x ** rw [indicatorConstLp_eq_toSpanSingleton_compLp hs h\u03bcs x] ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E' \u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>     \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x ** have h_comp :=\n  condexpL2_comp_continuousLinearMap \u211d \ud835\udd5c hm (toSpanSingleton \u211d x)\n    (indicatorConstLp 2 hs h\u03bcs (1 : \u211d)) ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E' h_comp :   \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc]     \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>     \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x ** refine' h_comp.trans _ ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2075 : CompleteSpace E inst\u271d\u00b9\u2074 : NormedAddCommGroup E' inst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E' inst\u271d\u00b9\u00b2 : CompleteSpace E' inst\u271d\u00b9\u00b9 : NormedSpace \u211d E' inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup G inst\u271d\u2077 : NormedAddCommGroup G' inst\u271d\u2076 : NormedSpace \u211d G' inst\u271d\u2075 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 E'' : Type u_8 \ud835\udd5c' : Type u_9 inst\u271d\u2074 : IsROrC \ud835\udd5c' inst\u271d\u00b3 : NormedAddCommGroup E'' inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E'' inst\u271d\u00b9 : CompleteSpace E'' inst\u271d : NormedSpace \u211d E'' hm : m \u2264 m0 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x : E' h_comp :   \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc]     \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u22a2 \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>     \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x ** exact (toSpanSingleton \u211d x).coeFn_compLp _ ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_measure_prod_mk_left ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s \u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) ** have : \u2200 x, MeasurableSet (Prod.mk x \u207b\u00b9' s) := fun x => measurable_prod_mk_left hs ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s this : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s) \u22a2 Measurable fun x => \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) ** simp only [\u2190 @iSup_restrict_spanningSets _ _ \u03bd, this] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s this : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s) \u22a2 Measurable fun x => \u2a06 i, \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk x \u207b\u00b9' s) ** apply measurable_iSup ** case hf \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s this : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s) \u22a2 \u2200 (i : \u2115), Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s) ** intro i ** case hf \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s this : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s) i : \u2115 \u22a2 Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s) ** haveI := Fact.mk (measure_spanningSets_lt_top \u03bd i) ** case hf \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd s : Set (\u03b1 \u00d7 \u03b2) hs : MeasurableSet s this\u271d : \u2200 (x : \u03b1), MeasurableSet (Prod.mk x \u207b\u00b9' s) i : \u2115 this : Fact (\u2191\u2191\u03bd (spanningSets \u03bd i) < \u22a4) \u22a2 Measurable fun b => \u2191\u2191(Measure.restrict \u03bd (spanningSets \u03bd i)) (Prod.mk b \u207b\u00b9' s) ** exact measurable_measure_prod_mk_left_finite hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrableOn_Ioi_comp_rpow_iff ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 \u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0) ** let S := Ioi (0 : \u211d) ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 \u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0) ** have a1 : \u2200 x : \u211d, x \u2208 S \u2192 HasDerivWithinAt (fun t : \u211d => t ^ p) (p * x ^ (p - 1)) S x :=\n  fun x hx => (hasDerivAt_rpow_const (Or.inl (mem_Ioi.mp hx).ne')).hasDerivWithinAt ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S \u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0) ** have := integrableOn_image_iff_integrableOn_abs_deriv_smul measurableSet_Ioi a1 a2 f ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : IntegrableOn f ((fun t => t ^ p) '' Ioi 0) \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0) \u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0) ** rw [a3] at this ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : IntegrableOn f S \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0) \u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194 IntegrableOn f (Ioi 0) ** rw [this] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : IntegrableOn f S \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0) \u22a2 IntegrableOn (fun x => (|p| * x ^ (p - 1)) \u2022 f (x ^ p)) (Ioi 0) \u2194     IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0) ** refine' integrableOn_congr_fun (fun x hx => _) measurableSet_Ioi ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : IntegrableOn f S \u2194 IntegrableOn (fun x => |p * x ^ (p - 1)| \u2022 f (x ^ p)) (Ioi 0) x : \u211d hx : x \u2208 Ioi 0 \u22a2 (|p| * x ^ (p - 1)) \u2022 f (x ^ p) = |p * x ^ (p - 1)| \u2022 f (x ^ p) ** simp_rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x \u22a2 InjOn (fun x => x ^ p) S ** rcases lt_or_gt_of_ne hp with (h | h) ** case inr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p > 0 \u22a2 InjOn (fun x => x ^ p) S ** exact StrictMonoOn.injOn fun x hx y _hy hxy => rpow_lt_rpow (mem_Ioi.mp hx).le hxy h ** case inl E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p < 0 \u22a2 InjOn (fun x => x ^ p) S ** apply StrictAntiOn.injOn ** case inl.H E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p < 0 \u22a2 StrictAntiOn (fun x => x ^ p) S ** intro x hx y hy hxy ** case inl.H E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p < 0 x : \u211d hx : x \u2208 S y : \u211d hy : y \u2208 S hxy : x < y \u22a2 (fun x => x ^ p) y < (fun x => x ^ p) x ** rw [\u2190 inv_lt_inv (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), \u2190 rpow_neg (le_of_lt hx), \u2190\n  rpow_neg (le_of_lt hy)] ** case inl.H E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p < 0 x : \u211d hx : x \u2208 S y : \u211d hy : y \u2208 S hxy : x < y \u22a2 x ^ (-p) < y ^ (-p) ** exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h) ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S \u22a2 (fun t => t ^ p) '' S = S ** ext1 x ** case h E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d \u22a2 x \u2208 (fun t => t ^ p) '' S \u2194 x \u2208 S ** rw [mem_image] ** case h E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d \u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2194 x \u2208 S ** constructor ** case h.mp E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d \u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2192 x \u2208 S ** rintro \u27e8y, hy, rfl\u27e9 ** case h.mp.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S y : \u211d hy : y \u2208 S \u22a2 y ^ p \u2208 S ** exact rpow_pos_of_pos hy p ** case h.mpr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d \u22a2 x \u2208 S \u2192 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x ** intro hx ** case h.mpr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d hx : x \u2208 S \u22a2 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x ** refine' \u27e8x ^ (1 / p), rpow_pos_of_pos hx _, _\u27e9 ** case h.mpr E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d hx : x \u2208 S \u22a2 (x ^ (1 / p)) ^ p = x ** rw [\u2190 rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.exist_disjoint_covering_families ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 \u22a2 \u2203 s,     (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227       range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j) ** cases isEmpty_or_nonempty \u03b2 ** case inr \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 \u22a2 \u2203 s,     (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227       range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j) ** let p : TauPackage \u03b2 \u03b1 :=\n  { q with\n    \u03c4\n    one_lt_tau := h\u03c4 } ** case inr \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } \u22a2 \u2203 s,     (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227       range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j) ** let s := fun i : Fin N =>\n  \u22c3 (k : Ordinal.{u}) (_ : k < p.lastStep) (_ : p.color k = i), ({p.index k} : Set \u03b2) ** case inr \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} \u22a2 \u2203 s,     (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227       range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j) ** refine' \u27e8s, fun i => _, _\u27e9 ** case inl \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : IsEmpty \u03b2 \u22a2 \u2203 s,     (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227       range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j) ** refine' \u27e8fun _ => \u2205, fun _ => pairwiseDisjoint_empty, _\u27e9 ** case inl \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : IsEmpty \u03b2 \u22a2 range q.c \u2286 \u22c3 i, \u22c3 j \u2208 (fun x => \u2205) i, ball (BallPackage.c q j) (BallPackage.r q j) ** rw [\u2190 image_univ, eq_empty_of_isEmpty (univ : Set \u03b2)] ** case inl \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : IsEmpty \u03b2 \u22a2 q.c '' \u2205 \u2286 \u22c3 i, \u22c3 j \u2208 (fun x => \u2205) i, ball (BallPackage.c q j) (BallPackage.r q j) ** simp ** case inr.refine'_1 \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N \u22a2 PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j) ** intro x hx y hy x_ne_y ** case inr.refine'_1 \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N x : \u03b2 hx : x \u2208 s i y : \u03b2 hy : y \u2208 s i x_ne_y : x \u2260 y \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) x y ** obtain \u27e8jx, jx_lt, jxi, rfl\u27e9 :\n  \u2203 jx : Ordinal, jx < p.lastStep \u2227 p.color jx = i \u2227 x = p.index jx := by\n  simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hx ** case inr.refine'_1.intro.intro.intro \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N y : \u03b2 hy : y \u2208 s i jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i x_ne_y : index p jx \u2260 y \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) y ** obtain \u27e8jy, jy_lt, jyi, rfl\u27e9 :\n  \u2203 jy : Ordinal, jy < p.lastStep \u2227 p.color jy = i \u2227 y = p.index jy := by\n  simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hy ** case inr.refine'_1.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) ** wlog jxy : jx \u2264 jy generalizing jx jy ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx \u2264 jy \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) ** replace jxy : jx < jy ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) ** let A : Set \u2115 :=\n  \u22c3 (j : { j // j < jy })\n    (_ : (closedBall (p.c (p.index j)) (p.r (p.index j)) \u2229\n      closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),\n    {p.color j} ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) ** have color_j : p.color jy = sInf (univ \\ A) := by rw [TauPackage.color] ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) ** have h : p.color jy \u2208 univ \\ A := by\n  rw [color_j]\n  apply csInf_mem\n  refine' \u27e8N, _\u27e9\n  simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and,\n    mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk]\n  intro k hk _\n  exact (p.color_lt (hk.trans jy_lt) hN).ne' ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) h : color p jy \u2208 univ \\ A \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) ** simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and,\n  mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk] at h ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) h :   \u2200 (x : Ordinal.{u}),     x < jy \u2192       Set.Nonempty           (closedBall               (BallPackage.c q                 (index                   {                     toBallPackage :=                       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                     \u03c4 := \u03c4, one_lt_tau := h\u03c4 }                   x))               (BallPackage.r q                 (index                   {                     toBallPackage :=                       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                     \u03c4 := \u03c4, one_lt_tau := h\u03c4 }                   x)) \u2229             closedBall               (BallPackage.c q                 (index                   {                     toBallPackage :=                       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                     \u03c4 := \u03c4, one_lt_tau := h\u03c4 }                   jy))               (BallPackage.r q                 (index                   {                     toBallPackage :=                       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                     \u03c4 := \u03c4, one_lt_tau := h\u03c4 }                   jy))) \u2192         \u00accolor               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               jy =             color               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               x \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) ** specialize h jx jxy ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) h :   Set.Nonempty       (closedBall           (BallPackage.c q             (index               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               jx))           (BallPackage.r q             (index               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               jx)) \u2229         closedBall           (BallPackage.c q             (index               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               jy))           (BallPackage.r q             (index               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               jy))) \u2192     \u00accolor           {             toBallPackage :=               { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                 r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },             \u03c4 := \u03c4, one_lt_tau := h\u03c4 }           jy =         color           {             toBallPackage :=               { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                 r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },             \u03c4 := \u03c4, one_lt_tau := h\u03c4 }           jx \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) ** contrapose! h ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) h : \u00ac(Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) \u22a2 Set.Nonempty       (closedBall           (BallPackage.c q             (index               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               jx))           (BallPackage.r q             (index               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               jx)) \u2229         closedBall           (BallPackage.c q             (index               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               jy))           (BallPackage.r q             (index               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               jy))) \u2227     color         {           toBallPackage :=             { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,               r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },           \u03c4 := \u03c4, one_lt_tau := h\u03c4 }         jy =       color         {           toBallPackage :=             { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,               r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },           \u03c4 := \u03c4, one_lt_tau := h\u03c4 }         jx ** simpa only [jxi, jyi, and_true_iff, eq_self_iff_true, \u2190 not_disjoint_iff_nonempty_inter] using h ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N x : \u03b2 hx : x \u2208 s i y : \u03b2 hy : y \u2208 s i x_ne_y : x \u2260 y \u22a2 \u2203 jx, jx < lastStep p \u2227 color p jx = \u2191i \u2227 x = index p jx ** simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hx ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N y : \u03b2 hy : y \u2208 s i jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i x_ne_y : index p jx \u2260 y \u22a2 \u2203 jy, jy < lastStep p \u2227 color p jy = \u2191i \u2227 y = index p jy ** simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hy ** case inr.refine'_1.intro.intro.intro.intro.intro.intro.inr \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy this :   \u2200 (jx : Ordinal.{u}),     jx < lastStep p \u2192       color p jx = \u2191i \u2192         index p jx \u2208 s i \u2192           \u2200 (jy : Ordinal.{u}),             jy < lastStep p \u2192               color p jy = \u2191i \u2192                 index p jy \u2208 s i \u2192                   index p jx \u2260 index p jy \u2192                     jx \u2264 jy \u2192                       (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx)                         (index p jy) jxy : \u00acjx \u2264 jy \u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy) ** exact (this jy jy_lt jyi hy jx jx_lt jxi hx x_ne_y.symm (le_of_not_le jxy)).symm ** case jxy \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx \u2264 jy \u22a2 jx < jy ** rcases lt_or_eq_of_le jxy with (H | rfl) ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} \u22a2 color p jy = sInf (univ \\ A) ** rw [TauPackage.color] ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) \u22a2 color p jy \u2208 univ \\ A ** rw [color_j] ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) \u22a2 sInf (univ \\ A) \u2208 univ \\ A ** apply csInf_mem ** case hs \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) \u22a2 Set.Nonempty (univ \\ A) ** refine' \u27e8N, _\u27e9 ** case hs \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) \u22a2 N \u2208 univ \\ A ** simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and,\n  mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk] ** case hs \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) \u22a2 \u2200 (x : Ordinal.{u}),     x < jy \u2192       Set.Nonempty           (closedBall               (BallPackage.c q                 (index                   {                     toBallPackage :=                       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                     \u03c4 := \u03c4, one_lt_tau := h\u03c4 }                   x))               (BallPackage.r q                 (index                   {                     toBallPackage :=                       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                     \u03c4 := \u03c4, one_lt_tau := h\u03c4 }                   x)) \u2229             closedBall               (BallPackage.c q                 (index                   {                     toBallPackage :=                       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                     \u03c4 := \u03c4, one_lt_tau := h\u03c4 }                   jy))               (BallPackage.r q                 (index                   {                     toBallPackage :=                       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                     \u03c4 := \u03c4, one_lt_tau := h\u03c4 }                   jy))) \u2192         \u00acN =             color               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               x ** intro k hk _ ** case hs \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} i : Fin N jx : Ordinal.{u} jx_lt : jx < lastStep p jxi : color p jx = \u2191i hx : index p jx \u2208 s i jy : Ordinal.{u} jy_lt : jy < lastStep p jyi : color p jy = \u2191i hy : index p jy \u2208 s i x_ne_y : index p jx \u2260 index p jy jxy : jx < jy A : Set \u2115 :=   \u22c3 j,     \u22c3 (_ :       Set.Nonempty         (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229           closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),       {color p \u2191j} color_j : color p jy = sInf (univ \\ A) k : Ordinal.{u} hk : k < jy a\u271d :   Set.Nonempty     (closedBall         (BallPackage.c q           (index             {               toBallPackage :=                 { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                   r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },               \u03c4 := \u03c4, one_lt_tau := h\u03c4 }             k))         (BallPackage.r q           (index             {               toBallPackage :=                 { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                   r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },               \u03c4 := \u03c4, one_lt_tau := h\u03c4 }             k)) \u2229       closedBall         (BallPackage.c q           (index             {               toBallPackage :=                 { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                   r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },               \u03c4 := \u03c4, one_lt_tau := h\u03c4 }             jy))         (BallPackage.r q           (index             {               toBallPackage :=                 { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                   r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },               \u03c4 := \u03c4, one_lt_tau := h\u03c4 }             jy))) \u22a2 \u00acN =       color         {           toBallPackage :=             { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,               r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },           \u03c4 := \u03c4, one_lt_tau := h\u03c4 }         k ** exact (p.color_lt (hk.trans jy_lt) hN).ne' ** case inr.refine'_2 \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} \u22a2 range q.c \u2286 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j) ** refine' range_subset_iff.2 fun b => _ ** case inr.refine'_2 \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} b : \u03b2 \u22a2 BallPackage.c q b \u2208 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j) ** obtain \u27e8a, ha\u27e9 :\n  \u2203 a : Ordinal, a < p.lastStep \u2227 dist (p.c b) (p.c (p.index a)) < p.r (p.index a) := by\n  simpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists,\n    Subtype.coe_mk] using p.mem_iUnionUpTo_lastStep b ** case inr.refine'_2.intro \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} b : \u03b2 a : Ordinal.{u} ha :   a < lastStep p \u2227     dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <       BallPackage.r p.toBallPackage (index p a) \u22a2 BallPackage.c q b \u2208 \u22c3 i, \u22c3 j \u2208 s i, ball (BallPackage.c q j) (BallPackage.r q j) ** simp only [exists_prop, mem_iUnion, mem_ball, mem_singleton_iff, biUnion_and', exists_eq_left,\n  iUnion_exists, exists_and_left] ** case inr.refine'_2.intro \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} b : \u03b2 a : Ordinal.{u} ha :   a < lastStep p \u2227     dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <       BallPackage.r p.toBallPackage (index p a) \u22a2 \u2203 i i_1,     color           {             toBallPackage :=               { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                 r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },             \u03c4 := \u03c4, one_lt_tau := h\u03c4 }           i_1 =         \u2191i \u2227       i_1 <           lastStep             {               toBallPackage :=                 { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                   r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },               \u03c4 := \u03c4, one_lt_tau := h\u03c4 } \u2227         dist (BallPackage.c q b)             (BallPackage.c q               (index                 {                   toBallPackage :=                     { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                       r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                   \u03c4 := \u03c4, one_lt_tau := h\u03c4 }                 i_1)) <           BallPackage.r q             (index               {                 toBallPackage :=                   { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,                     r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },                 \u03c4 := \u03c4, one_lt_tau := h\u03c4 }               i_1) ** exact \u27e8\u27e8p.color a, p.color_lt ha.1 hN\u27e9, a, rfl, ha\u27e9 ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 \u03b2 : Type u N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) q : BallPackage \u03b2 \u03b1 h\u271d : Nonempty \u03b2 p : TauPackage \u03b2 \u03b1 :=   {     toBallPackage :=       { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,         r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },     \u03c4 := \u03c4, one_lt_tau := h\u03c4 } s : Fin N \u2192 Set \u03b2 := fun i => \u22c3 k, \u22c3 (_ : k < lastStep p), \u22c3 (_ : color p k = \u2191i), {index p k} b : \u03b2 \u22a2 \u2203 a,     a < lastStep p \u2227       dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <         BallPackage.r p.toBallPackage (index p a) ** simpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists,\n  Subtype.coe_mk] using p.mem_iUnionUpTo_lastStep b ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PartrecToTM2.contSupp_supports ** S : Finset \u039b' k : Cont' H : contSupp k \u2286 S \u22a2 Supports (contSupp k) S ** induction k ** case cons\u2081 S : Finset \u039b' a\u271d\u00b9 : Code a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.cons\u2081 a\u271d\u00b9 a\u271d)) S  case cons\u2082 S : Finset \u039b' a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.cons\u2082 a\u271d)) S  case comp S : Finset \u039b' a\u271d\u00b9 : Code a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.comp a\u271d\u00b9 a\u271d)) S  case fix S : Finset \u039b' a\u271d\u00b9 : Code a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S ** case cons\u2081 f k IH =>\n  have H\u2081 := H; rw [contSupp_cons\u2081] at H\u2081; have H\u2082 := Finset.union_subset_right H\u2081\n  refine' trStmts\u2081_supports' (trNormal_supports H\u2082) H\u2081 fun h => _\n  refine' supports_union.2 \u27e8codeSupp'_supports H\u2082, _\u27e9\n  simp only [codeSupp, contSupp_cons\u2082, Finset.union_subset_iff] at H\u2082\n  exact trStmts\u2081_supports' (head_supports H\u2082.2.2) (Finset.union_subset_right h) IH ** case cons\u2082 S : Finset \u039b' a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.cons\u2082 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.cons\u2082 a\u271d)) S  case comp S : Finset \u039b' a\u271d\u00b9 : Code a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.comp a\u271d\u00b9 a\u271d)) S  case fix S : Finset \u039b' a\u271d\u00b9 : Code a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S ** case cons\u2082 k IH =>\n  have H' := H; rw [contSupp_cons\u2082] at H'\n  exact trStmts\u2081_supports' (head_supports <| Finset.union_subset_right H') H' IH ** case comp S : Finset \u039b' a\u271d\u00b9 : Code a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.comp a\u271d\u00b9 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.comp a\u271d\u00b9 a\u271d)) S  case fix S : Finset \u039b' a\u271d\u00b9 : Code a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S ** case comp f k IH =>\n  have H' := H; rw [contSupp_comp] at H'; have H\u2082 := Finset.union_subset_right H'\n  exact supports_union.2 \u27e8codeSupp'_supports H', IH H\u2082\u27e9 ** case fix S : Finset \u039b' a\u271d\u00b9 : Code a\u271d : Cont' a_ih\u271d : contSupp a\u271d \u2286 S \u2192 Supports (contSupp a\u271d) S H : contSupp (Cont'.fix a\u271d\u00b9 a\u271d) \u2286 S \u22a2 Supports (contSupp (Cont'.fix a\u271d\u00b9 a\u271d)) S ** case fix f k IH =>\n  rw [contSupp] at H\n  exact supports_union.2 \u27e8codeSupp'_supports H, IH (Finset.union_subset_right H)\u27e9 ** case halt S : Finset \u039b' H : contSupp Cont'.halt \u2286 S \u22a2 Supports (contSupp Cont'.halt) S ** simp [contSupp_halt, Supports] ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.cons\u2081 f k) \u2286 S \u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S ** have H\u2081 := H ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H H\u2081 : contSupp (Cont'.cons\u2081 f k) \u2286 S \u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S ** rw [contSupp_cons\u2081] at H\u2081 ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.cons\u2081 f k) \u2286 S H\u2081 :   trStmts\u2081         (move\u2082 (fun x => false) main aux           (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main             (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a       codeSupp f (Cont'.cons\u2082 k) \u2286     S \u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S ** have H\u2082 := Finset.union_subset_right H\u2081 ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.cons\u2081 f k) \u2286 S H\u2081 :   trStmts\u2081         (move\u2082 (fun x => false) main aux           (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main             (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a       codeSupp f (Cont'.cons\u2082 k) \u2286     S H\u2082 : codeSupp f (Cont'.cons\u2082 k) \u2286 S \u22a2 Supports (contSupp (Cont'.cons\u2081 f k)) S ** refine' trStmts\u2081_supports' (trNormal_supports H\u2082) H\u2081 fun h => _ ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.cons\u2081 f k) \u2286 S H\u2081 :   trStmts\u2081         (move\u2082 (fun x => false) main aux           (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main             (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a       codeSupp f (Cont'.cons\u2082 k) \u2286     S H\u2082 : codeSupp f (Cont'.cons\u2082 k) \u2286 S h : codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) \u2286 S \u22a2 Supports (codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k)) S ** refine' supports_union.2 \u27e8codeSupp'_supports H\u2082, _\u27e9 ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.cons\u2081 f k) \u2286 S H\u2081 :   trStmts\u2081         (move\u2082 (fun x => false) main aux           (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main             (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a       codeSupp f (Cont'.cons\u2082 k) \u2286     S H\u2082 : codeSupp f (Cont'.cons\u2082 k) \u2286 S h : codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) \u2286 S \u22a2 Supports (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) S ** simp only [codeSupp, contSupp_cons\u2082, Finset.union_subset_iff] at H\u2082 ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.cons\u2081 f k) \u2286 S H\u2081 :   trStmts\u2081         (move\u2082 (fun x => false) main aux           (move\u2082 (fun s => decide (s = \u0393'.cons\u2097)) stack main             (move\u2082 (fun x => false) aux stack (trNormal f (Cont'.cons\u2082 k))))) \u222a       codeSupp f (Cont'.cons\u2082 k) \u2286     S h : codeSupp' f (Cont'.cons\u2082 k) \u222a (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) \u2286 S H\u2082 : codeSupp' f (Cont'.cons\u2082 k) \u2286 S \u2227 trStmts\u2081 (head stack (\u039b'.ret k)) \u2286 S \u2227 contSupp k \u2286 S \u22a2 Supports (trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k) S ** exact trStmts\u2081_supports' (head_supports H\u2082.2.2) (Finset.union_subset_right h) IH ** S : Finset \u039b' k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.cons\u2082 k) \u2286 S \u22a2 Supports (contSupp (Cont'.cons\u2082 k)) S ** have H' := H ** S : Finset \u039b' k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H H' : contSupp (Cont'.cons\u2082 k) \u2286 S \u22a2 Supports (contSupp (Cont'.cons\u2082 k)) S ** rw [contSupp_cons\u2082] at H' ** S : Finset \u039b' k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.cons\u2082 k) \u2286 S H' : trStmts\u2081 (head stack (\u039b'.ret k)) \u222a contSupp k \u2286 S \u22a2 Supports (contSupp (Cont'.cons\u2082 k)) S ** exact trStmts\u2081_supports' (head_supports <| Finset.union_subset_right H') H' IH ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.comp f k) \u2286 S \u22a2 Supports (contSupp (Cont'.comp f k)) S ** have H' := H ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H H' : contSupp (Cont'.comp f k) \u2286 S \u22a2 Supports (contSupp (Cont'.comp f k)) S ** rw [contSupp_comp] at H' ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.comp f k) \u2286 S H' : codeSupp f k \u2286 S \u22a2 Supports (contSupp (Cont'.comp f k)) S ** have H\u2082 := Finset.union_subset_right H' ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.comp f k) \u2286 S H' : codeSupp f k \u2286 S H\u2082 : contSupp k \u2286 S \u22a2 Supports (contSupp (Cont'.comp f k)) S ** exact supports_union.2 \u27e8codeSupp'_supports H', IH H\u2082\u27e9 ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : contSupp (Cont'.fix f k) \u2286 S \u22a2 Supports (contSupp (Cont'.fix f k)) S ** rw [contSupp] at H ** S : Finset \u039b' f : Code k : Cont' IH : contSupp k \u2286 S \u2192 Supports (contSupp k) S H : codeSupp' (Code.fix f) k \u222a contSupp k \u2286 S \u22a2 Supports (contSupp (Cont'.fix f k)) S ** exact supports_union.2 \u27e8codeSupp'_supports H, IH (Finset.union_subset_right H)\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.disjoint_erase_comm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t u v : Finset \u03b1 a b : \u03b1 \u22a2 _root_.Disjoint (erase s a) t \u2194 _root_.Disjoint s (erase t a) ** simp_rw [erase_eq, disjoint_sdiff_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaar ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s h'f : Measurable f \u22a2 Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =     Measure.restrict \u03bc (f '' s) ** apply Measure.ext fun t ht => ?_ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s h'f : Measurable f t : Set E ht : MeasurableSet t \u22a2 \u2191\u2191(Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) t =     \u2191\u2191(Measure.restrict \u03bc (f '' s)) t ** rw [map_apply h'f ht, withDensity_apply _ (h'f ht), Measure.restrict_apply ht,\n  restrict_restrict (h'f ht),\n  lintegral_abs_det_fderiv_eq_addHaar_image \u03bc ((h'f ht).inter hs)\n    (fun x hx => (hf' x hx.2).mono (inter_subset_right _ _)) (hf.mono (inter_subset_right _ _)),\n  image_preimage_inter] ** Qed",
        "informal": ""
    },
    {
        "formal": "Besicovitch.exist_finset_disjoint_balls_large_measure ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** rcases le_or_lt (\u03bc s) 0 with (h\u03bcs | h\u03bcs) ** case inr \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** cases isEmpty_or_nonempty \u03b1 ** case inr.inr \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** have Npos : N \u2260 0 := by\n  rintro rfl\n  inhabit \u03b1\n  exact not_isEmpty_of_nonempty _ hN ** case inr.inr \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** obtain \u27e8o, so, omeas, \u03bco\u27e9 : \u2203 o : Set \u03b1, s \u2286 o \u2227 MeasurableSet o \u2227 \u03bc o = \u03bc s :=\n  exists_measurable_superset \u03bc s ** case inr.inr.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** let a : BallPackage s \u03b1 :=\n  { c := fun x => x\n    r := fun x => r x\n    rpos := fun x => rpos x x.2\n    r_bound := 1\n    r_le := fun x => rle x x.2 } ** case inr.inr.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** rcases exist_disjoint_covering_families h\u03c4 hN a with \u27e8u, hu, hu'\u27e9 ** case inr.inr.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** have u_count : \u2200 i, (u i).Countable := by\n  intro i\n  refine' (hu i).countable_of_nonempty_interior fun j _ => _\n  have : (ball (j : \u03b1) (r j)).Nonempty := nonempty_ball.2 (a.rpos _)\n  exact this.mono ball_subset_interior_closedBall ** case inr.inr.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** let v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : s) (_ : x \u2208 u i), closedBall x (r x) ** case inr.inr.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** have A : s = \u22c3 i : Fin N, s \u2229 v i := by\n  refine' Subset.antisymm _ (iUnion_subset fun i => inter_subset_left _ _)\n  intro x hx\n  obtain \u27e8i, y, hxy, h'\u27e9 :\n      \u2203 (i : Fin N) (i_1 : \u21a5s), i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1) := by\n    have : x \u2208 range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq]\n    simpa only [mem_iUnion, bex_def] using hu' this\n  refine' mem_iUnion.2 \u27e8i, \u27e8hx, _\u27e9\u27e9\n  simp only [exists_prop, mem_iUnion, SetCoe.exists, exists_and_right, Subtype.coe_mk]\n  exact \u27e8y, \u27e8y.2, by simpa only [Subtype.coe_eta]\u27e9, ball_subset_closedBall h'\u27e9 ** case inr.inr.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** obtain \u27e8i, -, hi\u27e9 : \u2203 (i : Fin N), i \u2208 Finset.univ \u2227 \u03bc s / N \u2264 \u03bc (s \u2229 v i) := by\n  apply ENNReal.exists_le_of_sum_le _ S\n  exact \u27e8\u27e80, bot_lt_iff_ne_bot.2 Npos\u27e9, Finset.mem_univ _\u27e9 ** case inr.inr.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i) \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** replace hi : \u03bc s / (N + 1) < \u03bc (s \u2229 v i) ** case inr.inr.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** obtain \u27e8w, hw\u27e9 :\n  \u2203 w : Finset (u i), \u03bc s / (N + 1) <\n    \u2211 x : u i in w, \u03bc (o \u2229 closedBall (x : \u03b1) (r (x : \u03b1))) := by\n  have C : HasSum (fun x : u i => \u03bc (o \u2229 closedBall x (r x))) (\u03bc (o \u2229 v i)) := by\n    rw [B]; exact ENNReal.summable.hasSum\n  have : \u03bc s / (N + 1) < \u03bc (o \u2229 v i) := hi.trans_le (measure_mono (inter_subset_inter_left _ so))\n  exact ((tendsto_order.1 C).1 _ this).exists ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** refine' \u27e8Finset.image (fun x : u i => x) w, _, _, _\u27e9 ** case inl \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : \u2191\u2191\u03bc s \u2264 0 \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** have : \u03bc s = 0 := le_bot_iff.1 h\u03bcs ** case inl \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : \u2191\u2191\u03bc s \u2264 0 this : \u2191\u2191\u03bc s = 0 \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** refine' \u27e8\u2205, by simp only [Finset.coe_empty, empty_subset], _, _\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : \u2191\u2191\u03bc s \u2264 0 this : \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2205 \u2286 s ** simp only [Finset.coe_empty, empty_subset] ** case inl.refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : \u2191\u2191\u03bc s \u2264 0 this : \u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 \u2205, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s ** simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,\n  nonpos_iff_eq_zero, mul_zero] ** case inl.refine'_2 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : \u2191\u2191\u03bc s \u2264 0 this : \u2191\u2191\u03bc s = 0 \u22a2 PairwiseDisjoint \u2191\u2205 fun x => closedBall x (r x) ** simp only [Finset.coe_empty, pairwiseDisjoint_empty] ** case inr.inl \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : IsEmpty \u03b1 \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** simp only [eq_empty_of_isEmpty s, measure_empty] at h\u03bcs ** case inr.inl \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u271d : IsEmpty \u03b1 h\u03bcs : 0 < 0 \u22a2 \u2203 t,     \u2191t \u2286 s \u2227       \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227 PairwiseDisjoint \u2191t fun x => closedBall x (r x) ** exact (lt_irrefl _ h\u03bcs).elim ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 \u22a2 N \u2260 0 ** rintro rfl ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc \u03c4 : \u211d h\u03c4 : 1 < \u03c4 s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 hN : IsEmpty (SatelliteConfig \u03b1 0 \u03c4) \u22a2 False ** inhabit \u03b1 ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc \u03c4 : \u211d h\u03c4 : 1 < \u03c4 s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 hN : IsEmpty (SatelliteConfig \u03b1 0 \u03c4) inhabited_h : Inhabited \u03b1 \u22a2 False ** exact not_isEmpty_of_nonempty _ hN ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) \u22a2 \u2200 (i : Fin N), Set.Countable (u i) ** intro i ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) i : Fin N \u22a2 Set.Countable (u i) ** refine' (hu i).countable_of_nonempty_interior fun j _ => _ ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) i : Fin N j : \u2191s x\u271d : j \u2208 u i \u22a2 Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j))) ** have : (ball (j : \u03b1) (r j)).Nonempty := nonempty_ball.2 (a.rpos _) ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) i : Fin N j : \u2191s x\u271d : j \u2208 u i this : Set.Nonempty (ball (\u2191j) (r \u2191j)) \u22a2 Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j))) ** exact this.mono ball_subset_interior_closedBall ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) \u22a2 s = \u22c3 i, s \u2229 v i ** refine' Subset.antisymm _ (iUnion_subset fun i => inter_subset_left _ _) ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) \u22a2 s \u2286 \u22c3 i, s \u2229 v i ** intro x hx ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) x : \u03b1 hx : x \u2208 s \u22a2 x \u2208 \u22c3 i, s \u2229 v i ** obtain \u27e8i, y, hxy, h'\u27e9 :\n    \u2203 (i : Fin N) (i_1 : \u21a5s), i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1) := by\n  have : x \u2208 range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq]\n  simpa only [mem_iUnion, bex_def] using hu' this ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) x : \u03b1 hx : x \u2208 s i : Fin N y : \u2191s hxy : y \u2208 u i h' : x \u2208 ball (\u2191y) (r \u2191y) \u22a2 x \u2208 \u22c3 i, s \u2229 v i ** refine' mem_iUnion.2 \u27e8i, \u27e8hx, _\u27e9\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) x : \u03b1 hx : x \u2208 s i : Fin N y : \u2191s hxy : y \u2208 u i h' : x \u2208 ball (\u2191y) (r \u2191y) \u22a2 x \u2208 v i ** simp only [exists_prop, mem_iUnion, SetCoe.exists, exists_and_right, Subtype.coe_mk] ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) x : \u03b1 hx : x \u2208 s i : Fin N y : \u2191s hxy : y \u2208 u i h' : x \u2208 ball (\u2191y) (r \u2191y) \u22a2 \u2203 x_1, (\u2203 x, { val := x_1, property := (_ : x_1 \u2208 s) } \u2208 u i) \u2227 x \u2208 closedBall x_1 (r x_1) ** exact \u27e8y, \u27e8y.2, by simpa only [Subtype.coe_eta]\u27e9, ball_subset_closedBall h'\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) x : \u03b1 hx : x \u2208 s \u22a2 \u2203 i i_1, i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1) ** have : x \u2208 range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) x : \u03b1 hx : x \u2208 s this : x \u2208 range a.c \u22a2 \u2203 i i_1, i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1) ** simpa only [mem_iUnion, bex_def] using hu' this ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) x : \u03b1 hx : x \u2208 s \u22a2 x \u2208 range a.c ** simpa only [Subtype.range_coe_subtype, setOf_mem_eq] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) x : \u03b1 hx : x \u2208 s i : Fin N y : \u2191s hxy : y \u2208 u i h' : x \u2208 ball (\u2191y) (r \u2191y) \u22a2 { val := \u2191y, property := (_ : \u2191y \u2208 s) } \u2208 u i ** simpa only [Subtype.coe_eta] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i \u22a2 \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N = \u2191\u2191\u03bc s ** simp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i \u22a2 \u2191N * (\u2191\u2191\u03bc s / \u2191N) = \u2191\u2191\u03bc s ** rw [ENNReal.mul_div_cancel'] ** case h0 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i \u22a2 \u2191N \u2260 0 ** simp only [Npos, Ne.def, Nat.cast_eq_zero, not_false_iff] ** case hI \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i \u22a2 \u2191N \u2260 \u22a4 ** exact ENNReal.nat_ne_top _ ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i \u22a2 \u2191\u2191\u03bc s \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) ** conv_lhs => rw [A] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i \u22a2 \u2191\u2191\u03bc (\u22c3 i, s \u2229 v i) \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) ** apply measure_iUnion_fintype_le ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) \u22a2 \u2203 i, i \u2208 Finset.univ \u2227 \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i) ** apply ENNReal.exists_le_of_sum_le _ S ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) \u22a2 Finset.Nonempty Finset.univ ** exact \u27e8\u27e80, bot_lt_iff_ne_bot.2 Npos\u27e9, Finset.mem_univ _\u27e9 ** case hi \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i) \u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) ** apply lt_of_lt_of_le _ hi ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i) \u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc s / \u2191N ** apply (ENNReal.mul_lt_mul_left h\u03bcs.ne' (measure_lt_top \u03bc s).ne).2 ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i) \u22a2 (\u2191N + 1)\u207b\u00b9 < (\u2191N)\u207b\u00b9 ** rw [ENNReal.inv_lt_inv] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i) \u22a2 \u2191N < \u2191N + 1 ** conv_lhs => rw [\u2190 add_zero (N : \u211d\u22650\u221e)] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i) \u22a2 \u2191N + 0 < \u2191N + 1 ** exact ENNReal.add_lt_add_left (ENNReal.nat_ne_top N) zero_lt_one ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) \u22a2 \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** have : o \u2229 v i = \u22c3 (x : s) (_ : x \u2208 u i), o \u2229 closedBall x (r x) := by simp only [inter_iUnion] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) this : o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x) \u22a2 \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** rw [this, measure_biUnion (u_count i)] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) \u22a2 o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x) ** simp only [inter_iUnion] ** case hd \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) this : o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x) \u22a2 PairwiseDisjoint (u i) fun x => o \u2229 closedBall (\u2191x) (r \u2191x) ** exact (hu i).mono fun k => inter_subset_right _ _ ** case h \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) this : o \u2229 v i = \u22c3 x \u2208 u i, o \u2229 closedBall (\u2191x) (r \u2191x) \u22a2 \u2200 (b : \u2191s), b \u2208 u i \u2192 MeasurableSet (o \u2229 closedBall (\u2191b) (r \u2191b)) ** exact fun b _ => omeas.inter measurableSet_closedBall ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** have C : HasSum (fun x : u i => \u03bc (o \u2229 closedBall x (r x))) (\u03bc (o \u2229 v i)) := by\n  rw [B]; exact ENNReal.summable.hasSum ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) C : HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i)) \u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** have : \u03bc s / (N + 1) < \u03bc (o \u2229 v i) := hi.trans_le (measure_mono (inter_subset_inter_left _ so)) ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) C : HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i)) this : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (o \u2229 v i) \u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** exact ((tendsto_order.1 C).1 _ this).exists ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i)) ** rw [B] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) ** exact ENNReal.summable.hasSum ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2191(Finset.image (fun x => \u2191\u2191x) w) \u2286 s ** simp only [image_subset_iff, Finset.coe_image] ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2191w \u2286 (fun x => \u2191\u2191x) \u207b\u00b9' s ** intro y _ ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) y : \u2191(u i) a\u271d : y \u2208 \u2191w \u22a2 y \u2208 (fun x => \u2191\u2191x) \u207b\u00b9' s ** simp only [Subtype.coe_prop, mem_preimage] ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 Finset.image (fun x => \u2191\u2191x) w, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s ** suffices H : \u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191x) (r \u2191x)) \u2264 N / (N + 1) * \u03bc s ** case H \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2191\u2191\u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s ** rw [\u2190 diff_inter_self_eq_diff,\n  measure_diff_le_iff_le_add _ (inter_subset_right _ _) (measure_lt_top \u03bc _).ne] ** case H \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2191\u2191\u03bc o \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s  \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 MeasurableSet ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) ** swap ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) H : \u2191\u2191\u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 x \u2208 Finset.image (fun x => \u2191\u2191x) w, closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s ** rw [Finset.set_biUnion_finset_image] ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) H : \u2191\u2191\u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u22a2 \u2191\u2191\u03bc (s \\ \u22c3 y \u2208 w, closedBall (\u2191\u2191y) (r \u2191\u2191y)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s ** exact le_trans (measure_mono (diff_subset_diff so (Subset.refl _))) H ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 MeasurableSet ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) ** apply MeasurableSet.inter _ omeas ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 MeasurableSet (\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** haveI : Encodable (u i) := (u_count i).toEncodable ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) this : Encodable \u2191(u i) \u22a2 MeasurableSet (\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) ** exact MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => measurableSet_closedBall ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2191\u2191\u03bc o = 1 / (\u2191N + 1) * \u2191\u2191\u03bc s + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s ** rw [\u03bco, \u2190 add_mul, ENNReal.div_add_div_same, add_comm, ENNReal.div_self, one_mul] <;> simp ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 1 / (\u2191N + 1) * \u2191\u2191\u03bc s + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s ** refine' add_le_add _ le_rfl ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 1 / (\u2191N + 1) * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) ** rw [div_eq_mul_inv, one_mul, mul_comm, \u2190 div_eq_mul_inv] ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) \u2264 \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) ** apply hw.le.trans (le_of_eq _) ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc ((\u22c3 x \u2208 w, closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) ** rw [\u2190 Finset.set_biUnion_coe, inter_comm _ o, inter_iUnion\u2082, Finset.set_biUnion_coe,\n  measure_biUnion_finset] ** case hd \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 PairwiseDisjoint \u2191w fun x => o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x) ** have : (w : Set (u i)).PairwiseDisjoint\n    fun b : u i => closedBall (b : \u03b1) (r (b : \u03b1)) := by\n  intro k _ l _ hkl; exact hu i k.2 l.2 (Subtype.val_injective.ne hkl) ** case hd \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) this : PairwiseDisjoint \u2191w fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b) \u22a2 PairwiseDisjoint \u2191w fun x => o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x) ** exact this.mono fun k => inter_subset_right _ _ ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 PairwiseDisjoint \u2191w fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b) ** intro k _ l _ hkl ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) k : \u2191(u i) a\u271d\u00b9 : k \u2208 \u2191w l : \u2191(u i) a\u271d : l \u2208 \u2191w hkl : k \u2260 l \u22a2 (Disjoint on fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b)) k l ** exact hu i k.2 l.2 (Subtype.val_injective.ne hkl) ** case hm \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 \u2200 (b : \u2191(u i)), b \u2208 w \u2192 MeasurableSet (o \u2229 closedBall (\u2191\u2191b) (r \u2191\u2191b)) ** intro b _ ** case hm \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) b : \u2191(u i) a\u271d : b \u2208 w \u22a2 MeasurableSet (o \u2229 closedBall (\u2191\u2191b) (r \u2191\u2191b)) ** apply omeas.inter measurableSet_closedBall ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u22a2 PairwiseDisjoint \u2191(Finset.image (fun x => \u2191\u2191x) w) fun x => closedBall x (r x) ** intro k hk l hl hkl ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) k : \u03b1 hk : k \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) l : \u03b1 hl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) hkl : k \u2260 l \u22a2 (Disjoint on fun x => closedBall x (r x)) k l ** obtain \u27e8k', _, rfl\u27e9 : \u2203 k' : u i, k' \u2208 w \u2227 \u2191k' = k := by\n  simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hk ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) l : \u03b1 hl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) k' : \u2191(u i) left\u271d : k' \u2208 w hk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) hkl : \u2191\u2191k' \u2260 l \u22a2 (Disjoint on fun x => closedBall x (r x)) (\u2191\u2191k') l ** obtain \u27e8l', _, rfl\u27e9 : \u2203 l' : u i, l' \u2208 w \u2227 \u2191l' = l := by\n  simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hl ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) k' : \u2191(u i) left\u271d\u00b9 : k' \u2208 w hk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) l' : \u2191(u i) left\u271d : l' \u2208 w hl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) hkl : \u2191\u2191k' \u2260 \u2191\u2191l' \u22a2 (Disjoint on fun x => closedBall x (r x)) \u2191\u2191k' \u2191\u2191l' ** have k'nel' : (k' : s) \u2260 l' := by intro h; rw [h] at hkl; exact hkl rfl ** case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) k' : \u2191(u i) left\u271d\u00b9 : k' \u2208 w hk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) l' : \u2191(u i) left\u271d : l' \u2208 w hl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) hkl : \u2191\u2191k' \u2260 \u2191\u2191l' k'nel' : \u2191k' \u2260 \u2191l' \u22a2 (Disjoint on fun x => closedBall x (r x)) \u2191\u2191k' \u2191\u2191l' ** exact hu i k'.2 l'.2 k'nel' ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) k : \u03b1 hk : k \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) l : \u03b1 hl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) hkl : k \u2260 l \u22a2 \u2203 k', k' \u2208 w \u2227 \u2191\u2191k' = k ** simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hk ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) l : \u03b1 hl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) k' : \u2191(u i) left\u271d : k' \u2208 w hk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) hkl : \u2191\u2191k' \u2260 l \u22a2 \u2203 l', l' \u2208 w \u2227 \u2191\u2191l' = l ** simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hl ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) k' : \u2191(u i) left\u271d\u00b9 : k' \u2208 w hk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) l' : \u2191(u i) left\u271d : l' \u2208 w hl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) hkl : \u2191\u2191k' \u2260 \u2191\u2191l' \u22a2 \u2191k' \u2260 \u2191l' ** intro h ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) k' : \u2191(u i) left\u271d\u00b9 : k' \u2208 w hk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) l' : \u2191(u i) left\u271d : l' \u2208 w hl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) hkl : \u2191\u2191k' \u2260 \u2191\u2191l' h : \u2191k' = \u2191l' \u22a2 False ** rw [h] at hkl ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 \u03b2 : Type u inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc N : \u2115 \u03c4 : \u211d h\u03c4 : 1 < \u03c4 hN : IsEmpty (SatelliteConfig \u03b1 N \u03c4) s : Set \u03b1 r : \u03b1 \u2192 \u211d rpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x rle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1 h\u03bcs : 0 < \u2191\u2191\u03bc s h\u271d : Nonempty \u03b1 Npos : N \u2260 0 o : Set \u03b1 so : s \u2286 o omeas : MeasurableSet o \u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s a : BallPackage (\u2191s) \u03b1 :=   { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,     r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) } u : Fin N \u2192 Set \u2191s hu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j) hu' : range a.c \u2286 \u22c3 i, \u22c3 j \u2208 u i, ball (BallPackage.c a j) (BallPackage.r a j) u_count : \u2200 (i : Fin N), Set.Countable (u i) v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 x \u2208 u i, closedBall (\u2191x) (r \u2191x) A : s = \u22c3 i, s \u2229 v i S : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i) i : Fin N hi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i) B : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) w : Finset \u2191(u i) hw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) k' : \u2191(u i) left\u271d\u00b9 : k' \u2208 w hk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) l' : \u2191(u i) left\u271d : l' \u2208 w hl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w) hkl : \u2191\u2191l' \u2260 \u2191\u2191l' h : \u2191k' = \u2191l' \u22a2 False ** exact hkl rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "String.utf8GetAux_of_valid ** cs cs' : List Char i p : Nat hp : i + utf8Len cs = p \u22a2 utf8GetAux (cs ++ cs') { byteIdx := i } { byteIdx := p } = List.headD cs' default ** match cs, cs' with\n| [], [] => rfl\n| [], c::cs' => simp [\u2190 hp, utf8GetAux]\n| c::cs, cs' =>\n  simp [utf8GetAux, -List.headD_eq_head?]; rw [if_neg]\n  case hnc => simp [\u2190 hp, Pos.ext_iff]; exact ne_self_add_add_csize\n  refine utf8GetAux_of_valid cs cs' ?_\n  simpa [Nat.add_assoc, Nat.add_comm] using hp ** cs cs' : List Char i p : Nat hp : i + utf8Len [] = p \u22a2 utf8GetAux ([] ++ []) { byteIdx := i } { byteIdx := p } = List.headD [] default ** rfl ** cs cs'\u271d : List Char i p : Nat c : Char cs' : List Char hp : i + utf8Len [] = p \u22a2 utf8GetAux ([] ++ c :: cs') { byteIdx := i } { byteIdx := p } = List.headD (c :: cs') default ** simp [\u2190 hp, utf8GetAux] ** cs\u271d cs'\u271d : List Char i p : Nat c : Char cs cs' : List Char hp : i + utf8Len (c :: cs) = p \u22a2 utf8GetAux (c :: cs ++ cs') { byteIdx := i } { byteIdx := p } = List.headD cs' default ** simp [utf8GetAux, -List.headD_eq_head?] ** cs\u271d cs'\u271d : List Char i p : Nat c : Char cs cs' : List Char hp : i + utf8Len (c :: cs) = p \u22a2 (if { byteIdx := i } = { byteIdx := p } then c else utf8GetAux (cs ++ cs') ({ byteIdx := i } + c) { byteIdx := p }) =     List.headD cs' default ** rw [if_neg] ** cs\u271d cs'\u271d : List Char i p : Nat c : Char cs cs' : List Char hp : i + utf8Len (c :: cs) = p \u22a2 utf8GetAux (cs ++ cs') ({ byteIdx := i } + c) { byteIdx := p } = List.headD cs' default  case hnc cs\u271d cs'\u271d : List Char i p : Nat c : Char cs cs' : List Char hp : i + utf8Len (c :: cs) = p \u22a2 \u00ac{ byteIdx := i } = { byteIdx := p } ** case hnc => simp [\u2190 hp, Pos.ext_iff]; exact ne_self_add_add_csize ** cs\u271d cs'\u271d : List Char i p : Nat c : Char cs cs' : List Char hp : i + utf8Len (c :: cs) = p \u22a2 utf8GetAux (cs ++ cs') ({ byteIdx := i } + c) { byteIdx := p } = List.headD cs' default ** refine utf8GetAux_of_valid cs cs' ?_ ** cs\u271d cs'\u271d : List Char i p : Nat c : Char cs cs' : List Char hp : i + utf8Len (c :: cs) = p \u22a2 { byteIdx := i }.byteIdx + csize c + utf8Len cs = p ** simpa [Nat.add_assoc, Nat.add_comm] using hp ** cs\u271d cs'\u271d : List Char i p : Nat c : Char cs cs' : List Char hp : i + utf8Len (c :: cs) = p \u22a2 \u00ac{ byteIdx := i } = { byteIdx := p } ** simp [\u2190 hp, Pos.ext_iff] ** cs\u271d cs'\u271d : List Char i p : Nat c : Char cs cs' : List Char hp : i + utf8Len (c :: cs) = p \u22a2 \u00aci = i + (utf8Len cs + csize c) ** exact ne_self_add_add_csize ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Path.zoom_insert ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t' : RBNode \u03b1 v : \u03b1 path : Path \u03b1 t : RBNode \u03b1 ht : Balanced t c n H : zoom (cmp v) t root = (t', path) \u22a2 setBlack (insert path t' v) = setBlack (RBNode.insert cmp t v) ** have \u27e8_, _, ht', hp'\u27e9 := ht.zoom .root H ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering t' : RBNode \u03b1 v : \u03b1 path : Path \u03b1 t : RBNode \u03b1 ht : Balanced t c n H : zoom (cmp v) t root = (t', path) w\u271d\u00b9 : RBColor w\u271d : Nat ht' : Balanced t' w\u271d\u00b9 w\u271d hp' : Path.Balanced c n path w\u271d\u00b9 w\u271d \u22a2 setBlack (insert path t' v) = setBlack (RBNode.insert cmp t v) ** cases ht' with simp [insert]\n| nil => simp [insertNew_eq_insert H, setBlack_idem]\n| red hl hr => rw [\u2190 ins_eq_fill hp' (.red hl hr), insert_setBlack]; exact (zoom_ins H).symm\n| black hl hr => rw [\u2190 ins_eq_fill hp' (.black hl hr), insert_setBlack]; exact (zoom_ins H).symm ** case nil \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 path : Path \u03b1 t : RBNode \u03b1 ht : Balanced t c n H : zoom (cmp v) t root = (nil, path) hp' : Path.Balanced c n path black 0 \u22a2 setBlack (insertNew path v) = setBlack (RBNode.insert cmp t v) ** simp [insertNew_eq_insert H, setBlack_idem] ** case red \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 path : Path \u03b1 t : RBNode \u03b1 ht : Balanced t c n w\u271d : Nat x\u271d y\u271d : RBNode \u03b1 v\u271d : \u03b1 H : zoom (cmp v) t root = (node red x\u271d v\u271d y\u271d, path) hp' : Path.Balanced c n path red w\u271d hl : Balanced x\u271d black w\u271d hr : Balanced y\u271d black w\u271d \u22a2 setBlack (fill path (node red x\u271d v y\u271d)) = setBlack (RBNode.insert cmp t v) ** rw [\u2190 ins_eq_fill hp' (.red hl hr), insert_setBlack] ** case red \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 path : Path \u03b1 t : RBNode \u03b1 ht : Balanced t c n w\u271d : Nat x\u271d y\u271d : RBNode \u03b1 v\u271d : \u03b1 H : zoom (cmp v) t root = (node red x\u271d v\u271d y\u271d, path) hp' : Path.Balanced c n path red w\u271d hl : Balanced x\u271d black w\u271d hr : Balanced y\u271d black w\u271d \u22a2 ins path (node red x\u271d v y\u271d) = setBlack (RBNode.ins cmp v t) ** exact (zoom_ins H).symm ** case black \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 path : Path \u03b1 t : RBNode \u03b1 ht : Balanced t c n x\u271d : RBNode \u03b1 c\u2081\u271d : RBColor n\u271d : Nat y\u271d : RBNode \u03b1 c\u2082\u271d : RBColor v\u271d : \u03b1 hl : Balanced x\u271d c\u2081\u271d n\u271d hr : Balanced y\u271d c\u2082\u271d n\u271d H : zoom (cmp v) t root = (node black x\u271d v\u271d y\u271d, path) hp' : Path.Balanced c n path black (n\u271d + 1) \u22a2 setBlack (fill path (node black x\u271d v y\u271d)) = setBlack (RBNode.insert cmp t v) ** rw [\u2190 ins_eq_fill hp' (.black hl hr), insert_setBlack] ** case black \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 path : Path \u03b1 t : RBNode \u03b1 ht : Balanced t c n x\u271d : RBNode \u03b1 c\u2081\u271d : RBColor n\u271d : Nat y\u271d : RBNode \u03b1 c\u2082\u271d : RBColor v\u271d : \u03b1 hl : Balanced x\u271d c\u2081\u271d n\u271d hr : Balanced y\u271d c\u2082\u271d n\u271d H : zoom (cmp v) t root = (node black x\u271d v\u271d y\u271d, path) hp' : Path.Balanced c n path black (n\u271d + 1) \u22a2 ins path (node black x\u271d v y\u271d) = setBlack (RBNode.ins cmp v t) ** exact (zoom_ins H).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityMeasure.toFiniteMeasure_normalize_eq_self ** \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0\u271d : MeasurableSpace \u03a9 \u03bc\u271d : FiniteMeasure \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u22a2 normalize (ProbabilityMeasure.toFiniteMeasure \u03bc) = \u03bc ** apply ProbabilityMeasure.eq_of_forall_apply_eq ** case h \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0\u271d : MeasurableSpace \u03a9 \u03bc\u271d : FiniteMeasure \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u22a2 \u2200 (s : Set \u03a9),     MeasurableSet s \u2192       (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize (ProbabilityMeasure.toFiniteMeasure \u03bc)) s)) s =         (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s ** intro s _s_mble ** case h \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0\u271d : MeasurableSpace \u03a9 \u03bc\u271d : FiniteMeasure \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 s : Set \u03a9 _s_mble : MeasurableSet s \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(normalize (ProbabilityMeasure.toFiniteMeasure \u03bc)) s)) s =     (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s ** rw [\u03bc.toFiniteMeasure.normalize_eq_of_nonzero \u03bc.toFiniteMeasure_nonzero s] ** case h \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0\u271d : MeasurableSpace \u03a9 \u03bc\u271d : FiniteMeasure \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 s : Set \u03a9 _s_mble : MeasurableSet s \u22a2 (mass (ProbabilityMeasure.toFiniteMeasure \u03bc))\u207b\u00b9 *       (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(ProbabilityMeasure.toFiniteMeasure \u03bc) s)) s =     (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s ** simp only [ProbabilityMeasure.mass_toFiniteMeasure, inv_one, one_mul] ** case h \u03a9 : Type u_1 inst\u271d : Nonempty \u03a9 m0\u271d : MeasurableSpace \u03a9 \u03bc\u271d : FiniteMeasure \u03a9 m0 : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 s : Set \u03a9 _s_mble : MeasurableSet s \u22a2 ENNReal.toNNReal (\u2191\u2191\u2191(ProbabilityMeasure.toFiniteMeasure \u03bc) s) = ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s) ** congr ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.im ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p \u22a2 Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p ** have : \u2200 x, \u2016IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016 := by\n  intro x\n  rw [one_mul]\n  exact IsROrC.norm_im_le_norm (f x) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p this : \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016 \u22a2 Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p ** refine' hf.of_le_mul _ (eventually_of_forall this) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p this : \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016 \u22a2 AEStronglyMeasurable (fun x => \u2191IsROrC.im (f x)) \u03bc ** exact IsROrC.continuous_im.comp_aestronglyMeasurable hf.1 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p \u22a2 \u2200 (x : \u03b1), \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016 ** intro x ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p x : \u03b1 \u22a2 \u2016\u2191IsROrC.im (f x)\u2016 \u2264 1 * \u2016f x\u2016 ** rw [one_mul] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p x : \u03b1 \u22a2 \u2016\u2191IsROrC.im (f x)\u2016 \u2264 \u2016f x\u2016 ** exact IsROrC.norm_im_le_norm (f x) ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec'.bind ** n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Vector \u2115 (n + 1) \u2192. \u2115 hf : Partrec' f hg : Partrec' g i : Fin (n + 1) \u22a2 Partrec' ((fun i => Fin.cases f (fun i v => \u2191(some (Vector.get v i))) i) i) ** refine' Fin.cases _ (fun i => _) i <;> simp [*] ** case refine'_2 n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Vector \u2115 (n + 1) \u2192. \u2115 hf : Partrec' f hg : Partrec' g i\u271d : Fin (n + 1) i : Fin n \u22a2 Partrec' fun v => Part.some (Vector.get v i) ** exact prim (Nat.Primrec'.get _) ** n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Vector \u2115 (n + 1) \u2192. \u2115 hf : Partrec' f hg : Partrec' g v : Vector \u2115 n \u22a2 (mOfFn fun i => Fin.cases f (fun i v => \u2191(some (Vector.get v i))) i v) >>= g = Part.bind (f v) fun a => g (a ::\u1d65 v) ** simp [mOfFn, Part.bind_assoc, pure] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.nonempty_inter_of_measure_lt_add' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t\u271d : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t u : Set \u03b1 hs : MeasurableSet s h's : s \u2286 u h't : t \u2286 u h : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u2191\u2191\u03bc t \u22a2 Set.Nonempty (s \u2229 t) ** rw [add_comm] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t\u271d : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t u : Set \u03b1 hs : MeasurableSet s h's : s \u2286 u h't : t \u2286 u h : \u2191\u2191\u03bc u < \u2191\u2191\u03bc t + \u2191\u2191\u03bc s \u22a2 Set.Nonempty (s \u2229 t) ** rw [inter_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t\u271d : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t u : Set \u03b1 hs : MeasurableSet s h's : s \u2286 u h't : t \u2286 u h : \u2191\u2191\u03bc u < \u2191\u2191\u03bc t + \u2191\u2191\u03bc s \u22a2 Set.Nonempty (t \u2229 s) ** exact nonempty_inter_of_measure_lt_add \u03bc hs h't h's h ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.measurable_ceil ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : LinearOrderedSemiring R inst\u271d\u2074 : FloorSemiring R inst\u271d\u00b3 : TopologicalSpace R inst\u271d\u00b2 : OrderTopology R inst\u271d\u00b9 : MeasurableSpace R inst\u271d : OpensMeasurableSpace R f : \u03b1 \u2192 R n : R \u22a2 MeasurableSet (ceil \u207b\u00b9' {\u2308n\u2309\u208a}) ** cases' eq_or_ne \u2308n\u2309\u208a 0 with h h <;> simp_all [h, Nat.preimage_ceil_of_ne_zero, -ceil_eq_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.unifIntegrable_of ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 UnifIntegrable f p \u03bc ** set g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => (hf i).choose ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u22a2 UnifIntegrable f p \u03bc ** refine'\n  (unifIntegrable_of' \u03bc hp hp' (fun i => (Exists.choose_spec <| hf i).1) fun \u03b5 h\u03b5 => _).ae_eq\n    fun i => (Exists.choose_spec <| hf i).2.symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 C,     0 < C \u2227       \u2200 (i : \u03b9),         snorm             (indicator {x | C \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}               (Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)))             p \u03bc \u2264           ENNReal.ofReal \u03b5 ** obtain \u27e8C, hC\u27e9 := h \u03b5 h\u03b5 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hCg : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 C,     0 < C \u2227       \u2200 (i : \u03b9),         snorm             (indicator {x | C \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}               (Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)))             p \u03bc \u2264           ENNReal.ofReal \u03b5 ** refine' \u27e8max C 1, lt_max_of_lt_right one_pos, fun i => le_trans (snorm_mono fun x => _) (hCg i)\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hCg : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 x : \u03b1 \u22a2 \u2016indicator {x | max C 1 \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}         (Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc)) x\u2016 \u2264     \u2016indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x\u2016 ** rw [norm_indicator_eq_indicator_norm, norm_indicator_eq_indicator_norm] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hCg : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 x : \u03b1 \u22a2 indicator {x | max C 1 \u2264 \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x\u2016\u208a}       (fun a => \u2016Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) a\u2016) x \u2264     indicator {x | C \u2264 \u2016g i x\u2016\u208a} (fun a => \u2016g i a\u2016) x ** exact Set.indicator_le_indicator_of_subset\n  (fun x hx => Set.mem_setOf_eq \u25b8 le_trans (le_max_left _ _) hx) (fun _ => norm_nonneg _) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** intro i ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 \u22a2 snorm (indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' le_trans (le_of_eq <| snorm_congr_ae _) (hC i) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 \u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) =\u1d50[\u03bc] indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) ** filter_upwards [(Exists.choose_spec <| hf i).2] with x hx ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 x : \u03b1 hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x \u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x = indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x ** by_cases hfx : x \u2208 { x | C \u2264 \u2016f i x\u2016\u208a } ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 x : \u03b1 hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x hfx : x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} \u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x = indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x ** rw [Set.indicator_of_mem hfx, Set.indicator_of_mem, hx] ** case pos.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 x : \u03b1 hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x hfx : x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} \u22a2 x \u2208 {x | C \u2264 \u2016g i x\u2016\u208a} ** rwa [Set.mem_setOf, hx] at hfx ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 x : \u03b1 hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x hfx : \u00acx \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} \u22a2 indicator {x | C \u2264 \u2016g i x\u2016\u208a} (g i) x = indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i) x ** rw [Set.indicator_of_not_mem hfx, Set.indicator_of_not_mem] ** case neg.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g\u271d : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 := fun i => Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 i : \u03b9 x : \u03b1 hx : f i x = Exists.choose (_ : AEStronglyMeasurable (f i) \u03bc) x hfx : \u00acx \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} \u22a2 \u00acx \u2208 {x | C \u2264 \u2016g i x\u2016\u208a} ** rwa [Set.mem_setOf, hx] at hfx ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.Heap.realSize_tail ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 \u22a2 realSize (tail le s) = realSize s - 1 ** simp only [Heap.tail] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 \u22a2 realSize (Option.getD (tail? le s) nil) = realSize s - 1 ** match eq : s.tail? le with\n| none => cases s with cases eq | nil => rfl\n| some tl => simp [Heap.realSize_tail? eq]; rfl ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s : Heap \u03b1 eq : tail? le s = none \u22a2 realSize (Option.getD none nil) = realSize s - 1 ** cases s with cases eq | nil => rfl ** case nil.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool \u22a2 realSize (Option.getD none nil) = realSize nil - 1 ** rfl ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s tl : Heap \u03b1 eq : tail? le s = some tl \u22a2 realSize (Option.getD (some tl) nil) = realSize s - 1 ** simp [Heap.realSize_tail? eq] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s tl : Heap \u03b1 eq : tail? le s = some tl \u22a2 realSize (Option.getD (some tl) nil) = realSize tl ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.upcrossings_ae_lt_top' ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, upcrossings a b f \u03c9 < \u22a4 ** refine' ae_lt_top (hf.adapted.measurable_upcrossings hab) _ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b \u22a2 \u222b\u207b (x : \u03a9), upcrossings a b f x \u2202\u03bc \u2260 \u22a4 ** have := hf.mul_lintegral_upcrossings_le_lintegral_pos_part a b ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u22a2 \u222b\u207b (x : \u03a9), upcrossings a b f x \u2202\u03bc \u2260 \u22a4 ** rw [mul_comm, \u2190 ENNReal.le_div_iff_mul_le] at this ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) \u22a2 \u222b\u207b (x : \u03a9), upcrossings a b f x \u2202\u03bc \u2260 \u22a4 ** refine' (lt_of_le_of_lt this (ENNReal.div_lt_top _ _)).ne ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ \u22a2 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u2260 \u22a4 ** refine' ne_of_lt (iSup_lt_iff.2 \u27e8R + \u2016a\u2016\u208a * \u03bc Set.univ, ENNReal.add_lt_top.2\n  \u27e8ENNReal.coe_lt_top, ENNReal.mul_lt_top ENNReal.coe_lt_top.ne (measure_ne_top _ _)\u27e9,\n  fun n => le_trans _ (hR' n)\u27e9) ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ n : \u2115 \u22a2 \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f n \u03c9 - a)\u207a \u2202\u03bc \u2264 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc ** refine' lintegral_mono fun \u03c9 => _ ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ n : \u2115 \u03c9 : \u03a9 \u22a2 ENNReal.ofReal (f n \u03c9 - a)\u207a \u2264 \u2191\u2016f n \u03c9 - a\u2016\u208a ** rw [ENNReal.ofReal_le_iff_le_toReal, ENNReal.coe_toReal, coe_nnnorm] ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ n : \u2115 \u03c9 : \u03a9 \u22a2 (f n \u03c9 - a)\u207a \u2264 \u2016f n \u03c9 - a\u2016  case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ n : \u2115 \u03c9 : \u03a9 \u22a2 \u2191\u2016f n \u03c9 - a\u2016\u208a \u2260 \u22a4 ** by_cases hnonneg : 0 \u2264 f n \u03c9 - a ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) \u22a2 \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ ** simp_rw [snorm_one_eq_lintegral_nnnorm] at hbdd ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R \u22a2 \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ ** intro n ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R n : \u2115 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ ** refine' (lintegral_mono _ : \u222b\u207b \u03c9, \u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b \u03c9, \u2016f n \u03c9\u2016\u208a + \u2016a\u2016\u208a \u2202\u03bc).trans _ ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R n : \u2115 \u22a2 (fun \u03c9 => \u2191\u2016f n \u03c9 - a\u2016\u208a) \u2264 fun \u03c9 => \u2191\u2016f n \u03c9\u2016\u208a + \u2191\u2016a\u2016\u208a ** intro \u03c9 ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R n : \u2115 \u03c9 : \u03a9 \u22a2 (fun \u03c9 => \u2191\u2016f n \u03c9 - a\u2016\u208a) \u03c9 \u2264 (fun \u03c9 => \u2191\u2016f n \u03c9\u2016\u208a + \u2191\u2016a\u2016\u208a) \u03c9 ** simp_rw [sub_eq_add_neg, \u2190 nnnorm_neg a, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe] ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R n : \u2115 \u03c9 : \u03a9 \u22a2 \u2016f n \u03c9 + -a\u2016\u208a \u2264 \u2016f n \u03c9\u2016\u208a + \u2016-a\u2016\u208a ** exact nnnorm_add_le _ _ ** case refine'_2 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R n : \u2115 \u22a2 \u222b\u207b (a_1 : \u03a9), \u2191\u2016f n a_1\u2016\u208a + \u2191\u2016a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ ** simp_rw [lintegral_add_right _ measurable_const, lintegral_const] ** case refine'_2 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hbdd : \u2200 (n : \u2115), \u222b\u207b (x : \u03a9), \u2191\u2016f n x\u2016\u208a \u2202\u03bc \u2264 \u2191R n : \u2115 \u22a2 \u222b\u207b (a : \u03a9), \u2191\u2016f n a\u2016\u208a \u2202\u03bc + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ ** exact add_le_add (hbdd _) le_rfl ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ n : \u2115 \u03c9 : \u03a9 hnonneg : 0 \u2264 f n \u03c9 - a \u22a2 (f n \u03c9 - a)\u207a \u2264 \u2016f n \u03c9 - a\u2016 ** rw [LatticeOrderedGroup.pos_of_nonneg _ hnonneg, Real.norm_eq_abs,\n  abs_of_nonneg hnonneg] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ n : \u2115 \u03c9 : \u03a9 hnonneg : \u00ac0 \u2264 f n \u03c9 - a \u22a2 (f n \u03c9 - a)\u207a \u2264 \u2016f n \u03c9 - a\u2016 ** rw [LatticeOrderedGroup.pos_of_nonpos _ (not_le.1 hnonneg).le] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ n : \u2115 \u03c9 : \u03a9 hnonneg : \u00ac0 \u2264 f n \u03c9 - a \u22a2 0 \u2264 \u2016f n \u03c9 - a\u2016 ** exact norm_nonneg _ ** case refine'_1 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9\u271d : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) hR' : \u2200 (n : \u2115), \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016f n \u03c9 - a\u2016\u208a \u2202\u03bc \u2264 \u2191R + \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc Set.univ n : \u2115 \u03c9 : \u03a9 \u22a2 \u2191\u2016f n \u03c9 - a\u2016\u208a \u2260 \u22a4 ** simp only [Ne.def, ENNReal.coe_ne_top, not_false_iff] ** case refine'_2 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 (\u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc) / ENNReal.ofReal (b - a) \u22a2 ENNReal.ofReal (b - a) \u2260 0 ** simp only [hab, Ne.def, ENNReal.ofReal_eq_zero, sub_nonpos, not_le] ** case h0 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : (\u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc) * ENNReal.ofReal (b - a) \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u22a2 ENNReal.ofReal (b - a) \u2260 0 \u2228 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u2260 0 ** simp only [hab, Ne.def, ENNReal.ofReal_eq_zero, sub_nonpos, not_le, true_or_iff] ** case ht \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R hab : a < b this : (\u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc) * ENNReal.ofReal (b - a) \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u22a2 ENNReal.ofReal (b - a) \u2260 \u22a4 \u2228 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc \u2260 \u22a4 ** simp only [Ne.def, ENNReal.ofReal_ne_top, not_false_iff, true_or_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.Regular.map ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : OpensMeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : BorelSpace \u03b2 inst\u271d : Regular \u03bc f : \u03b1 \u2243\u209c \u03b2 \u22a2 Regular (map (\u2191f) \u03bc) ** haveI := OuterRegular.map f \u03bc ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : OpensMeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : BorelSpace \u03b2 inst\u271d : Regular \u03bc f : \u03b1 \u2243\u209c \u03b2 this : OuterRegular (map (\u2191f) \u03bc) \u22a2 Regular (map (\u2191f) \u03bc) ** haveI := IsFiniteMeasureOnCompacts.map \u03bc f ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : OpensMeasurableSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : T2Space \u03b2 inst\u271d\u00b9 : BorelSpace \u03b2 inst\u271d : Regular \u03bc f : \u03b1 \u2243\u209c \u03b2 this\u271d : OuterRegular (map (\u2191f) \u03bc) this : IsFiniteMeasureOnCompacts (map (\u2191f) \u03bc) \u22a2 Regular (map (\u2191f) \u03bc) ** exact\n  \u27e8Regular.innerRegular.map f.toEquiv f.measurable.aemeasurable\n      (fun U hU => hU.preimage f.continuous) (fun K hK => hK.image f.continuous)\n      (fun K hK => hK.measurableSet) fun U hU => hU.measurableSet\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.restrict_map_withDensity_abs_det_fderiv_eq_addHaar ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 Measure.map (Set.restrict s f)       (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) ** obtain \u27e8u, u_meas, uf\u27e9 : \u2203 u, Measurable u \u2227 EqOn u f s := by\n  classical\n  refine' \u27e8piecewise s f 0, _, piecewise_eqOn _ _ _\u27e9\n  refine' ContinuousOn.measurable_piecewise _ continuous_zero.continuousOn hs\n  have : DifferentiableOn \u211d f s := fun x hx => (hf' x hx).differentiableWithinAt\n  exact this.continuousOn ** case intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s \u22a2 Measure.map (Set.restrict s f)       (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) ** have u' : \u2200 x \u2208 s, HasFDerivWithinAt u (f' x) s x := fun x hx =>\n  (hf' x hx).congr (fun y hy => uf hy) (uf hx) ** case intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s u' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x \u22a2 Measure.map (Set.restrict s f)       (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) ** set F : s \u2192 E := u \u2218 (\u2191) with hF ** case intro.intro E : Type u_1 F\u271d : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F\u271d inst\u271d\u00b3 : NormedSpace \u211d F\u271d s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s u' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x F : \u2191s \u2192 E := u \u2218 Subtype.val hF : F = u \u2218 Subtype.val \u22a2 Measure.map (Set.restrict s f)       (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) ** have A :\n  Measure.map F (comap (\u2191) (\u03bc.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =\n    \u03bc.restrict (u '' s) := by\n  rw [hF, \u2190 Measure.map_map u_meas measurable_subtype_coe, map_comap_subtype_coe hs,\n    restrict_withDensity hs]\n  exact map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs u' (hf.congr uf.symm) u_meas ** case intro.intro E : Type u_1 F\u271d : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F\u271d inst\u271d\u00b3 : NormedSpace \u211d F\u271d s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s u' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x F : \u2191s \u2192 E := u \u2218 Subtype.val hF : F = u \u2218 Subtype.val A :   Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (u '' s) \u22a2 Measure.map (Set.restrict s f)       (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) ** rw [uf.image_eq] at A ** case intro.intro E : Type u_1 F\u271d : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F\u271d inst\u271d\u00b3 : NormedSpace \u211d F\u271d s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s u' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x F : \u2191s \u2192 E := u \u2218 Subtype.val hF : F = u \u2218 Subtype.val A :   Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) \u22a2 Measure.map (Set.restrict s f)       (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) ** have : F = s.restrict f := by\n  ext x\n  exact uf x.2 ** case intro.intro E : Type u_1 F\u271d : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F\u271d inst\u271d\u00b3 : NormedSpace \u211d F\u271d s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s u' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x F : \u2191s \u2192 E := u \u2218 Subtype.val hF : F = u \u2218 Subtype.val A :   Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) this : F = Set.restrict s f \u22a2 Measure.map (Set.restrict s f)       (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) ** rwa [this] at A ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 \u2203 u, Measurable u \u2227 EqOn u f s ** classical\nrefine' \u27e8piecewise s f 0, _, piecewise_eqOn _ _ _\u27e9\nrefine' ContinuousOn.measurable_piecewise _ continuous_zero.continuousOn hs\nhave : DifferentiableOn \u211d f s := fun x hx => (hf' x hx).differentiableWithinAt\nexact this.continuousOn ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 \u2203 u, Measurable u \u2227 EqOn u f s ** refine' \u27e8piecewise s f 0, _, piecewise_eqOn _ _ _\u27e9 ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 Measurable (piecewise s f 0) ** refine' ContinuousOn.measurable_piecewise _ continuous_zero.continuousOn hs ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s \u22a2 ContinuousOn f s ** have : DifferentiableOn \u211d f s := fun x hx => (hf' x hx).differentiableWithinAt ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s this : DifferentiableOn \u211d f s \u22a2 ContinuousOn f s ** exact this.continuousOn ** E : Type u_1 F\u271d : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F\u271d inst\u271d\u00b3 : NormedSpace \u211d F\u271d s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s u' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x F : \u2191s \u2192 E := u \u2218 Subtype.val hF : F = u \u2218 Subtype.val \u22a2 Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (u '' s) ** rw [hF, \u2190 Measure.map_map u_meas measurable_subtype_coe, map_comap_subtype_coe hs,\n  restrict_withDensity hs] ** E : Type u_1 F\u271d : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F\u271d inst\u271d\u00b3 : NormedSpace \u211d F\u271d s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s u' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x F : \u2191s \u2192 E := u \u2218 Subtype.val hF : F = u \u2218 Subtype.val \u22a2 Measure.map u (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =     Measure.restrict \u03bc (u '' s) ** exact map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs u' (hf.congr uf.symm) u_meas ** E : Type u_1 F\u271d : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F\u271d inst\u271d\u00b3 : NormedSpace \u211d F\u271d s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s u' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x F : \u2191s \u2192 E := u \u2218 Subtype.val hF : F = u \u2218 Subtype.val A :   Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) \u22a2 F = Set.restrict s f ** ext x ** case h E : Type u_1 F\u271d : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F\u271d inst\u271d\u00b3 : NormedSpace \u211d F\u271d s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x hf : InjOn f s u : E \u2192 E u_meas : Measurable u uf : EqOn u f s u' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x F : \u2191s \u2192 E := u \u2218 Subtype.val hF : F = u \u2218 Subtype.val A :   Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =     Measure.restrict \u03bc (f '' s) x : \u2191s \u22a2 F x = Set.restrict s f x ** exact uf x.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.restrictNonposSeq_lt ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 n : \u2115 hn :   \u00acrestrict s (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u2264       restrict 0 (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u22a2 1 /       (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s             (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +         1) <     \u2191s (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n)) ** rw [restrictNonposSeq_succ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 n : \u2115 hn :   \u00acrestrict s (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u2264       restrict 0 (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) \u22a2 1 /       (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s             (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +         1) <     \u2191s       (MeasureTheory.SignedMeasure.someExistsOneDivLT s         (i \\ \u22c3 k, \u22c3 (_ : k \u2264 n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) ** apply someExistsOneDivLT_lt hn ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.Cofix.abs_repr ** n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : Cofix F \u03b1 \u22a2 Quot.mk Mcongr (repr x) = x ** let R := fun x y : Cofix F \u03b1 => abs (repr y) = x ** n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x \u22a2 Quot.mk Mcongr (repr x) = x ** refine' Cofix.bisim\u2082 R _ _ _ rfl ** n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n x : Cofix F \u03b1 R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x \u22a2 \u2200 (x y : Cofix F \u03b1), R x y \u2192 LiftR' (RelLast' \u03b1 R) (dest x) (dest y) ** clear x ** n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x \u22a2 \u2200 (x y : Cofix F \u03b1), R x y \u2192 LiftR' (RelLast' \u03b1 R) (dest x) (dest y) ** rintro x y h ** n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x x y : Cofix F \u03b1 h : R x y \u22a2 LiftR' (RelLast' \u03b1 R) (dest x) (dest y) ** subst h ** n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x y : Cofix F \u03b1 \u22a2 LiftR' (RelLast' \u03b1 R) (dest (abs (repr y))) (dest y) ** dsimp [Cofix.dest, Cofix.abs] ** n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x y : Cofix F \u03b1 \u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (repr y) = x)     ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr y)))     (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))       (_ :         \u2200 (x y : M (P F) \u03b1),           Mcongr x y \u2192             (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =               (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)       y) ** induction y using Quot.ind ** case mk n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x a\u271d : M (P F) \u03b1 \u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (repr y) = x)     ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr (Quot.mk Mcongr a\u271d))))     (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))       (_ :         \u2200 (x y : M (P F) \u03b1),           Mcongr x y \u2192             (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =               (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)       (Quot.mk Mcongr a\u271d)) ** simp only [Cofix.repr, M.dest_corec, abs_map, MvQPF.abs_repr, Function.comp] ** case mk n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x a\u271d : M (P F) \u03b1 \u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)     ((TypeVec.id ::: Quot.mk Mcongr) <$$>       (TypeVec.id ::: M.corec (P F) fun x => MvQPF.repr (dest x)) <$$> dest (Quot.mk Mcongr a\u271d))     ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d)) ** conv =>\n  congr\n  rfl\n  rw [Cofix.dest] ** case mk n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x a\u271d : M (P F) \u03b1 \u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)     ((TypeVec.id ::: Quot.mk Mcongr) <$$>       (TypeVec.id :::           M.corec (P F) fun x =>             MvQPF.repr               (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))                 (_ :                   \u2200 (x y : M (P F) \u03b1),                     Mcongr x y \u2192                       (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =                         (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)                 x)) <$$>         Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))           (_ :             \u2200 (x y : M (P F) \u03b1),               Mcongr x y \u2192                 (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =                   (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)           (Quot.mk Mcongr a\u271d))     ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d)) ** rw [MvFunctor.map_map, MvFunctor.map_map, \u2190appendFun_comp_id, \u2190appendFun_comp_id] ** case mk n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x a\u271d : M (P F) \u03b1 \u22a2 LiftR' (RelLast' \u03b1 fun x y => Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) y) = x)     ((TypeVec.id :::         (Quot.mk Mcongr \u2218             M.corec (P F) fun x =>               MvQPF.repr                 (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))                   (_ :                     \u2200 (x y : M (P F) \u03b1),                       Mcongr x y \u2192                         (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =                           (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)                   x)) \u2218           Quot.mk Mcongr) <$$>       MvQPF.abs (M.dest (P F) a\u271d))     ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) a\u271d)) ** apply liftR_map_last ** case mk.hh n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x a\u271d : M (P F) \u03b1 \u22a2 \u2200 (x : M (P F) \u03b1),     Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) (Quot.mk Mcongr x)) =       ((Quot.mk Mcongr \u2218             M.corec (P F) fun x =>               MvQPF.repr                 (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))                   (_ :                     \u2200 (x y : M (P F) \u03b1),                       Mcongr x y \u2192                         (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =                           (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)                   x)) \u2218           Quot.mk Mcongr)         x ** intros ** case mk.hh n : \u2115 F\u271d : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F\u271d q\u271d : MvQPF F\u271d F : TypeVec.{u} (n + 1) \u2192 Type u inst\u271d : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n R : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => abs (repr y) = x a\u271d x\u271d : M (P F) \u03b1 \u22a2 Quot.mk Mcongr (M.corec (P F) (fun x => MvQPF.repr (dest x)) (Quot.mk Mcongr x\u271d)) =     ((Quot.mk Mcongr \u2218           M.corec (P F) fun x =>             MvQPF.repr               (Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x))                 (_ :                   \u2200 (x y : M (P F) \u03b1),                     Mcongr x y \u2192                       (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) x =                         (fun x => (TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) x)) y)                 x)) \u2218         Quot.mk Mcongr)       x\u271d ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "IsUnifLocDoublingMeasure.closedBall_mem_vitaliFamily_of_dist_le_mul ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r \u22a2 closedBall y r \u2208 VitaliFamily.setsAt (vitaliFamily \u03bc K) x ** let R := scalingScaleOf \u03bc (max (4 * K + 3) 3) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) \u22a2 closedBall y r \u2208 VitaliFamily.setsAt (vitaliFamily \u03bc K) x ** simp only [vitaliFamily, VitaliFamily.enlarge, Vitali.vitaliFamily, mem_union, mem_setOf_eq,\n  isClosed_ball, true_and_iff, (nonempty_ball.2 rpos).mono ball_subset_interior_closedBall,\n  measurableSet_closedBall] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) \u22a2 (\u2203 r_1,       closedBall y r \u2286 closedBall x r_1 \u2227         \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228     \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4) ** by_cases H : closedBall y r \u2286 closedBall x (R / 4) ** case pos \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) \u22a2 (\u2203 r_1,       closedBall y r \u2286 closedBall x r_1 \u2227         \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228     \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4)  case neg \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : \u00acclosedBall y r \u2286 closedBall x (R / 4) \u22a2 (\u2203 r_1,       closedBall y r \u2286 closedBall x r_1 \u2227         \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228     \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4) ** swap ** case pos \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) \u22a2 (\u2203 r_1,       closedBall y r \u2286 closedBall x r_1 \u2227         \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228     \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4) ** left ** case pos.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) \u22a2 \u2203 r_1,     closedBall y r \u2286 closedBall x r_1 \u2227       \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** rcases le_or_lt r R with (hr | hr) ** case neg \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : \u00acclosedBall y r \u2286 closedBall x (R / 4) \u22a2 (\u2203 r_1,       closedBall y r \u2286 closedBall x r_1 \u2227         \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r)) \u2228     \u00acclosedBall y r \u2286 closedBall x (scalingScaleOf \u03bc (max (4 * K + 3) 3) / 4) ** exact Or.inr H ** case pos.h.inl \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R \u22a2 \u2203 r_1,     closedBall y r \u2286 closedBall x r_1 \u2227       \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** refine' \u27e8(K + 1) * r, _\u27e9 ** case pos.h.inl \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R \u22a2 closedBall y r \u2286 closedBall x ((K + 1) * r) \u2227     \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** constructor ** case pos.h.inl.left \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R \u22a2 closedBall y r \u2286 closedBall x ((K + 1) * r) ** apply closedBall_subset_closedBall' ** case pos.h.inl.left.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R \u22a2 r + dist y x \u2264 (K + 1) * r ** rw [dist_comm] ** case pos.h.inl.left.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R \u22a2 r + dist x y \u2264 (K + 1) * r ** linarith ** case pos.h.inl.right \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R \u22a2 \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** have I1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r) := by\n  apply closedBall_subset_closedBall'\n  linarith ** case pos.h.inl.right \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R I1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r) \u22a2 \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** have I2 : closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r) := by\n  apply closedBall_subset_closedBall\n  exact mul_le_mul_of_nonneg_right (le_max_left _ _) rpos.le ** case pos.h.inl.right \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R I1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r) I2 : closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r) \u22a2 \u2191\u2191\u03bc (closedBall x (3 * ((K + 1) * r))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** apply (measure_mono (I1.trans I2)).trans ** case pos.h.inl.right \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R I1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r) I2 : closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r) \u22a2 \u2191\u2191\u03bc (closedBall y (max (4 * K + 3) 3 * r)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** exact measure_mul_le_scalingConstantOf_mul _\n  \u27e8zero_lt_three.trans_le (le_max_right _ _), le_rfl\u27e9 hr ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R \u22a2 closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r) ** apply closedBall_subset_closedBall' ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R \u22a2 3 * ((K + 1) * r) + dist x y \u2264 (4 * K + 3) * r ** linarith ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R I1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r) \u22a2 closedBall y ((4 * K + 3) * r) \u2286 closedBall y (max (4 * K + 3) 3 * r) ** apply closedBall_subset_closedBall ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : r \u2264 R I1 : closedBall x (3 * ((K + 1) * r)) \u2286 closedBall y ((4 * K + 3) * r) \u22a2 (4 * K + 3) * r \u2264 max (4 * K + 3) 3 * r ** exact mul_le_mul_of_nonneg_right (le_max_left _ _) rpos.le ** case pos.h.inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : R < r \u22a2 \u2203 r_1,     closedBall y r \u2286 closedBall x r_1 \u2227       \u2191\u2191\u03bc (closedBall x (3 * r_1)) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** refine' \u27e8R / 4, H, _\u27e9 ** case pos.h.inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : R < r \u22a2 \u2191\u2191\u03bc (closedBall x (3 * (R / 4))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** have : closedBall x (3 * (R / 4)) \u2286 closedBall y r := by\n  apply closedBall_subset_closedBall'\n  have A : y \u2208 closedBall y r := mem_closedBall_self rpos.le\n  have B := mem_closedBall'.1 (H A)\n  linarith ** case pos.h.inr \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : R < r this : closedBall x (3 * (R / 4)) \u2286 closedBall y r \u22a2 \u2191\u2191\u03bc (closedBall x (3 * (R / 4))) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** apply (measure_mono this).trans _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : R < r this : closedBall x (3 * (R / 4)) \u2286 closedBall y r \u22a2 \u2191\u2191\u03bc (closedBall y r) \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) * \u2191\u2191\u03bc (closedBall y r) ** refine' le_mul_of_one_le_left (zero_le _) _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : R < r this : closedBall x (3 * (R / 4)) \u2286 closedBall y r \u22a2 1 \u2264 \u2191(scalingConstantOf \u03bc (max (4 * K + 3) 3)) ** exact ENNReal.one_le_coe_iff.2 (le_max_right _ _) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : R < r \u22a2 closedBall x (3 * (R / 4)) \u2286 closedBall y r ** apply closedBall_subset_closedBall' ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : R < r \u22a2 3 * (R / 4) + dist x y \u2264 r ** have A : y \u2208 closedBall y r := mem_closedBall_self rpos.le ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : R < r A : y \u2208 closedBall y r \u22a2 3 * (R / 4) + dist x y \u2264 r ** have B := mem_closedBall'.1 (H A) ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : IsUnifLocDoublingMeasure \u03bc inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bc K : \u211d x y : \u03b1 r : \u211d h : dist x y \u2264 K * r rpos : 0 < r R : \u211d := scalingScaleOf \u03bc (max (4 * K + 3) 3) H : closedBall y r \u2286 closedBall x (R / 4) hr : R < r A : y \u2208 closedBall y r B : dist x y \u2264 R / 4 \u22a2 3 * (R / 4) + dist x y \u2264 r ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.strong_law_aux7 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** obtain \u27e8c, -, cone, clim\u27e9 :\n    \u2203 c : \u2115 \u2192 \u211d, StrictAnti c \u2227 (\u2200 n : \u2115, 1 < c n) \u2227 Tendsto c atTop (\ud835\udcdd 1) :=\n  exists_seq_strictAnti_tendsto (1 : \u211d) ** case intro.intro.intro \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u2115 \u2192 \u211d cone : \u2200 (n : \u2115), 1 < c n clim : Tendsto c atTop (\ud835\udcdd 1) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** have : \u2200 k, \u2200\u1d50 \u03c9,\n    Tendsto (fun n : \u2115 => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd \ud835\udd3c[X 0]) :=\n  fun k => strong_law_aux6 X hint hindep hident hnonneg (cone k) ** case intro.intro.intro \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u2115 \u2192 \u211d cone : \u2200 (n : \u2115), 1 < c n clim : Tendsto c atTop (\ud835\udcdd 1) this :   \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** filter_upwards [ae_all_iff.2 this] with \u03c9 h\u03c9 ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u2115 \u2192 \u211d cone : \u2200 (n : \u2115), 1 < c n clim : Tendsto c atTop (\ud835\udcdd 1) this :   \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u22a2 Tendsto (fun n => (\u2211 i in range n, X i \u03c9) / \u2191n) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** apply tendsto_div_of_monotone_of_tendsto_div_floor_pow _ _ _ c cone clim _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u2115 \u2192 \u211d cone : \u2200 (n : \u2115), 1 < c n clim : Tendsto c atTop (\ud835\udcdd 1) this :   \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u22a2 Monotone fun n => \u2211 i in range n, X i \u03c9 ** intro m n hmn ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u2115 \u2192 \u211d cone : \u2200 (n : \u2115), 1 < c n clim : Tendsto c atTop (\ud835\udcdd 1) this :   \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) m n : \u2115 hmn : m \u2264 n \u22a2 (fun n => \u2211 i in range n, X i \u03c9) m \u2264 (fun n => \u2211 i in range n, X i \u03c9) n ** exact sum_le_sum_of_subset_of_nonneg (range_mono hmn) fun i _ _ => hnonneg i \u03c9 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u2115 \u2192 \u211d cone : \u2200 (n : \u2115), 1 < c n clim : Tendsto c atTop (\ud835\udcdd 1) this :   \u2200 (k : \u2115), \u2200\u1d50 (\u03c9 : \u03a9), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u03c9 : \u03a9 h\u03c9 : \u2200 (i : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac i ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac i ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) \u22a2 \u2200 (k : \u2115), Tendsto (fun n => (\u2211 i in range \u230ac k ^ n\u230b\u208a, X i \u03c9) / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (\u222b (a : \u03a9), X 0 a)) ** exact h\u03c9 ** Qed",
        "informal": ""
    },
    {
        "formal": "map_restrict_ae_le_map_indicator_ae ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 f : \u03b1 \u2192 \u03b2 inst\u271d : Zero \u03b2 hs : MeasurableSet s \u22a2 Filter.map f (ae (Measure.restrict \u03bc s)) \u2264 Filter.map (indicator s f) (ae \u03bc) ** intro t ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 f : \u03b1 \u2192 \u03b2 inst\u271d : Zero \u03b2 hs : MeasurableSet s t : Set \u03b2 \u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s)) ** by_cases ht : (0 : \u03b2) \u2208 t ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 f : \u03b1 \u2192 \u03b2 inst\u271d : Zero \u03b2 hs : MeasurableSet s t : Set \u03b2 ht : \u00ac0 \u2208 t \u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s)) ** rw [mem_map_indicator_ae_iff_of_zero_nmem ht, mem_map_restrict_ae_iff hs] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 f : \u03b1 \u2192 \u03b2 inst\u271d : Zero \u03b2 hs : MeasurableSet s t : Set \u03b2 ht : \u00ac0 \u2208 t \u22a2 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0 \u2192 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u2229 s) = 0 ** exact fun h => measure_mono_null ((Set.inter_subset_left _ _).trans (Set.subset_union_left _ _)) h ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 f : \u03b1 \u2192 \u03b2 inst\u271d : Zero \u03b2 hs : MeasurableSet s t : Set \u03b2 ht : 0 \u2208 t \u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s)) ** rw [mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem ht hs] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 f : \u03b1 \u2192 \u03b2 inst\u271d : Zero \u03b2 hs : MeasurableSet s t : Set \u03b2 ht : 0 \u2208 t \u22a2 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s)) \u2192 t \u2208 Filter.map f (ae (Measure.restrict \u03bc s)) ** exact id ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.VectorMeasure.of_union ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : T2Space M v : VectorMeasure \u03b1 M f : \u2115 \u2192 Set \u03b1 A B : Set \u03b1 h : Disjoint A B hA : MeasurableSet A hB : MeasurableSet B \u22a2 \u2191v (A \u222a B) = \u2191v A + \u2191v B ** rw [Set.union_eq_iUnion, of_disjoint_iUnion, tsum_fintype, Fintype.sum_bool, cond, cond] ** case hf\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : T2Space M v : VectorMeasure \u03b1 M f : \u2115 \u2192 Set \u03b1 A B : Set \u03b1 h : Disjoint A B hA : MeasurableSet A hB : MeasurableSet B \u22a2 \u2200 (i : Bool), MeasurableSet (bif i then A else B)  case hf\u2082 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : T2Space M v : VectorMeasure \u03b1 M f : \u2115 \u2192 Set \u03b1 A B : Set \u03b1 h : Disjoint A B hA : MeasurableSet A hB : MeasurableSet B \u22a2 Pairwise (Disjoint on fun b => bif b then A else B) ** exacts [fun b => Bool.casesOn b hB hA, pairwise_disjoint_on_bool.2 h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpIndL1Fin_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \u211d x : G \u22a2 condexpIndL1Fin hm hs h\u03bcs (c \u2022 x) = c \u2022 condexpIndL1Fin hm hs h\u03bcs x ** ext1 ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \u211d x : G \u22a2 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs (c \u2022 x)) =\u1d50[\u03bc] \u2191\u2191(c \u2022 condexpIndL1Fin hm hs h\u03bcs x) ** refine' (Mem\u2112p.coeFn_toLp q).trans _ ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \u211d x : G \u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (c \u2022 x)) =\u1d50[\u03bc] \u2191\u2191(c \u2022 condexpIndL1Fin hm hs h\u03bcs x) ** refine' EventuallyEq.trans _ (Lp.coeFn_smul _ _).symm ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \u211d x : G \u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (c \u2022 x)) =\u1d50[\u03bc] c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) ** rw [condexpIndSMul_smul hs h\u03bcs c x] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \u211d x : G \u22a2 \u2191\u2191(c \u2022 condexpIndSMul hm hs h\u03bcs x) =\u1d50[\u03bc] c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) ** refine' (Lp.coeFn_smul _ _).trans _ ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \u211d x : G \u22a2 c \u2022 \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) =\u1d50[\u03bc] c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) ** refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h\u03bcs x).mono fun y hy => _ ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : \u211d x : G y : \u03b1 hy : \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) y = \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) y \u22a2 (c \u2022 \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)) y = (c \u2022 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x)) y ** rw [Pi.smul_apply, Pi.smul_apply, hy] ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_subtype_mk_eq_iff ** \u03b1 : Sort u_1 \u03b2 : Sort u_2 \u03b3 : Sort u_3 p q : \u03b1 \u2192 Prop a : Subtype p b : \u03b1 \u22a2 (\u2203 h, { val := b, property := h } = a) \u2194 b = \u2191a ** simp only [@eq_comm _ b, exists_eq_subtype_mk_iff, @eq_comm _ _ a] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBSet.mem_insert ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp \u22a2 v' \u2208 insert t v \u2194 v' \u2208 t \u2228 cmp v v' = Ordering.eq ** refine \u27e8fun h => ?_, fun | .inl h => mem_insert_of_mem _ h | .inr h => mem_insert_of_eq _ h\u27e9 ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp h : v' \u2208 insert t v \u22a2 v' \u2208 t \u2228 cmp v v' = Ordering.eq ** let \u27e8_, h\u2081, h\u2082\u27e9 := mem_iff_mem_toList.1 h ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp h : v' \u2208 insert t v w\u271d : \u03b1 h\u2081 : w\u271d \u2208 toList (insert t v) h\u2082 : cmp v' w\u271d = Ordering.eq \u22a2 v' \u2208 t \u2228 cmp v v' = Ordering.eq ** match mem_toList_insert.1 h\u2081 with\n| .inl \u27e8h\u2083, _\u27e9 => exact .inl <| mem_iff_mem_toList.2 \u27e8_, h\u2083, h\u2082\u27e9\n| .inr rfl => exact .inr <| OrientedCmp.cmp_eq_eq_symm.1 h\u2082 ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp w\u271d : \u03b1 h\u2082 : cmp v' w\u271d = Ordering.eq v : \u03b1 h\u2083 : w\u271d \u2208 toList t right\u271d : find? t v \u2260 some w\u271d h : v' \u2208 insert t v h\u2081 : w\u271d \u2208 toList (insert t v) \u22a2 v' \u2208 t \u2228 cmp v v' = Ordering.eq ** exact .inl <| mem_iff_mem_toList.2 \u27e8_, h\u2083, h\u2082\u27e9 ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp w\u271d : \u03b1 h\u2082 : cmp v' w\u271d = Ordering.eq h : v' \u2208 insert t w\u271d h\u2081 : w\u271d \u2208 toList (insert t w\u271d) \u22a2 v' \u2208 t \u2228 cmp w\u271d v' = Ordering.eq ** exact .inr <| OrientedCmp.cmp_eq_eq_symm.1 h\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.atEnd ** l m r : List Char p : Nat \u22a2 Substring.atEnd         { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },           stopPos := { byteIdx := utf8Len l + utf8Len m } }         { byteIdx := p } =       true \u2194     p = utf8Len m ** simp [Substring.atEnd, Pos.ext_iff, Nat.add_left_cancel_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Iic_eq_cons_Iio ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrderBot \u03b1 b : \u03b1 \u22a2 Iic b = cons b (Iio b) (_ : \u00acb \u2208 Iio b) ** classical rw [cons_eq_insert, Iio_insert] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : LocallyFiniteOrderBot \u03b1 b : \u03b1 \u22a2 Iic b = cons b (Iio b) (_ : \u00acb \u2208 Iio b) ** rw [cons_eq_insert, Iio_insert] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.extract.go\u2081_add_right_cancel ** s : List Char i b e n : Nat \u22a2 go\u2081 s { byteIdx := i + n } { byteIdx := b + n } { byteIdx := e + n } =     go\u2081 s { byteIdx := i } { byteIdx := b } { byteIdx := e } ** apply utf8InductionOn s \u27e8i\u27e9 \u27e8b\u27e9 (motive := fun s i =>\n  go\u2081 s \u27e8i.byteIdx + n\u27e9 \u27e8b + n\u27e9 \u27e8e + n\u27e9 = go\u2081 s i \u27e8b\u27e9 \u27e8e\u27e9) <;>\nsimp [go\u2081] ** case eq s : List Char i b e n : Nat \u22a2 \u2200 (c : Char) (cs : List Char),     go\u2082 (c :: cs) { byteIdx := b + n } { byteIdx := e + n } = go\u2082 (c :: cs) { byteIdx := b } { byteIdx := e } ** intro c cs ** case eq s : List Char i b e n : Nat c : Char cs : List Char \u22a2 go\u2082 (c :: cs) { byteIdx := b + n } { byteIdx := e + n } = go\u2082 (c :: cs) { byteIdx := b } { byteIdx := e } ** apply go\u2082_add_right_cancel ** case ind s : List Char i b e n : Nat \u22a2 \u2200 (c : Char) (cs : List Char) (i : Pos),     \u00aci = { byteIdx := b } \u2192       go\u2081 cs { byteIdx := i.byteIdx + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =           go\u2081 cs (i + c) { byteIdx := b } { byteIdx := e } \u2192         (if { byteIdx := i.byteIdx + n } = { byteIdx := b + n } then             go\u2082 (c :: cs) { byteIdx := i.byteIdx + n } { byteIdx := e + n }           else go\u2081 cs ({ byteIdx := i.byteIdx + n } + c) { byteIdx := b + n } { byteIdx := e + n }) =           if i = { byteIdx := b } then go\u2082 (c :: cs) i { byteIdx := e }           else go\u2081 cs (i + c) { byteIdx := b } { byteIdx := e } ** intro c cs \u27e8i\u27e9 h ih ** case ind s : List Char i\u271d b e n : Nat c : Char cs : List Char i : Nat h : \u00ac{ byteIdx := i } = { byteIdx := b } ih :   go\u2081 cs { byteIdx := { byteIdx := i }.byteIdx + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =     go\u2081 cs ({ byteIdx := i } + c) { byteIdx := b } { byteIdx := e } \u22a2 (if { byteIdx := { byteIdx := i }.byteIdx + n } = { byteIdx := b + n } then       go\u2082 (c :: cs) { byteIdx := { byteIdx := i }.byteIdx + n } { byteIdx := e + n }     else go\u2081 cs ({ byteIdx := { byteIdx := i }.byteIdx + n } + c) { byteIdx := b + n } { byteIdx := e + n }) =     if { byteIdx := i } = { byteIdx := b } then go\u2082 (c :: cs) { byteIdx := i } { byteIdx := e }     else go\u2081 cs ({ byteIdx := i } + c) { byteIdx := b } { byteIdx := e } ** simp [Pos.ext_iff, Pos.addChar_eq] at h ih \u22a2 ** case ind s : List Char i\u271d b e n : Nat c : Char cs : List Char i : Nat ih :   go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =     go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e } h : \u00aci = b \u22a2 (if i + n = b + n then go\u2082 (c :: cs) { byteIdx := i + n } { byteIdx := e + n }     else go\u2081 cs { byteIdx := i + n + csize c } { byteIdx := b + n } { byteIdx := e + n }) =     if i = b then go\u2082 (c :: cs) { byteIdx := i } { byteIdx := e }     else go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e } ** simp [Nat.add_right_cancel_iff, h] ** case ind s : List Char i\u271d b e n : Nat c : Char cs : List Char i : Nat ih :   go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =     go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e } h : \u00aci = b \u22a2 go\u2081 cs { byteIdx := i + n + csize c } { byteIdx := b + n } { byteIdx := e + n } =     go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e } ** rw [Nat.add_right_comm] ** case ind s : List Char i\u271d b e n : Nat c : Char cs : List Char i : Nat ih :   go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =     go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e } h : \u00aci = b \u22a2 go\u2081 cs { byteIdx := i + csize c + n } { byteIdx := b + n } { byteIdx := e + n } =     go\u2081 cs { byteIdx := i + csize c } { byteIdx := b } { byteIdx := e } ** exact ih ** Qed",
        "informal": ""
    },
    {
        "formal": "Complex.circleTransformDeriv_bound ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) \u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B ** obtain \u27e8r, hr, hrx\u27e9 := exists_lt_mem_ball_of_mem_ball hx ** case intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B ** obtain \u27e8\u03b5', h\u03b5', H\u27e9 := exists_ball_subset_ball hrx ** case intro.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r \u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B ** obtain \u27e8\u27e8\u27e8a, b\u27e9, \u27e8ha, hb\u27e9\u27e9, hab\u27e9 :=\n  abs_circleTransformBoundingFunction_le hr (pos_of_mem_ball hrx).le z ** case intro.intro.intro.intro.intro.mk.mk.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) \u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B ** let V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun _ => 1) \u03b8 ** case intro.intro.intro.intro.intro.mk.mk.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 \u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B ** have funccomp : ContinuousOn (fun r => abs (f r)) (sphere z R) := by\n  have cabs : ContinuousOn abs \u22a4 := by apply continuous_abs.continuousOn\n  apply cabs.comp hf; rw [MapsTo]; tauto ** case intro.intro.intro.intro.intro.mk.mk.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) \u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B ** have sbou :=\n  IsCompact.exists_isMaxOn (isCompact_sphere z R) (NormedSpace.sphere_nonempty.2 hR.le) funccomp ** case intro.intro.intro.intro.intro.mk.mk.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) sbou : \u2203 x, x \u2208 sphere z R \u2227 IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) x \u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B ** obtain \u27e8X, HX, HX2\u27e9 := sbou ** case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R HX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X \u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B ** refine' \u27e8abs (V b a) * abs (f X), \u03b5', h\u03b5', Subset.trans H (ball_subset_ball hr.le), _\u27e9 ** case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R HX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X \u22a2 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5' \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X) ** intro y v hv ** case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R HX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X y : \u211d v : \u2102 hv : v \u2208 ball x \u03b5' \u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X) ** obtain \u27e8y1, hy1, hfun\u27e9 :=\n  Periodic.exists_mem_Ico\u2080 (circleTransformDeriv_periodic R z v f) Real.two_pi_pos y ** case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R HX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X y : \u211d v : \u2102 hv : v \u2208 ball x \u03b5' y1 : \u211d hy1 : y1 \u2208 Ico 0 (2 * \u03c0) hfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1 \u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X) ** have hy2 : y1 \u2208 [[0, 2 * \u03c0]] := by\n  convert Ico_subset_Icc_self hy1 using 1\n  simp [uIcc_of_le Real.two_pi_pos.le] ** case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R HX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X y : \u211d v : \u2102 hv : v \u2208 ball x \u03b5' y1 : \u211d hy1 : y1 \u2208 Ico 0 (2 * \u03c0) hfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1 hy2 : y1 \u2208 [[0, 2 * \u03c0]] \u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X) ** simp only [IsMaxOn, IsMaxFilter, eventually_principal, mem_sphere_iff_norm, norm_eq_abs] at HX2 ** case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R y : \u211d v : \u2102 hv : v \u2208 ball x \u03b5' y1 : \u211d hy1 : y1 \u2208 Ico 0 (2 * \u03c0) hfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1 hy2 : y1 \u2208 [[0, 2 * \u03c0]] HX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X) \u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X) ** have := mul_le_mul (hab \u27e8\u27e8v, y1\u27e9, \u27e8ball_subset_closedBall (H hv), hy2\u27e9\u27e9)\n  (HX2 (circleMap z R y1) (circleMap_mem_sphere z hR.le y1)) (Complex.abs.nonneg _)\n  (Complex.abs.nonneg _) ** case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R y : \u211d v : \u2102 hv : v \u2208 ball x \u03b5' y1 : \u211d hy1 : y1 \u2208 Ico 0 (2 * \u03c0) hfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1 hy2 : y1 \u2208 [[0, 2 * \u03c0]] HX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X) this :   \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (v, y1), property := (_ : (v, y1).1 \u2208 closedBall z r \u2227 (v, y1).2 \u2208 [[0, 2 * \u03c0]]) }) *       \u2191abs (f (circleMap z R y1)) \u2264     \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) *       \u2191abs (f X) \u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X) ** simp_rw [hfun] ** case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R y : \u211d v : \u2102 hv : v \u2208 ball x \u03b5' y1 : \u211d hy1 : y1 \u2208 Ico 0 (2 * \u03c0) hfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1 hy2 : y1 \u2208 [[0, 2 * \u03c0]] HX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X) this :   \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (v, y1), property := (_ : (v, y1).1 \u2208 closedBall z r \u2227 (v, y1).2 \u2208 [[0, 2 * \u03c0]]) }) *       \u2191abs (f (circleMap z R y1)) \u2264     \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) *       \u2191abs (f X) \u22a2 \u2016circleTransformDeriv R z v f y1\u2016 \u2264 \u2191abs (circleTransformDeriv R z a (fun x => 1) b) * \u2191abs (f X) ** simp only [circleTransformBoundingFunction, circleTransformDeriv, norm_eq_abs,\n  Algebra.id.smul_eq_mul, deriv_circleMap, map_mul, abs_circleMap_zero, abs_I, mul_one, \u2190\n  mul_assoc, mul_inv_rev, inv_I, abs_neg, abs_inv, abs_ofReal, one_mul, abs_two, abs_pow,\n  mem_ball, gt_iff_lt, Subtype.coe_mk, SetCoe.forall, mem_prod, mem_closedBall, and_imp,\n  Prod.forall, NormedSpace.sphere_nonempty, mem_sphere_iff_norm] at * ** case intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R \u03b5' : \u211d H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 y : \u211d v : \u2102 hv : v \u2208 ball x \u03b5' y1 : \u211d hy1 : y1 \u2208 Ico 0 (2 * \u03c0) hy2 : y1 \u2208 [[0, 2 * \u03c0]] HX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X) hx : dist x z < R hrx : dist x z < r h\u03b5' : 0 < \u03b5' hab :   \u2200 (a_1 : \u2102) (b_1 : \u211d),     dist a_1 z \u2264 r \u2192       b_1 \u2208 [[0, 2 * \u03c0]] \u2192         \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b_1 - a_1) ^ 2)\u207b\u00b9 \u2264           \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9 HX : \u2191abs (X - z) = R hfun :   -I * (\u2191\u03c0)\u207b\u00b9 * 2\u207b\u00b9 * circleMap 0 R y * I * ((circleMap z R y - v) ^ 2)\u207b\u00b9 * f (circleMap z R y) =     -I * (\u2191\u03c0)\u207b\u00b9 * 2\u207b\u00b9 * circleMap 0 R y1 * I * ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * f (circleMap z R y1) this :   \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * \u2191abs (f (circleMap z R y1)) \u2264     \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9 * \u2191abs (f X) \u22a2 \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * \u2191abs (f (circleMap z R y1)) \u2264     \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9 * \u2191abs (f X) ** exact this ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 \u22a2 ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) ** have cabs : ContinuousOn abs \u22a4 := by apply continuous_abs.continuousOn ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 cabs : ContinuousOn \u2191abs \u22a4 \u22a2 ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) ** apply cabs.comp hf ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 cabs : ContinuousOn \u2191abs \u22a4 \u22a2 MapsTo f (sphere z R) \u22a4 ** rw [MapsTo] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 cabs : ContinuousOn \u2191abs \u22a4 \u22a2 \u2200 \u2983x : \u2102\u2984, x \u2208 sphere z R \u2192 f x \u2208 \u22a4 ** tauto ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 \u22a2 ContinuousOn \u2191abs \u22a4 ** apply continuous_abs.continuousOn ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R HX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X y : \u211d v : \u2102 hv : v \u2208 ball x \u03b5' y1 : \u211d hy1 : y1 \u2208 Ico 0 (2 * \u03c0) hfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1 \u22a2 y1 \u2208 [[0, 2 * \u03c0]] ** convert Ico_subset_Icc_self hy1 using 1 ** case h.e'_5 E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R : \u211d hR : 0 < R z x : \u2102 f : \u2102 \u2192 \u2102 hx : x \u2208 ball z R hf : ContinuousOn f (sphere z R) r : \u211d hr : r < R hrx : x \u2208 ball z r \u03b5' : \u211d h\u03b5' : \u03b5' > 0 H : ball x \u03b5' \u2286 ball z r a : \u2102 b : \u211d ha : (a, b).1 \u2208 closedBall z r hb : (a, b).2 \u2208 [[0, 2 * \u03c0]] hab :   \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),     \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264       \u2191abs         (circleTransformBoundingFunction R z           \u2191{ val := (a, b), property := (_ : (a, b).1 \u2208 closedBall z r \u2227 (a, b).2 \u2208 [[0, 2 * \u03c0]]) }) V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8 funccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R) X : \u2102 HX : X \u2208 sphere z R HX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X y : \u211d v : \u2102 hv : v \u2208 ball x \u03b5' y1 : \u211d hy1 : y1 \u2208 Ico 0 (2 * \u03c0) hfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1 \u22a2 [[0, 2 * \u03c0]] = Icc 0 (2 * \u03c0) ** simp [uIcc_of_le Real.two_pi_pos.le] ** Qed",
        "informal": ""
    },
    {
        "formal": "snd_integral ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2078 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedSpace \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace E f : \u03b1 \u2192 E \u00d7 F hf : Integrable f \u22a2 (\u222b (x : \u03b1), f x \u2202\u03bc).2 = \u222b (x : \u03b1), (f x).2 \u2202\u03bc ** rw [\u2190 Prod.fst_swap, swap_integral] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2078 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedSpace \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace E f : \u03b1 \u2192 E \u00d7 F hf : Integrable f \u22a2 (\u222b (x : \u03b1), Prod.swap (f x) \u2202\u03bc).1 = \u222b (x : \u03b1), (f x).2 \u2202\u03bc ** exact fst_integral <| hf.snd.prod_mk hf.fst ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.Heap.WF.tail ** \u03b1 : Type u_1 s : Heap \u03b1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n : Nat hwf : WF le n s \u22a2 WF le 0 (Heap.tail le s) ** simp only [Heap.tail] ** \u03b1 : Type u_1 s : Heap \u03b1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n : Nat hwf : WF le n s \u22a2 WF le 0 (Option.getD (Heap.tail? le s) Heap.nil) ** match eq : s.tail? le with\n| none => exact Heap.WF.nil\n| some tl => exact hwf.tail? eq ** \u03b1 : Type u_1 s : Heap \u03b1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n : Nat hwf : WF le n s eq : Heap.tail? le s = none \u22a2 WF le 0 (Option.getD none Heap.nil) ** exact Heap.WF.nil ** \u03b1 : Type u_1 s : Heap \u03b1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n : Nat hwf : WF le n s tl : Heap \u03b1 eq : Heap.tail? le s = some tl \u22a2 WF le 0 (Option.getD (some tl) Heap.nil) ** exact hwf.tail? eq ** Qed",
        "informal": ""
    },
    {
        "formal": "SatisfiesM_ReaderT_eq ** m : Type u_1 \u2192 Type u_2 \u03c1 \u03b1\u271d : Type u_1 p : \u03b1\u271d \u2192 Prop x : ReaderT \u03c1 m \u03b1\u271d inst\u271d : Monad m a : ReaderT \u03c1 m { a // p a } \u22a2 Subtype.val <$> a = x \u2194 \u2200 (x_1 : \u03c1), Subtype.val <$> a x_1 = x x_1 ** exact \u27e8fun eq _ => eq \u25b8 rfl, funext\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.not_monotoneOn_not_antitoneOn_iff_exists_lt_lt ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LinearOrder \u03b2 f : \u03b1 \u2192 \u03b2 \u22a2 \u00acMonotoneOn f s \u2227 \u00acAntitoneOn f s \u2194 \u2203 a x b x c x, a < b \u2227 b < c \u2227 (f a < f b \u2227 f c < f b \u2228 f b < f a \u2227 f b < f c) ** simp [monotoneOn_iff_monotone, antitoneOn_iff_antitone, and_assoc, exists_and_left,\n  not_monotone_not_antitone_iff_exists_lt_lt, @and_left_comm (_ \u2208 s)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.UniformIntegrable.ae_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : UniformIntegrable f p \u03bc hfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n \u22a2 UniformIntegrable g p \u03bc ** obtain \u27e8hfm, hunif, C, hC\u27e9 := hf ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n hfm : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hunif : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C \u22a2 UniformIntegrable g p \u03bc ** refine' \u27e8fun i => (hfm i).congr (hfg i), (unifIntegrable_congr_ae hfg).1 hunif, C, fun i => _\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n hfm : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hunif : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C i : \u03b9 \u22a2 snorm (g i) p \u03bc \u2264 \u2191C ** rw [\u2190 snorm_congr_ae (hfg i)] ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f g : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hfg : \u2200 (n : \u03b9), f n =\u1d50[\u03bc] g n hfm : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hunif : UnifIntegrable f p \u03bc C : \u211d\u22650 hC : \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C i : \u03b9 \u22a2 snorm (f i) p \u03bc \u2264 \u2191C ** exact hC i ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.uIcc_injective_right ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : DistribLattice \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x a b c : \u03b1 h : (fun b => [[b, a]]) b = (fun b => [[b, a]]) c \u22a2 b = c ** rw [ext_iff] at h ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : DistribLattice \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x a b c : \u03b1 h : \u2200 (a_1 : \u03b1), a_1 \u2208 (fun b => [[b, a]]) b \u2194 a_1 \u2208 (fun b => [[b, a]]) c \u22a2 b = c ** exact eq_of_mem_uIcc_of_mem_uIcc ((h _).1 left_mem_uIcc) ((h _).2 left_mem_uIcc) ** Qed",
        "informal": ""
    },
    {
        "formal": "QPF.Cofix.bisim_rel ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y \u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y ** let r' (x y) := x = y \u2228 r x y ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y \u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y ** intro x y rxy ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F rxy : r x y \u22a2 x = y ** apply Cofix.bisim_aux r' ** case a F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F rxy : r x y \u22a2 r' x y ** right ** case a.h F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F rxy : r x y \u22a2 r x y ** exact rxy ** case h' F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F rxy : r x y \u22a2 \u2200 (x : Cofix F), r' x x ** intro x ** case h' F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y : Cofix F rxy : r x\u271d y x : Cofix F \u22a2 r' x x ** left ** case h'.h F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y : Cofix F rxy : r x\u271d y x : Cofix F \u22a2 x = x ** rfl ** case h F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F rxy : r x y \u22a2 \u2200 (x y : Cofix F), r' x y \u2192 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y ** intro x y r'xy ** case h F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F rxy : r x\u271d y\u271d x y : Cofix F r'xy : r' x y \u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y ** cases' r'xy with r'xy r'xy ** case h.inr F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F rxy : r x\u271d y\u271d x y : Cofix F r'xy : r x y \u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y ** have : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h ** case h.inr F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F rxy : r x\u271d y\u271d x y : Cofix F r'xy : r x y this : \u2200 (x y : Cofix F), r x y \u2192 r' x y \u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y ** rw [\u2190 Quot.factor_mk_eq _ _ this] ** case h.inr F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F rxy : r x\u271d y\u271d x y : Cofix F r'xy : r x y this : \u2200 (x y : Cofix F), r x y \u2192 r' x y \u22a2 (Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$> dest x =     (Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$> dest y ** dsimp ** case h.inr F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F rxy : r x\u271d y\u271d x y : Cofix F r'xy : r x y this : \u2200 (x y : Cofix F), r x y \u2192 r' x y \u22a2 (Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$> dest x =     (Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$> dest y ** rw [@comp_map _ _ q _ _ _ (Quot.mk r), @comp_map _ _ q _ _ _ (Quot.mk r)] ** case h.inr F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F rxy : r x\u271d y\u271d x y : Cofix F r'xy : r x y this : \u2200 (x y : Cofix F), r x y \u2192 r' x y \u22a2 Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this <$> Quot.mk r <$> dest x =     Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this <$> Quot.mk r <$> dest y ** rw [h _ _ r'xy] ** case h.inl F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F r : Cofix F \u2192 Cofix F \u2192 Prop h : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y r' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F rxy : r x\u271d y\u271d x y : Cofix F r'xy : x = y \u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y ** rw [r'xy] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.iIndepSet.condexp_indicator_filtrationOfSet_ae_eq ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc \u03b9 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b2 : LinearOrder \u03b9 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : BorelSpace \u03b2 f : \u03b9 \u2192 \u03a9 \u2192 \u03b2 i j : \u03b9 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (n : \u03b9), MeasurableSet (s n) hs : iIndepSet s hij : i < j \u22a2 \u03bc[Set.indicator (s j) fun x => 1|\u2191(filtrationOfSet hsm) i] =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j)) ** rw [Filtration.filtrationOfSet_eq_natural (\u03b2 := \u211d) hsm] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc \u03b9 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b2 : LinearOrder \u03b9 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : BorelSpace \u03b2 f : \u03b9 \u2192 \u03a9 \u2192 \u03b2 i j : \u03b9 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (n : \u03b9), MeasurableSet (s n) hs : iIndepSet s hij : i < j \u22a2 \u03bc[Set.indicator (s j) fun x =>         1|\u2191(Filtration.natural (fun i => Set.indicator (s i) fun x => 1)             (_ : \u2200 (i : \u03b9), StronglyMeasurable (Set.indicator (s i) 1)))         i] =\u1d50[\u03bc]     fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j)) ** refine' (iIndepFun.condexp_natural_ae_eq_of_lt _ hs.iIndepFun_indicator hij).trans _ ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc \u03b9 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b2 : LinearOrder \u03b9 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : BorelSpace \u03b2 f : \u03b9 \u2192 \u03a9 \u2192 \u03b2 i j : \u03b9 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (n : \u03b9), MeasurableSet (s n) hs : iIndepSet s hij : i < j \u22a2 (fun x => \u222b (x : \u03a9), Set.indicator (s j) (fun _\u03c9 => 1) x \u2202\u03bc) =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j)) ** simp only [integral_indicator_const _ (hsm _), Algebra.id.smul_eq_mul, mul_one] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsProbabilityMeasure \u03bc \u03b9 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b2 : LinearOrder \u03b9 m\u03b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : BorelSpace \u03b2 f : \u03b9 \u2192 \u03a9 \u2192 \u03b2 i j : \u03b9 s : \u03b9 \u2192 Set \u03a9 hsm : \u2200 (n : \u03b9), MeasurableSet (s n) hs : iIndepSet s hij : i < j \u22a2 (fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j))) =\u1d50[\u03bc] fun x => ENNReal.toReal (\u2191\u2191\u03bc (s j)) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real_nonneg ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 r : \u211d hr : 0 \u2264 r \u22a2 toJordanDecomposition (r \u2022 s) = r \u2022 toJordanDecomposition s ** lift r to \u211d\u22650 using hr ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 r : \u211d\u22650 \u22a2 toJordanDecomposition (\u2191r \u2022 s) = \u2191r \u2022 toJordanDecomposition s ** rw [JordanDecomposition.coe_smul, \u2190 toJordanDecomposition_smul] ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 r : \u211d\u22650 \u22a2 toJordanDecomposition (\u2191r \u2022 s) = toJordanDecomposition (r \u2022 s) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.bitwise_xor ** \u22a2 bitwise xor = Int.xor ** funext m n ** case h.h m n : \u2124 \u22a2 bitwise xor m n = Int.xor m n ** cases' m with m m <;> cases' n with n n <;> try {rfl}\n  <;> simp only [bitwise, natBitwise, Bool.not_false, Bool.or_true, Bool.bne_eq_xor,\n    cond_false, cond_true, lor, Nat.ldiff, Bool.and_true, negSucc.injEq, Bool.false_xor,\n    Bool.true_xor, Bool.and_false, Nat.land, Bool.not_true, ldiff,\n    HOr.hOr, OrOp.or, Nat.lor, Int.xor, HXor.hXor, Xor.xor, Nat.xor] ** case h.h.ofNat.negSucc m n : \u2115 \u22a2 Nat.bitwise (fun x y => !xor x !y) m n = Nat.bitwise xor m n ** congr ** case h.h.ofNat.negSucc.e_f m n : \u2115 \u22a2 (fun x y => !xor x !y) = xor ** funext x y ** case h.h.ofNat.negSucc.e_f.h.h m n : \u2115 x y : Bool \u22a2 (!xor x !y) = xor x y ** cases x <;> cases y <;> rfl ** case h.h.negSucc.ofNat m n : \u2115 \u22a2 Nat.bitwise (fun x y => !xor (!x) y) m n = Nat.bitwise xor m n ** congr ** case h.h.negSucc.ofNat.e_f m n : \u2115 \u22a2 (fun x y => !xor (!x) y) = xor ** funext x y ** case h.h.negSucc.ofNat.e_f.h.h m n : \u2115 x y : Bool \u22a2 (!xor (!x) y) = xor x y ** cases x <;> cases y <;> rfl ** case h.h.negSucc.negSucc m n : \u2115 \u22a2 \u2191(Nat.bitwise (fun x y => xor (!x) !y) m n) = \u2191(Nat.bitwise xor m n) ** congr ** case h.h.negSucc.negSucc.e_a.e_f m n : \u2115 \u22a2 (fun x y => xor (!x) !y) = xor ** funext x y ** case h.h.negSucc.negSucc.e_a.e_f.h.h m n : \u2115 x y : Bool \u22a2 (xor (!x) !y) = xor x y ** cases x <;> cases y <;> rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 h : MeasurableSet s \u22a2 \u2191(restrict \u03bc s) = \u2191(OuterMeasure.restrict s) \u2191\u03bc ** simp_rw [restrict, restrict\u2097, liftLinear, LinearMap.coe_mk, AddHom.coe_mk,\n  toMeasure_toOuterMeasure, OuterMeasure.restrict_trim h, \u03bc.trimmed] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.tr_reaches ** \u03c3\u2081 : Type u_1 \u03c3\u2082 : Type u_2 f\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081 f\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082 tr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop H : Respects f\u2081 f\u2082 tr a\u2081 : \u03c3\u2081 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 b\u2081 : \u03c3\u2081 ab : Reaches f\u2081 a\u2081 b\u2081 \u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082 ** rcases reflTransGen_iff_eq_or_transGen.1 ab with (rfl | ab) ** case inl \u03c3\u2081 : Type u_1 \u03c3\u2082 : Type u_2 f\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081 f\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082 tr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop H : Respects f\u2081 f\u2082 tr a\u2082 : \u03c3\u2082 b\u2081 : \u03c3\u2081 aa : tr b\u2081 a\u2082 ab : Reaches f\u2081 b\u2081 b\u2081 \u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082 ** exact \u27e8_, aa, ReflTransGen.refl\u27e9 ** case inr \u03c3\u2081 : Type u_1 \u03c3\u2082 : Type u_2 f\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081 f\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082 tr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop H : Respects f\u2081 f\u2082 tr a\u2081 : \u03c3\u2081 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 b\u2081 : \u03c3\u2081 ab\u271d : Reaches f\u2081 a\u2081 b\u2081 ab : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 b\u2081 \u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082 ** have \u27e8b\u2082, bb, h\u27e9 := tr_reaches\u2081 H aa ab ** case inr \u03c3\u2081 : Type u_1 \u03c3\u2082 : Type u_2 f\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081 f\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082 tr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop H : Respects f\u2081 f\u2082 tr a\u2081 : \u03c3\u2081 a\u2082 : \u03c3\u2082 aa : tr a\u2081 a\u2082 b\u2081 : \u03c3\u2081 ab\u271d : Reaches f\u2081 a\u2081 b\u2081 ab : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 b\u2081 b\u2082 : \u03c3\u2082 bb : tr b\u2081 b\u2082 h : Reaches\u2081 f\u2082 a\u2082 b\u2082 \u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082 ** exact \u27e8b\u2082, bb, h.to_reflTransGen\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Primrec'.sqrt ** \u22a2 Primrec' fun v => Nat.sqrt (Vector.head v) ** suffices H : \u2200 n : \u2115, n.sqrt = n.rec 0 fun x y => if x.succ < y.succ * y.succ then y else y.succ ** case H  \u22a2 \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n ** introv ** case H n : \u2115 \u22a2 Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n ** symm ** case H n : \u2115 \u22a2 Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n ** induction' n with n IH ** case H.succ n : \u2115 IH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n \u22a2 Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) (Nat.succ n) =     Nat.sqrt (Nat.succ n) ** dsimp ** case H.succ n : \u2115 IH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n \u22a2 (if         Nat.succ n <           Nat.succ (Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n) *             Nat.succ (Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n) then       Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n     else Nat.succ (Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n)) =     Nat.sqrt (Nat.succ n) ** rw [IH] ** case H.succ n : \u2115 IH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n \u22a2 (if Nat.succ n < Nat.succ (Nat.sqrt n) * Nat.succ (Nat.sqrt n) then Nat.sqrt n else Nat.succ (Nat.sqrt n)) =     Nat.sqrt (Nat.succ n) ** split_ifs with h ** H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n \u22a2 Primrec' fun v => Nat.sqrt (Vector.head v) ** simp [H] ** H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n \u22a2 Primrec' fun v =>     Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) (Vector.head v) ** have :=\n  @prec' 1 _ _\n    (fun v => by\n      have x := v.head; have y := v.tail.head;\n        exact if x.succ < y.succ * y.succ then y else y.succ)\n    head (const 0) ?_ ** case refine_1 H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n \u22a2 Primrec' fun v =>     let_fun x := Vector.head v;     let_fun y := Vector.head (Vector.tail v);     if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y ** have x1 : @Primrec' 3 fun v => v.head.succ := succ.comp\u2081 _ head ** case refine_1 H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n x1 : Primrec' fun v => Nat.succ (Vector.head v) \u22a2 Primrec' fun v =>     let_fun x := Vector.head v;     let_fun y := Vector.head (Vector.tail v);     if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y ** have y1 : @Primrec' 3 fun v => v.tail.head.succ := succ.comp\u2081 _ (tail head) ** case refine_1 H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n x1 : Primrec' fun v => Nat.succ (Vector.head v) y1 : Primrec' fun v => Nat.succ (Vector.head (Vector.tail v)) \u22a2 Primrec' fun v =>     let_fun x := Vector.head v;     let_fun y := Vector.head (Vector.tail v);     if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y ** exact if_lt x1 (mul.comp\u2082 _ y1 y1) (tail head) y1 ** H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n v : Vector \u2115 (1 + 2) \u22a2 \u2115 ** have x := v.head ** H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n v : Vector \u2115 (1 + 2) x : \u2115 \u22a2 \u2115 ** have y := v.tail.head ** H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n v : Vector \u2115 (1 + 2) x y : \u2115 \u22a2 \u2115 ** exact if x.succ < y.succ * y.succ then y else y.succ ** case refine_2 H : \u2200 (n : \u2115), Nat.sqrt n = Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n this :   Primrec' fun v =>     Nat.rec 0       (fun y IH =>         (fun v =>             let_fun x := Vector.head v;             let_fun y := Vector.head (Vector.tail v);             if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y)           (y ::\u1d65 IH ::\u1d65 v))       (Vector.head v) \u22a2 Primrec' fun v =>     Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) (Vector.head v) ** exact this ** case H.zero  \u22a2 Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) Nat.zero = Nat.sqrt Nat.zero ** simp ** case pos n : \u2115 IH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n h : Nat.succ n < Nat.succ (Nat.sqrt n) * Nat.succ (Nat.sqrt n) \u22a2 Nat.sqrt n = Nat.sqrt (Nat.succ n) ** exact le_antisymm (Nat.sqrt_le_sqrt (Nat.le_succ _)) (Nat.lt_succ_iff.1 <| Nat.sqrt_lt.2 h) ** case neg n : \u2115 IH : Nat.rec 0 (fun x y => if Nat.succ x < Nat.succ y * Nat.succ y then y else Nat.succ y) n = Nat.sqrt n h : \u00acNat.succ n < Nat.succ (Nat.sqrt n) * Nat.succ (Nat.sqrt n) \u22a2 Nat.succ (Nat.sqrt n) = Nat.sqrt (Nat.succ n) ** exact\n  Nat.eq_sqrt.2 \u27e8not_lt.1 h, Nat.sqrt_lt.1 <| Nat.lt_succ_iff.2 <| Nat.sqrt_succ_le_succ_sqrt _\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.PartrecToTM2.codeSupp_fix ** f : Code k : Cont' \u22a2 codeSupp (Code.fix f) k = trStmts\u2081 (trNormal (Code.fix f) k) \u222a codeSupp f (Cont'.fix f k) ** simp [codeSupp, codeSupp', contSupp, Finset.union_assoc, Finset.union_left_comm,\n  Finset.union_left_idem] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc \u22a2 borel X \u2264 OuterMeasure.caratheodory \u03bc ** rw [borel_eq_generateFrom_isClosed] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc \u22a2 MeasurableSpace.generateFrom {s | IsClosed s} \u2264 OuterMeasure.caratheodory \u03bc ** refine' MeasurableSpace.generateFrom_le fun t ht => \u03bc.isCaratheodory_iff_le.2 fun s => _ ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** set S : \u2115 \u2192 Set X := fun n => {x \u2208 s | (\u2191n)\u207b\u00b9 \u2264 infEdist x t} ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** have n0 : \u2200 {n : \u2115}, (n\u207b\u00b9 : \u211d\u22650\u221e) \u2260 0 := fun {n} => ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _) ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** have Ssep : \u2200 n, IsMetricSeparated (S n) t := fun n =>\n  \u27e8n\u207b\u00b9, n0, fun x hx y hy => hx.2.trans <| infEdist_le_edist_of_mem hy\u27e9 ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** have Ssep' : \u2200 n, IsMetricSeparated (S n) (s \u2229 t) := fun n =>\n  (Ssep n).mono Subset.rfl (inter_subset_right _ _) ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** have S_sub : \u2200 n, S n \u2286 s \\ t := fun n =>\n  subset_inter (inter_subset_left _ _) (Ssep n).subset_compl_right ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** have hSs : \u2200 n, \u03bc (s \u2229 t) + \u03bc (S n) \u2264 \u03bc s := fun n =>\n  calc\n    \u03bc (s \u2229 t) + \u03bc (S n) = \u03bc (s \u2229 t \u222a S n) := Eq.symm <| hm _ _ <| (Ssep' n).symm\n    _ \u2264 \u03bc (s \u2229 t \u222a s \\ t) := \u03bc.mono <| union_subset_union_right _ <| S_sub n\n    _ = \u03bc s := by rw [inter_union_diff] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** have iUnion_S : \u22c3 n, S n = s \\ t := by\n  refine' Subset.antisymm (iUnion_subset S_sub) _\n  rintro x \u27e8hxs, hxt\u27e9\n  rw [mem_iff_infEdist_zero_of_closed ht] at hxt\n  rcases ENNReal.exists_inv_nat_lt hxt with \u27e8n, hn\u27e9\n  exact mem_iUnion.2 \u27e8n, hxs, hn.le\u27e9 ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** by_cases htop : \u03bc (s \\ t) = \u221e ** case neg \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** suffices : \u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u03bc (S n) ** case neg \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 this : \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n) \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s  case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 \u22a2 \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n) ** calc\n  \u03bc (s \u2229 t) + \u03bc (s \\ t) = \u03bc (s \u2229 t) + \u03bc (\u22c3 n, S n) := by rw [iUnion_S]\n  _ \u2264 \u03bc (s \u2229 t) + \u2a06 n, \u03bc (S n) := (add_le_add le_rfl this)\n  _ = \u2a06 n, \u03bc (s \u2229 t) + \u03bc (S n) := ENNReal.add_iSup\n  _ \u2264 \u03bc s := iSup_le hSs ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 \u22a2 \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n) ** have : \u2200 n, S n \u2286 S (n + 1) := fun n x hx =>\n  \u27e8hx.1, le_trans (ENNReal.inv_le_inv.2 <| Nat.cast_le.2 n.le_succ) hx.2\u27e9 ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t n : \u2115 \u22a2 \u2191\u03bc (s \u2229 t \u222a s \\ t) = \u2191\u03bc s ** rw [inter_union_diff] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s \u22a2 \u22c3 n, S n = s \\ t ** refine' Subset.antisymm (iUnion_subset S_sub) _ ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s \u22a2 s \\ t \u2286 \u22c3 n, S n ** rintro x \u27e8hxs, hxt\u27e9 ** case intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s x : X hxs : x \u2208 s hxt : \u00acx \u2208 t \u22a2 x \u2208 \u22c3 n, S n ** rw [mem_iff_infEdist_zero_of_closed ht] at hxt ** case intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s x : X hxs : x \u2208 s hxt : \u00acinfEdist x t = 0 \u22a2 x \u2208 \u22c3 n, S n ** rcases ENNReal.exists_inv_nat_lt hxt with \u27e8n, hn\u27e9 ** case intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s x : X hxs : x \u2208 s hxt : \u00acinfEdist x t = 0 n : \u2115 hn : (\u2191n)\u207b\u00b9 < infEdist x t \u22a2 x \u2208 \u22c3 n, S n ** exact mem_iUnion.2 \u27e8n, hxs, hn.le\u27e9 ** case pos \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u2191\u03bc (s \\ t) = \u22a4 \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** rw [htop, add_top, \u2190 htop] ** case pos \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u2191\u03bc (s \\ t) = \u22a4 \u22a2 \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s ** exact \u03bc.mono (diff_subset _ _) ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 this : \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n) \u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) = \u2191\u03bc (s \u2229 t) + \u2191\u03bc (\u22c3 n, S n) ** rw [iUnion_S] ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 this : \u2200 (n : \u2115), S n \u2286 S (n + 1) \u22a2 \u2191\u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u2191\u03bc (S n) ** refine' (\u03bc.iUnion_nat_of_monotone_of_tsum_ne_top this _).le ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 this : \u2200 (n : \u2115), S n \u2286 S (n + 1) \u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (k + 1) \\ S k) \u2260 \u22a4 ** clear this ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 \u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (k + 1) \\ S k) \u2260 \u22a4 ** rw [\u2190 tsum_even_add_odd ENNReal.summable ENNReal.summable, ENNReal.add_ne_top] ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 \u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4 \u2227 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4 ** suffices : \u2200 a, (\u2211' k : \u2115, \u03bc (S (2 * k + 1 + a) \\ S (2 * k + a))) \u2260 \u221e ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 this : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4 \u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4 \u2227 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4  case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 \u22a2 \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4 ** exact \u27e8by simpa using this 0, by simpa using this 1\u27e9 ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 \u22a2 \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4 ** refine' fun r => ne_top_of_le_ne_top htop _ ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r : \u2115 \u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + r) \\ S (2 * k + r)) \u2264 \u2191\u03bc (s \\ t) ** rw [\u2190 iUnion_S, ENNReal.tsum_eq_iSup_nat, iSup_le_iff] ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r : \u2115 \u22a2 \u2200 (i : \u2115), \u2211 a in Finset.range i, \u2191\u03bc (S (2 * a + 1 + r) \\ S (2 * a + r)) \u2264 \u2191\u03bc (\u22c3 n, S n) ** intro n ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n : \u2115 \u22a2 \u2211 a in Finset.range n, \u2191\u03bc (S (2 * a + 1 + r) \\ S (2 * a + r)) \u2264 \u2191\u03bc (\u22c3 n, S n) ** rw [\u2190 hm.finset_iUnion_of_pairwise_separated] ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n : \u2115 \u22a2 \u2200 (i : \u2115),     i \u2208 Finset.range n \u2192       \u2200 (j : \u2115),         j \u2208 Finset.range n \u2192           i \u2260 j \u2192 IsMetricSeparated (S (2 * i + 1 + r) \\ S (2 * i + r)) (S (2 * j + 1 + r) \\ S (2 * j + r)) ** suffices : \u2200 i j, i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r)) ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n : \u2115 this : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r)) \u22a2 \u2200 (i : \u2115),     i \u2208 Finset.range n \u2192       \u2200 (j : \u2115),         j \u2208 Finset.range n \u2192           i \u2260 j \u2192 IsMetricSeparated (S (2 * i + 1 + r) \\ S (2 * i + r)) (S (2 * j + 1 + r) \\ S (2 * j + r))  case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n : \u2115 \u22a2 \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r)) ** exact fun i _ j _ hij =>\n  hij.lt_or_lt.elim\n    (fun h => (this i j h).mono (inter_subset_left _ _) fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9)\n    fun h => (this j i h).symm.mono (fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9) (inter_subset_left _ _) ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n : \u2115 \u22a2 \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r)) ** intro i j hj ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j \u22a2 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r)) ** have A : ((\u2191(2 * j + r))\u207b\u00b9 : \u211d\u22650\u221e) < (\u2191(2 * i + 1 + r))\u207b\u00b9 := by\n  rw [ENNReal.inv_lt_inv, Nat.cast_lt]; linarith ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j A : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 \u22a2 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r)) ** refine' \u27e8(\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9, by simpa using A, fun x hx y hy => _\u27e9 ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j A : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 x : X hx : x \u2208 S (2 * i + 1 + r) y : X hy : y \u2208 s \\ S (2 * j + r) \u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y ** have : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 := not_le.1 fun hle => hy.2 \u27e8hy.1, hle\u27e9 ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j A : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 x : X hx : x \u2208 S (2 * i + 1 + r) y : X hy : y \u2208 s \\ S (2 * j + r) this : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 \u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y ** rcases infEdist_lt_iff.mp this with \u27e8z, hzt, hyz\u27e9 ** case this.intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j A : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 x : X hx : x \u2208 S (2 * i + 1 + r) y : X hy : y \u2208 s \\ S (2 * j + r) this : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 z : X hzt : z \u2208 t hyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9 \u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y ** have hxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z := le_infEdist.1 hx.2 _ hzt ** case this.intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j A : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 x : X hx : x \u2208 S (2 * i + 1 + r) y : X hy : y \u2208 s \\ S (2 * j + r) this : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 z : X hzt : z \u2208 t hyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9 hxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z \u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y ** apply ENNReal.le_of_add_le_add_right hyz.ne_top ** case this.intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j A : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 x : X hx : x \u2208 S (2 * i + 1 + r) y : X hy : y \u2208 s \\ S (2 * j + r) this : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 z : X hzt : z \u2208 t hyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9 hxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z \u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + edist y z \u2264 edist x y + edist y z ** refine' le_trans _ (edist_triangle _ _ _) ** case this.intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j A : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 x : X hx : x \u2208 S (2 * i + 1 + r) y : X hy : y \u2208 s \\ S (2 * j + r) this : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 z : X hzt : z \u2208 t hyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9 hxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z \u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + edist y z \u2264 edist x z ** refine' (add_le_add le_rfl hyz.le).trans (Eq.trans_le _ hxz) ** case this.intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j A : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 x : X hx : x \u2208 S (2 * i + 1 + r) y : X hy : y \u2208 s \\ S (2 * j + r) this : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 z : X hzt : z \u2208 t hyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9 hxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z \u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + (\u2191(2 * j + r))\u207b\u00b9 = (\u2191(2 * i + 1 + r))\u207b\u00b9 ** rw [tsub_add_cancel_of_le A.le] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 this : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4 \u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4 ** simpa using this 0 ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 this : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4 \u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4 ** simpa using this 1 ** case this \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n : \u2115 \u22a2 \u2191\u03bc (\u22c3 i \u2208 Finset.range n, S (2 * i + 1 + r) \\ S (2 * i + r)) \u2264 \u2191\u03bc (\u22c3 n, S n) ** exact \u03bc.mono (iUnion_subset fun i => iUnion_subset fun _ x hx => mem_iUnion.2 \u27e8_, hx.1\u27e9) ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n : \u2115 this : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r)) i : \u2115 x\u271d\u00b9 : i \u2208 Finset.range n j : \u2115 x\u271d : j \u2208 Finset.range n hij : i \u2260 j h : i < j x : X hx : x \u2208 S (2 * j + 1 + r) \\ S (2 * j + r) \u22a2 x \u2208 s \\ S (2 * j + r) ** exact \u27e8hx.1.1, hx.2\u27e9 ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n : \u2115 this : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r)) i : \u2115 x\u271d\u00b9 : i \u2208 Finset.range n j : \u2115 x\u271d : j \u2208 Finset.range n hij : i \u2260 j h : j < i x : X hx : x \u2208 S (2 * i + 1 + r) \\ S (2 * i + r) \u22a2 x \u2208 s \\ S (2 * i + r) ** exact \u27e8hx.1.1, hx.2\u27e9 ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j \u22a2 (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 ** rw [ENNReal.inv_lt_inv, Nat.cast_lt] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j \u22a2 2 * i + 1 + r < 2 * j + r ** linarith ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y \u03bc : OuterMeasure X hm : IsMetric \u03bc t : Set X ht : t \u2208 {s | IsClosed s} s : Set X S : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t} n0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0 Ssep : \u2200 (n : \u2115), IsMetricSeparated (S n) t Ssep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t) S_sub : \u2200 (n : \u2115), S n \u2286 s \\ t hSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s iUnion_S : \u22c3 n, S n = s \\ t htop : \u00ac\u2191\u03bc (s \\ t) = \u22a4 r n i j : \u2115 hj : i < j A : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9 \u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2260 0 ** simpa using A ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.binomial_one_eq_bernoulli ** p : \u211d\u22650\u221e h : p \u2264 1 \u22a2 binomial p h 1 = map (fun x => bif x then 1 else 0) (bernoulli p h) ** ext i ** case h p : \u211d\u22650\u221e h : p \u2264 1 i : Fin (1 + 1) \u22a2 \u2191(binomial p h 1) i = \u2191(map (fun x => bif x then 1 else 0) (bernoulli p h)) i ** fin_cases i <;> simp [tsum_bool, binomial_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_integral_nonpos_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d s\u271d s : Set \u03b1 hs : MeasurableSet s hf : StronglyMeasurable f hfi : Integrable f \u22a2 \u222b (x : \u03b1) in {y | f y \u2264 0}, f x \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, f x \u2202\u03bc ** rw [\u2190 integral_indicator hs, \u2190\n  integral_indicator (hf.measurableSet_le stronglyMeasurable_const)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d s\u271d s : Set \u03b1 hs : MeasurableSet s hf : StronglyMeasurable f hfi : Integrable f \u22a2 \u222b (x : \u03b1), indicator {a | f a \u2264 0} (fun x => f x) x \u2202\u03bc \u2264 \u222b (x : \u03b1), indicator s (fun x => f x) x \u2202\u03bc ** exact\n  integral_mono (hfi.indicator (hf.measurableSet_le stronglyMeasurable_const))\n    (hfi.indicator hs) (indicator_nonpos_le_indicator s f) ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.addOrderOf_coe ** a n : \u2115 n0 : n \u2260 0 \u22a2 addOrderOf \u2191a = n / Nat.gcd n a ** cases' a with a ** case zero n : \u2115 n0 : n \u2260 0 \u22a2 addOrderOf \u2191Nat.zero = n / Nat.gcd n Nat.zero  case succ n : \u2115 n0 : n \u2260 0 a : \u2115 \u22a2 addOrderOf \u2191(Nat.succ a) = n / Nat.gcd n (Nat.succ a) ** simp only [Nat.zero_eq, Nat.cast_zero, addOrderOf_zero, Nat.gcd_zero_right, Nat.pos_of_ne_zero n0,\n  Nat.div_self] ** case succ n : \u2115 n0 : n \u2260 0 a : \u2115 \u22a2 addOrderOf \u2191(Nat.succ a) = n / Nat.gcd n (Nat.succ a) ** rw [\u2190 Nat.smul_one_eq_coe, addOrderOf_nsmul' _ a.succ_ne_zero, ZMod.addOrderOf_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.min_singleton ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 a : \u03b1 \u22a2 Finset.min {a} = \u2191a ** rw [\u2190 insert_emptyc_eq] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 a : \u03b1 \u22a2 Finset.min (insert a \u2205) = \u2191a ** exact min_insert ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** refine' tendsto_order.2 \u27e8fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun \u03b5 (\u03b5pos : 0 < \u03b5) => _\u27e9 ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** rcases eq_or_ne (\u03bc t) 0 with (h't | h't) ** case inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** obtain \u27e8n, npos, hn\u27e9 : \u2203 n : \u2115, 0 < n \u2227 \u03bc (t \\ closedBall 0 n) < \u03b5 / 2 * \u03bc t := by\n  have A :\n    Tendsto (fun n : \u2115 => \u03bc (t \\ closedBall 0 n)) atTop\n      (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ closedBall 0 n))) := by\n    have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e :=\n      \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9\n    refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N\n    exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))\n  have : \u22c2 n : \u2115, t \\ closedBall 0 n = \u2205 := by\n    simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,\n      iUnion_closedBall_nat, compl_univ, inter_empty]\n  simp only [this, measure_empty] at A\n  have I : 0 < \u03b5 / 2 * \u03bc t := ENNReal.mul_pos (ENNReal.half_pos \u03b5pos.ne').ne' h't\n  exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists ** case inr.intro.intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** have L :\n  Tendsto (fun r : \u211d => \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd 0) :=\n  tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 \u03bc s x h _ t h't n (Nat.cast_pos.2 npos)\n    (inter_subset_right _ _) ** case inr.intro.intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** filter_upwards [(tendsto_order.1 L).2 _ (ENNReal.half_pos \u03b5pos.ne'), self_mem_nhdsWithin] ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200 (a : \u211d),     \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5 / 2 \u2192       a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5 ** rintro r hr (rpos : 0 < r) ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** have I :\n  \u03bc (s \u2229 ({x} + r \u2022 t)) \u2264\n    \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({x} + r \u2022 (t \\ closedBall 0 n)) :=\n  calc\n    \u03bc (s \u2229 ({x} + r \u2022 t)) =\n        \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n)) \u222a s \u2229 ({x} + r \u2022 (t \\ closedBall 0 n))) :=\n      by rw [\u2190 inter_union_distrib_left, \u2190 add_union, \u2190 smul_set_union, inter_union_diff]\n    _ \u2264 \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc (s \u2229 ({x} + r \u2022 (t \\ closedBall 0 n))) :=\n      (measure_union_le _ _)\n    _ \u2264 \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({x} + r \u2022 (t \\ closedBall 0 n)) :=\n      add_le_add le_rfl (measure_mono (inter_subset_right _ _)) ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r I : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** calc\n  \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 t) \u2264\n      (\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({x} + r \u2022 (t \\ closedBall 0 n))) /\n        \u03bc ({x} + r \u2022 t) :=\n    mul_le_mul_right' I _\n  _ < \u03b5 / 2 + \u03b5 / 2 := by\n    rw [ENNReal.add_div]\n    apply ENNReal.add_lt_add hr _\n    rwa [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne',\n      ENNReal.div_lt_iff (Or.inl h't) (Or.inl h''t)]\n  _ = \u03b5 := ENNReal.add_halves _ ** case inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** apply eventually_of_forall fun r => ?_ ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** suffices H : \u03bc (s \u2229 ({x} + r \u2022 t)) = 0 ** case H E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0 ** apply le_antisymm _ (zero_le _) ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 0 ** calc\n  \u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u03bc ({x} + r \u2022 t) := measure_mono (inter_subset_right _ _)\n  _ = 0 := by\n    simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,\n      mul_zero] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d H : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0 \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** rw [H] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d H : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0 \u22a2 0 / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** simpa only [ENNReal.zero_div] using \u03b5pos ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d \u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) = 0 ** simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,\n  mul_zero] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** have A :\n  Tendsto (fun n : \u2115 => \u03bc (t \\ closedBall 0 n)) atTop\n    (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ closedBall 0 n))) := by\n  have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e :=\n    \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9\n  refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N\n  exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn)) ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** have : \u22c2 n : \u2115, t \\ closedBall 0 n = \u2205 := by\n  simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,\n    iUnion_closedBall_nat, compl_univ, inter_empty] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) this : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205 \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** simp only [this, measure_empty] at A ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 this : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0) \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** have I : 0 < \u03b5 / 2 * \u03bc t := ENNReal.mul_pos (ENNReal.half_pos \u03b5pos.ne').ne' h't ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 this : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0) I : 0 < \u03b5 / 2 * \u2191\u2191\u03bc t \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 \u22a2 Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) ** have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e :=\n  \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9 ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 N : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4 \u22a2 Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) ** refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 N : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4 m n : \u2115 hmn : m \u2264 n \u22a2 t \\ closedBall 0 \u2191n \u2264 t \\ closedBall 0 \u2191m ** exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn)) ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) \u22a2 \u22c2 n, t \\ closedBall 0 \u2191n = \u2205 ** simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,\n  iUnion_closedBall_nat, compl_univ, inter_empty] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n)) \u222a s \u2229 ({x} + r \u2022 (t \\ closedBall 0 \u2191n))) ** rw [\u2190 inter_union_distrib_left, \u2190 add_union, \u2190 smul_set_union, inter_union_diff] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r I : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) \u22a2 (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) <     \u03b5 / 2 + \u03b5 / 2 ** rw [ENNReal.add_div] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r I : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) +       \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) <     \u03b5 / 2 + \u03b5 / 2 ** apply ENNReal.add_lt_add hr _ ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r I : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) \u22a2 \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 ** rwa [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne',\n  ENNReal.div_lt_iff (Or.inl h't) (Or.inl h''t)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.cast_add ** \u03b1 : Type u_1 inst\u271d : Semiring \u03b1 m n : Num \u22a2 \u2191(m + n) = \u2191m + \u2191n ** rw [\u2190 cast_to_nat, add_to_nat, Nat.cast_add, cast_to_nat, cast_to_nat] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.sub_le_iff_le_add' ** a b c : Nat \u22a2 a - b \u2264 c \u2194 a \u2264 b + c ** rw [Nat.sub_le_iff_le_add, Nat.add_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.getElem?_eq_get? ** \u03b1 : Type u_1 l : List \u03b1 i : Nat \u22a2 l[i]? = get? l i ** unfold getElem? ** \u03b1 : Type u_1 l : List \u03b1 i : Nat \u22a2 (if h : (fun as i => i < length as) l i then some l[i] else none) = get? l i ** split ** case inl \u03b1 : Type u_1 l : List \u03b1 i : Nat h\u271d : i < length l \u22a2 some l[i] = get? l i ** exact (get?_eq_get \u2039_\u203a).symm ** case inr \u03b1 : Type u_1 l : List \u03b1 i : Nat h\u271d : \u00ac(fun as i => i < length as) l i \u22a2 none = get? l i ** exact (get?_eq_none.2 <| Nat.not_lt.1 \u2039_\u203a).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.divInt_mul_left ** n d a : Int a0 : a \u2260 0 \u22a2 a * n /. (a * d) = n /. d ** if d0 : d = 0 then simp [d0] else\nsimp [divInt_eq_iff (Int.mul_ne_zero a0 d0) d0, Int.mul_assoc, Int.mul_left_comm] ** n d a : Int a0 : a \u2260 0 d0 : d = 0 \u22a2 a * n /. (a * d) = n /. d ** simp [d0] ** n d a : Int a0 : a \u2260 0 d0 : \u00acd = 0 \u22a2 a * n /. (a * d) = n /. d ** simp [divInt_eq_iff (Int.mul_ne_zero a0 d0) d0, Int.mul_assoc, Int.mul_left_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.mul_pos_of_neg_of_neg ** a b : Int ha : a < 0 hb : b < 0 \u22a2 0 < a * b ** have : 0 * b < a * b := Int.mul_lt_mul_of_neg_right ha hb ** a b : Int ha : a < 0 hb : b < 0 this : 0 * b < a * b \u22a2 0 < a * b ** rwa [Int.zero_mul] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "measure_le_lintegral_thickenedIndicator ** \u03b1 : Type u_1 \u03b2 : Type u_2 E\u271d : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E\u271d inst\u271d : PseudoEMetricSpace \u03b1 \u03bc : Measure \u03b1 E : Set \u03b1 E_mble : MeasurableSet E \u03b4 : \u211d \u03b4_pos : 0 < \u03b4 \u22a2 \u2191\u2191\u03bc E \u2264 \u222b\u207b (a : \u03b1), \u2191(\u2191(thickenedIndicator \u03b4_pos E) a) \u2202\u03bc ** convert measure_le_lintegral_thickenedIndicatorAux \u03bc E_mble \u03b4 ** case h.e'_4.h.e'_4.h \u03b1 : Type u_1 \u03b2 : Type u_2 E\u271d : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E\u271d inst\u271d : PseudoEMetricSpace \u03b1 \u03bc : Measure \u03b1 E : Set \u03b1 E_mble : MeasurableSet E \u03b4 : \u211d \u03b4_pos : 0 < \u03b4 x\u271d : \u03b1 \u22a2 \u2191(\u2191(thickenedIndicator \u03b4_pos E) x\u271d) = thickenedIndicatorAux \u03b4 E x\u271d ** dsimp ** case h.e'_4.h.e'_4.h \u03b1 : Type u_1 \u03b2 : Type u_2 E\u271d : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E\u271d inst\u271d : PseudoEMetricSpace \u03b1 \u03bc : Measure \u03b1 E : Set \u03b1 E_mble : MeasurableSet E \u03b4 : \u211d \u03b4_pos : 0 < \u03b4 x\u271d : \u03b1 \u22a2 \u2191(ENNReal.toNNReal (thickenedIndicatorAux \u03b4 E x\u271d)) = thickenedIndicatorAux \u03b4 E x\u271d ** simp only [thickenedIndicatorAux_lt_top.ne, ENNReal.coe_toNNReal, Ne.def, not_false_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_iInf_ae ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e h_meas : \u2200 (n : \u2115), Measurable (f n) h_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n h_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4 fn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc fn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc h_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a n : \u2115 a : \u03b1 h : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a \u22a2 f n a \u2264 f 0 a ** induction' n with n ih ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e h_meas : \u2200 (n : \u2115), Measurable (f n) h_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n h_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4 fn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc fn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc h_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a a : \u03b1 h : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a \u22a2 f Nat.zero a \u2264 f 0 a ** exact le_rfl ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e h_meas : \u2200 (n : \u2115), Measurable (f n) h_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n h_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4 fn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc fn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc h_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a a : \u03b1 h : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a n : \u2115 ih : f n a \u2264 f 0 a \u22a2 f (Nat.succ n) a \u2264 f 0 a ** exact le_trans (h n) ih ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.compl_compl_image ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f : \u03b1 \u2192 \u03b2 s t : Set \u03b1 inst\u271d : BooleanAlgebra \u03b1 S : Set \u03b1 \u22a2 compl '' (compl '' S) = S ** rw [\u2190 image_comp, compl_comp_compl, image_id] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrableOn_image_iff_integrableOn_abs_deriv_smul ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f'\u271d : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set \u211d f f' : \u211d \u2192 \u211d hs : MeasurableSet s hf' : \u2200 (x : \u211d), x \u2208 s \u2192 HasDerivWithinAt f (f' x) s x hf : InjOn f s g : \u211d \u2192 F \u22a2 IntegrableOn g (f '' s) \u2194 IntegrableOn (fun x => |f' x| \u2022 g (f x)) s ** simpa only [det_one_smulRight] using\n  integrableOn_image_iff_integrableOn_abs_det_fderiv_smul volume hs\n    (fun x hx => (hf' x hx).hasFDerivWithinAt) hf g ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.withDensity_indicator_one ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 withDensity \u03bc (indicator s 1) = restrict \u03bc s ** rw [withDensity_indicator hs, withDensity_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "not_nat_primrec_ack_self ** h : Nat.Primrec fun n => ack n n \u22a2 False ** cases' exists_lt_ack_of_nat_primrec h with m hm ** case intro h : Nat.Primrec fun n => ack n n m : \u2115 hm : \u2200 (n : \u2115), ack n n < ack m n \u22a2 False ** exact (hm m).false ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.nat_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 p : \u2115 \u00d7 \u2115 \u22a2 (Nat.casesOn ((fun x x_1 => x - x_1) p.1 p.2) true fun b => false) =     (fun a b => decide ((fun x x_1 => x \u2264 x_1) a b)) p.1 p.2 ** dsimp [swap] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 p : \u2115 \u00d7 \u2115 \u22a2 Nat.rec true (fun n n_ih => false) (p.1 - p.2) = decide (p.1 \u2264 p.2) ** cases' e : p.1 - p.2 with n ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 p : \u2115 \u00d7 \u2115 e : p.1 - p.2 = Nat.zero \u22a2 Nat.rec true (fun n n_ih => false) Nat.zero = decide (p.1 \u2264 p.2) ** simp [tsub_eq_zero_iff_le.1 e] ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2074 : Primcodable \u03b1 inst\u271d\u00b3 : Primcodable \u03b2 inst\u271d\u00b2 : Primcodable \u03b3 inst\u271d\u00b9 : Primcodable \u03b4 inst\u271d : Primcodable \u03c3 p : \u2115 \u00d7 \u2115 n : \u2115 e : p.1 - p.2 = Nat.succ n \u22a2 Nat.rec true (fun n n_ih => false) (Nat.succ n) = decide (p.1 \u2264 p.2) ** simp [not_le.2 (Nat.lt_of_sub_eq_succ e)] ** Qed",
        "informal": ""
    },
    {
        "formal": "essSup_mono_measure ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : CompleteLattice \u03b2 f : \u03b1 \u2192 \u03b2 h\u03bc\u03bd : \u03bd \u226a \u03bc \u22a2 essSup f \u03bd \u2264 essSup f \u03bc ** refine' limsup_le_limsup_of_le (Measure.ae_le_iff_absolutelyContinuous.mpr h\u03bc\u03bd) _ _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : CompleteLattice \u03b2 f : \u03b1 \u2192 \u03b2 h\u03bc\u03bd : \u03bd \u226a \u03bc \u22a2 IsCoboundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bd) f  case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : CompleteLattice \u03b2 f : \u03b1 \u2192 \u03b2 h\u03bc\u03bd : \u03bd \u226a \u03bc \u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f ** all_goals isBoundedDefault ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : CompleteLattice \u03b2 f : \u03b1 \u2192 \u03b2 h\u03bc\u03bd : \u03bd \u226a \u03bc \u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f ** isBoundedDefault ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.card_Iic_finset ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 \u22a2 card (Iic s) = 2 ^ card s ** rw [Iic_eq_powerset, card_powerset] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.inter_insert_of_not_mem ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1 a\u271d b : \u03b1 s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 h : \u00aca \u2208 s\u2081 \u22a2 s\u2081 \u2229 insert a s\u2082 = s\u2081 \u2229 s\u2082 ** rw [inter_comm, insert_inter_of_not_mem h, inter_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =     ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) ** have one_lt_p : -1 < p - 1 := by linarith ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =     ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) ** have obs : \u2200 x : \u211d, (\u222b t : \u211d in (0)..x, t ^ (p - 1)) = x ^ p / p := by\n  intro x\n  rw [integral_rpow (Or.inl one_lt_p)]\n  simp [Real.zero_rpow p_pos.ne.symm] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 obs : \u2200 (x : \u211d), \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =     ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) ** set g := fun t : \u211d => t ^ (p - 1) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =     ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) ** have g_nn : \u2200\u1d50 t \u2202volume.restrict (Ioi (0 : \u211d)), 0 \u2264 g t := by\n  filter_upwards [self_mem_ae_restrict (measurableSet_Ioi : MeasurableSet (Ioi (0 : \u211d)))]\n  intro t t_pos\n  exact Real.rpow_nonneg_of_nonneg (mem_Ioi.mp t_pos).le (p - 1) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =     ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) ** have g_intble : \u2200 t > 0, IntervalIntegrable g volume 0 t := fun _ _ =>\n  intervalIntegral.intervalIntegrable_rpow' one_lt_p ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =     ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) ** have key := lintegral_comp_eq_lintegral_meas_le_mul \u03bc f_nn f_mble g_intble g_nn ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t key :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc =     ENNReal.ofReal p * \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (t ^ (p - 1)) ** rw [\u2190 key, \u2190 lintegral_const_mul'' (ENNReal.ofReal p)] <;> simp_rw [obs] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p \u22a2 -1 < p - 1 ** linarith ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 \u22a2 \u2200 (x : \u211d), \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p ** intro x ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 x : \u211d \u22a2 \u222b (t : \u211d) in 0 ..x, t ^ (p - 1) = x ^ p / p ** rw [integral_rpow (Or.inl one_lt_p)] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 x : \u211d \u22a2 (x ^ (p - 1 + 1) - 0 ^ (p - 1 + 1)) / (p - 1 + 1) = x ^ p / p ** simp [Real.zero_rpow p_pos.ne.symm] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p \u22a2 \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t ** filter_upwards [self_mem_ae_restrict (measurableSet_Ioi : MeasurableSet (Ioi (0 : \u211d)))] ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p \u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 0 \u2264 g a ** intro t t_pos ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p t : \u211d t_pos : t \u2208 Ioi 0 \u22a2 0 \u2264 g t ** exact Real.rpow_nonneg_of_nonneg (mem_Ioi.mp t_pos).le (p - 1) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t key :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9 ^ p) \u2202\u03bc = \u222b\u207b (a : \u03b1), ENNReal.ofReal p * ENNReal.ofReal (f a ^ p / p) \u2202\u03bc ** congr with \u03c9 ** case e_f.h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t key :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) \u03c9 : \u03b1 \u22a2 ENNReal.ofReal (f \u03c9 ^ p) = ENNReal.ofReal p * ENNReal.ofReal (f \u03c9 ^ p / p) ** rw [\u2190 ENNReal.ofReal_mul p_pos.le, mul_div_cancel' (f \u03c9 ^ p) p_pos.ne.symm] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t key :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 AEMeasurable fun \u03c9 => ENNReal.ofReal (f \u03c9 ^ p / p) ** have aux := (@measurable_const \u211d \u03b1 (by infer_instance) (by infer_instance) p).aemeasurable\n              (\u03bc := \u03bc) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t key :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) aux : AEMeasurable fun x => p \u22a2 AEMeasurable fun \u03c9 => ENNReal.ofReal (f \u03c9 ^ p / p) ** exact (Measurable.ennreal_ofReal (hf := measurable_id)).comp_aemeasurable\n  ((f_mble.pow aux).div_const p) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t key :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 MeasurableSpace \u211d ** infer_instance ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g\u271d : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 f_nn : 0 \u2264\u1da0[ae \u03bc] f f_mble : AEMeasurable f p : \u211d p_pos : 0 < p one_lt_p : -1 < p - 1 g : \u211d \u2192 \u211d := fun t => t ^ (p - 1) obs : \u2200 (x : \u211d), intervalIntegral g 0 x volume = x ^ p / p g_nn : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 g t g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t key :   \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) \u22a2 MeasurableSpace \u03b1 ** infer_instance ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.simpleFunc.toLp_sub ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F p : \u211d\u22650\u221e \u03bc : Measure \u03b1 f g : \u03b1 \u2192\u209b E hf : Mem\u2112p (\u2191f) p hg : Mem\u2112p (\u2191g) p \u22a2 toLp (f - g) (_ : Mem\u2112p (\u2191f - fun a => \u2191g a) p) = toLp f hf - toLp g hg ** simp only [sub_eq_add_neg, \u2190 toLp_neg, \u2190 toLp_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "AddCircle.measurePreserving_mk ** T : \u211d hT : Fact (0 < T) t : \u211d \u22a2 MeasurePreserving QuotientAddGroup.mk ** apply MeasurePreservingQuotientAddGroup.mk' ** case h\ud835\udcd5 T : \u211d hT : Fact (0 < T) t : \u211d \u22a2 IsAddFundamentalDomain { x // x \u2208 AddSubgroup.op (zmultiples T) } (Ioc t (t + T)) ** exact isAddFundamentalDomain_Ioc' hT.out t ** case h\ud835\udcd5_finite T : \u211d hT : Fact (0 < T) t : \u211d \u22a2 \u2191\u2191volume (Ioc t (t + T)) < \u22a4 ** simp ** case h T : \u211d hT : Fact (0 < T) t : \u211d \u22a2 \u2191\u2191volume (Ioc t (t + T) \u2229 \u2191(QuotientAddGroup.mk' (zmultiples T)) \u207b\u00b9' \u2191\u22a4) = \u2191(Real.toNNReal T) ** haveI : CompactSpace (\u211d \u29f8 zmultiples T) := inferInstanceAs (CompactSpace (AddCircle T)) ** case h T : \u211d hT : Fact (0 < T) t : \u211d this : CompactSpace (\u211d \u29f8 zmultiples T) \u22a2 \u2191\u2191volume (Ioc t (t + T) \u2229 \u2191(QuotientAddGroup.mk' (zmultiples T)) \u207b\u00b9' \u2191\u22a4) = \u2191(Real.toNNReal T) ** simp [\u2190 ENNReal.ofReal_coe_nnreal, Real.coe_toNNReal T hT.out.le, -Real.coe_toNNReal'] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.comap_comp ** \u03c3 : Type u_1 \u03c4 : Type u_2 \u03c5 : Type u_3 R : Type u_4 inst\u271d : CommSemiring R f : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c4 R g : MvPolynomial \u03c4 R \u2192\u2090[R] MvPolynomial \u03c5 R \u22a2 comap (AlgHom.comp g f) = comap f \u2218 comap g ** funext x ** case h \u03c3 : Type u_1 \u03c4 : Type u_2 \u03c5 : Type u_3 R : Type u_4 inst\u271d : CommSemiring R f : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c4 R g : MvPolynomial \u03c4 R \u2192\u2090[R] MvPolynomial \u03c5 R x : \u03c5 \u2192 R \u22a2 comap (AlgHom.comp g f) x = (comap f \u2218 comap g) x ** exact comap_comp_apply _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.stronglyMeasurable_condexp ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 \u22a2 StronglyMeasurable (\u03bc[f|m]) ** by_cases hm : m \u2264 m0 ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 \u22a2 StronglyMeasurable (\u03bc[f|m])  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : \u00acm \u2264 m0 \u22a2 StronglyMeasurable (\u03bc[f|m]) ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 \u22a2 StronglyMeasurable (\u03bc[f|m]) ** by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm) ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 StronglyMeasurable (\u03bc[f|m])  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 StronglyMeasurable (\u03bc[f|m]) ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 StronglyMeasurable (\u03bc[f|m]) ** haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 StronglyMeasurable (\u03bc[f|m]) ** rw [condexp_of_sigmaFinite hm] ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 StronglyMeasurable     (if Integrable f then       if StronglyMeasurable f then f       else AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)     else 0) ** split_ifs with hfi hfm ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : \u00acm \u2264 m0 \u22a2 StronglyMeasurable (\u03bc[f|m]) ** rw [condexp_of_not_le hm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : \u00acm \u2264 m0 \u22a2 StronglyMeasurable 0 ** exact stronglyMeasurable_zero ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 StronglyMeasurable (\u03bc[f|m]) ** rw [condexp_of_not_sigmaFinite hm h\u03bcm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 StronglyMeasurable 0 ** exact stronglyMeasurable_zero ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) hfi : Integrable f hfm : StronglyMeasurable f \u22a2 StronglyMeasurable f ** exact hfm ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) hfi : Integrable f hfm : \u00acStronglyMeasurable f \u22a2 StronglyMeasurable     (AEStronglyMeasurable'.mk \u2191\u2191(condexpL1 hm \u03bc f) (_ : AEStronglyMeasurable' m (\u2191\u2191(condexpL1 hm \u03bc f)) \u03bc)) ** exact AEStronglyMeasurable'.stronglyMeasurable_mk _ ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) hfi : \u00acIntegrable f \u22a2 StronglyMeasurable 0 ** exact stronglyMeasurable_zero ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpL1_of_aestronglyMeasurable' ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' s : Set \u03b1 hfm : AEStronglyMeasurable' m f \u03bc hfi : Integrable f \u22a2 \u2191\u2191(condexpL1 hm \u03bc f) =\u1d50[\u03bc] f ** rw [condexpL1_eq hfi] ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' s : Set \u03b1 hfm : AEStronglyMeasurable' m f \u03bc hfi : Integrable f \u22a2 \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1d50[\u03bc] f ** refine' EventuallyEq.trans _ (Integrable.coeFn_toL1 hfi) ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' s : Set \u03b1 hfm : AEStronglyMeasurable' m f \u03bc hfi : Integrable f \u22a2 \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1d50[\u03bc] \u2191\u2191(Integrable.toL1 f hfi) ** rw [condexpL1Clm_of_aestronglyMeasurable'] ** case hfm \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f g : \u03b1 \u2192 F' s : Set \u03b1 hfm : AEStronglyMeasurable' m f \u03bc hfi : Integrable f \u22a2 AEStronglyMeasurable' m (\u2191\u2191(Integrable.toL1 f hfi)) \u03bc ** exact AEStronglyMeasurable'.congr hfm (Integrable.coeFn_toL1 hfi).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec'.part_iff\u2082 ** f : \u2115 \u2192 \u2115 \u2192. \u2115 h : _root_.Partrec fun v => f (Vector.head v) (Vector.head (Vector.tail v)) v : \u2115 \u00d7 \u2115 \u22a2 f (Vector.head (v.1 ::\u1d65 v.2 ::\u1d65 nil)) (Vector.head (Vector.tail (v.1 ::\u1d65 v.2 ::\u1d65 nil))) = f v.1 v.2 ** simp only [head_cons, tail_cons] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure ** E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s \u22a2 \u2203 x x_1, \u2191x \u2208 s ** have h_vol : \u03bc F < \u03bc ((2\u207b\u00b9 : \u211d) \u2022 s) := by\n  rw [addHaar_smul_of_nonneg \u03bc (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9) s, \u2190\n    mul_lt_mul_right (pow_ne_zero (finrank \u211d E) (two_ne_zero' _)) (pow_ne_top two_ne_top),\n    mul_right_comm, ofReal_pow (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9), \u2190 ofReal_inv_of_pos zero_lt_two]\n  norm_num\n  rwa [\u2190 mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul] ** E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s h_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s) \u22a2 \u2203 x x_1, \u2191x \u2208 s ** obtain \u27e8x, y, hxy, h\u27e9 :=\n  exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol ** case intro.intro.intro E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s h_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s) x y : { x // x \u2208 L } hxy : x \u2260 y h : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s) \u22a2 \u2203 x x_1, \u2191x \u2208 s ** obtain \u27e8_, \u27e8v, hv, rfl\u27e9, w, hw, hvw\u27e9 := Set.not_disjoint_iff.mp h ** case intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s h_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s) x y : { x // x \u2208 L } hxy : x \u2260 y h : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s) v : E hv : v \u2208 2\u207b\u00b9 \u2022 s w : E hw : w \u2208 2\u207b\u00b9 \u2022 s hvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v \u22a2 \u2203 x x_1, \u2191x \u2208 s ** refine' \u27e8x - y, sub_ne_zero.2 hxy, _\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s h_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s) x y : { x // x \u2208 L } hxy : x \u2260 y h : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s) v : E hv : v \u2208 2\u207b\u00b9 \u2022 s w : E hw : w \u2208 2\u207b\u00b9 \u2022 s hvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v \u22a2 \u2191(x - y) \u2208 s ** rw [Set.mem_inv_smul_set_iff\u2080 (two_ne_zero' \u211d)] at hv hw ** case intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s h_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s) x y : { x // x \u2208 L } hxy : x \u2260 y h : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s) v : E hv : 2 \u2022 v \u2208 s w : E hw : 2 \u2022 w \u2208 s hvw : (fun x => y +\u1d65 x) w = (fun x_1 => x +\u1d65 x_1) v \u22a2 \u2191(x - y) \u2208 s ** simp_rw [AddSubgroup.vadd_def, vadd_eq_add, add_comm _ w, \u2190 sub_eq_sub_iff_add_eq_add, \u2190\n  AddSubgroup.coe_sub] at hvw ** case intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s h_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s) x y : { x // x \u2208 L } hxy : x \u2260 y h : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s) v : E hv : 2 \u2022 v \u2208 s w : E hw : 2 \u2022 w \u2208 s hvw : w - v = \u2191(x - y) \u22a2 \u2191(x - y) \u2208 s ** rw [\u2190 hvw, \u2190 inv_smul_smul\u2080 (two_ne_zero' \u211d) (_ - _), smul_sub, sub_eq_add_neg, smul_add] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h\u271d : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s h_vol : \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s) x y : { x // x \u2208 L } hxy : x \u2260 y h : \u00acDisjoint (x +\u1d65 2\u207b\u00b9 \u2022 s) (y +\u1d65 2\u207b\u00b9 \u2022 s) v : E hv : 2 \u2022 v \u2208 s w : E hw : 2 \u2022 w \u2208 s hvw : w - v = \u2191(x - y) \u22a2 2\u207b\u00b9 \u2022 2 \u2022 w + 2\u207b\u00b9 \u2022 -(2 \u2022 v) \u2208 s ** refine' h_conv hw (h_symm _ hv) _ _ _ <;> norm_num ** E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s \u22a2 \u2191\u2191\u03bc F < \u2191\u2191\u03bc (2\u207b\u00b9 \u2022 s) ** rw [addHaar_smul_of_nonneg \u03bc (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9) s, \u2190\n  mul_lt_mul_right (pow_ne_zero (finrank \u211d E) (two_ne_zero' _)) (pow_ne_top two_ne_top),\n  mul_right_comm, ofReal_pow (by norm_num : 0 \u2264 (2 : \u211d)\u207b\u00b9), \u2190 ofReal_inv_of_pos zero_lt_two] ** E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s \u22a2 \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < (ENNReal.ofReal 2)\u207b\u00b9 ^ finrank \u211d E * 2 ^ finrank \u211d E * \u2191\u2191\u03bc s ** norm_num ** E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s \u22a2 \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < 2\u207b\u00b9 ^ finrank \u211d E * 2 ^ finrank \u211d E * \u2191\u2191\u03bc s ** rwa [\u2190 mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul] ** E : Type u_1 L\u271d : Type u_2 inst\u271d\u2076 : MeasurableSpace E \u03bc : Measure E F s : Set E inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : FiniteDimensional \u211d E inst\u271d\u00b9 : IsAddHaarMeasure \u03bc L : AddSubgroup E inst\u271d : Countable { x // x \u2208 L } fund : IsAddFundamentalDomain { x // x \u2208 L } F h : \u2191\u2191\u03bc F * 2 ^ finrank \u211d E < \u2191\u2191\u03bc s h_symm : \u2200 (x : E), x \u2208 s \u2192 -x \u2208 s h_conv : Convex \u211d s \u22a2 0 \u2264 2\u207b\u00b9 ** norm_num ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_deriv_comp_mul_deriv' ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g'\u271d g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' g g' : \u211d \u2192 \u211d hf : ContinuousOn f [[a, b]] hff' : \u2200 (x : \u211d), x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hf' : ContinuousOn f' [[a, b]] hg : ContinuousOn g [[f a, f b]] hgg' : \u2200 (x : \u211d), x \u2208 Ioo (min (f a) (f b)) (max (f a) (f b)) \u2192 HasDerivWithinAt g (g' x) (Ioi x) x hg' : ContinuousOn g' (f '' [[a, b]]) \u22a2 \u222b (x : \u211d) in a..b, (g' \u2218 f) x * f' x = (g \u2218 f) b - (g \u2218 f) a ** simpa [mul_comm] using integral_deriv_comp_smul_deriv' hf hff' hf' hg hgg' hg' ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.encard_tsub_one_le_encard_diff_singleton ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 x : \u03b1 \u22a2 encard s - 1 \u2264 encard (s \\ {x}) ** rw [\u2190encard_singleton x] ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 x : \u03b1 \u22a2 encard s - encard {x} \u2264 encard (s \\ {x}) ** apply tsub_encard_le_encard_diff ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FinStronglyMeasurable.sup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b2 inst\u271d : ContinuousSup \u03b2 hf : FinStronglyMeasurable f \u03bc hg : FinStronglyMeasurable g \u03bc \u22a2 FinStronglyMeasurable (f \u2294 g) \u03bc ** refine'\n  \u27e8fun n => hf.approx n \u2294 hg.approx n, fun n => _, fun x =>\n    (hf.tendsto_approx x).sup_right_nhds (hg.tendsto_approx x)\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b2 inst\u271d : ContinuousSup \u03b2 hf : FinStronglyMeasurable f \u03bc hg : FinStronglyMeasurable g \u03bc n : \u2115 \u22a2 \u2191\u2191\u03bc (support \u2191((fun n => FinStronglyMeasurable.approx hf n \u2294 FinStronglyMeasurable.approx hg n) n)) < \u22a4 ** refine' (measure_mono (support_sup _ _)).trans_lt _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2074 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : SemilatticeSup \u03b2 inst\u271d : ContinuousSup \u03b2 hf : FinStronglyMeasurable f \u03bc hg : FinStronglyMeasurable g \u03bc n : \u2115 \u22a2 \u2191\u2191\u03bc       ((support fun x => \u2191(FinStronglyMeasurable.approx hf n) x) \u222a         support fun x => \u2191(FinStronglyMeasurable.approx hg n) x) <     \u22a4 ** exact measure_union_lt_top_iff.mpr \u27e8hf.fin_support_approx n, hg.fin_support_approx n\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1to1.tr_respects ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 l\u2081 : Option \u039b v : \u03c3 T : Tape \u0393 \u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081) (trCfg enc enc\u2080 { l := l\u2081, var := v, Tape := T })     (step M { l := l\u2081, var := v, Tape := T }) ** obtain \u27e8L, R, rfl\u27e9 := T.exists_mk' ** case intro.intro \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 l\u2081 : Option \u039b v : \u03c3 L R : ListBlank \u0393 \u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)     (trCfg enc enc\u2080 { l := l\u2081, var := v, Tape := Tape.mk' L R }) (step M { l := l\u2081, var := v, Tape := Tape.mk' L R }) ** cases' l\u2081 with l\u2081 ** case intro.intro.some \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v : \u03c3 L R : ListBlank \u0393 l\u2081 : \u039b \u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)     (trCfg enc enc\u2080 { l := some l\u2081, var := v, Tape := Tape.mk' L R })     (step M { l := some l\u2081, var := v, Tape := Tape.mk' L R }) ** suffices \u2200 q R, Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n    (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) by\n  refine' TransGen.head' rfl _\n  rw [trTape_mk']\n  exact this _ R ** case intro.intro.some \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v : \u03c3 L R : ListBlank \u0393 l\u2081 : \u039b \u22a2 \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) ** clear R l\u2081 ** case intro.intro.some \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v : \u03c3 L : ListBlank \u0393 \u22a2 \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) ** intro q R ** case intro.intro.some \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v : \u03c3 L : ListBlank \u0393 q : Stmt\u2081 R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) ** induction' q generalizing v L R ** case intro.intro.some.move \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b9 : Dir a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.move a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.write \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.write a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.load \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.branch \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R))) a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.goto \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))  case intro.intro.some.halt \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R))) ** case move d q IH =>\n  cases d <;>\n      simp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons,\n        Tape.move_left_mk', ListBlank.cons_head_tail, ListBlank.tail_cons,\n        trTape'_move_left enc0, trTape'_move_right enc0] <;>\n    apply IH ** case intro.intro.some.write \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.write a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.load \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.branch \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R))) a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.goto \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))  case intro.intro.some.halt \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R))) ** case write f q IH =>\n  simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n  refine' ReflTransGen.head rfl _\n  obtain \u27e8a, R, rfl\u27e9 := R.exists_cons\n  rw [tr, Tape.mk'_head, stepAux_write, ListBlank.head_cons, stepAux_move,\n    trTape'_move_left enc0, ListBlank.head_cons, ListBlank.tail_cons, Tape.write_mk']\n  apply IH ** case intro.intro.some.load \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3 a\u271d : Stmt\u2081 a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.branch \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R))) a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.goto \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))  case intro.intro.some.halt \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R))) ** case load a q IH =>\n  simp only [trNormal, stepAux_read dec enc0 encdec]\n  apply IH ** case intro.intro.some.branch \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool a\u271d\u00b9 a\u271d : Stmt\u2081 a_ih\u271d\u00b9 :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R))) a_ih\u271d :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))  case intro.intro.some.goto \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))  case intro.intro.some.halt \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R))) ** case branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n  simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n  cases p R.head v <;> [apply IH\u2082; apply IH\u2081] ** case intro.intro.some.goto \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))  case intro.intro.some.halt \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R))) ** case goto l =>\n  simp only [trNormal, stepAux_read dec enc0 encdec, stepAux, trCfg, trTape_mk']\n  apply ReflTransGen.refl ** case intro.intro.some.halt \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R))) ** case halt =>\n  simp only [trNormal, stepAux, trCfg, stepAux_move, trTape'_move_left enc0,\n    trTape'_move_right enc0, trTape_mk']\n  apply ReflTransGen.refl ** case intro.intro.none \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v : \u03c3 L R : ListBlank \u0393 \u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)     (trCfg enc enc\u2080 { l := none, var := v, Tape := Tape.mk' L R })     (step M { l := none, var := v, Tape := Tape.mk' L R }) ** exact rfl ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v : \u03c3 L R : ListBlank \u0393 l\u2081 : \u039b this :   \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) \u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)     (trCfg enc enc\u2080 { l := some l\u2081, var := v, Tape := Tape.mk' L R })     (step M { l := some l\u2081, var := v, Tape := Tape.mk' L R }) ** refine' TransGen.head' rfl _ ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v : \u03c3 L R : ListBlank \u0393 l\u2081 : \u039b this :   \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) \u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)     (stepAux (tr enc dec M (\u039b'.normal l\u2081)) v (trTape enc\u2080 (Tape.mk' L R)))     ((fun c\u2081 => trCfg enc enc\u2080 c\u2081) (stepAux (M l\u2081) v (Tape.mk' L R))) ** rw [trTape_mk'] ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v : \u03c3 L R : ListBlank \u0393 l\u2081 : \u039b this :   \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) \u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a) (stepAux (tr enc dec M (\u039b'.normal l\u2081)) v (trTape' enc\u2080 L R))     ((fun c\u2081 => trCfg enc enc\u2080 c\u2081) (stepAux (M l\u2081) v (Tape.mk' L R))) ** exact this _ R ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 d : Dir q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move d q)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.move d q) v (Tape.mk' L R))) ** cases d <;>\n    simp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons,\n      Tape.move_left_mk', ListBlank.cons_head_tail, ListBlank.tail_cons,\n      trTape'_move_left enc0, trTape'_move_right enc0] <;>\n  apply IH ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write f q)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.write f q) v (Tape.mk' L R))) ** simp only [trNormal, stepAux_read dec enc0 encdec, stepAux] ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) { l := some (\u039b'.write (f (ListBlank.head R) v) q), var := v, Tape := trTape' enc0 L R }     (trCfg enc enc0 (stepAux q v (Tape.write (f (Tape.mk' L R).head v) (Tape.mk' L R)))) ** refine' ReflTransGen.head rfl _ ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)     (stepAux (tr enc dec M (\u039b'.write (f (ListBlank.head R) v) q)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux q v (Tape.write (f (Tape.mk' L R).head v) (Tape.mk' L R)))) ** obtain \u27e8a, R, rfl\u27e9 := R.exists_cons ** case intro.intro \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) v : \u03c3 L : ListBlank \u0393 a : \u0393 R : ListBlank \u0393 \u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)     (stepAux (tr enc dec M (\u039b'.write (f (ListBlank.head (ListBlank.cons a R)) v) q)) v       (trTape' enc0 L (ListBlank.cons a R)))     (trCfg enc enc0       (stepAux q v (Tape.write (f (Tape.mk' L (ListBlank.cons a R)).head v) (Tape.mk' L (ListBlank.cons a R))))) ** rw [tr, Tape.mk'_head, stepAux_write, ListBlank.head_cons, stepAux_move,\n  trTape'_move_left enc0, ListBlank.head_cons, ListBlank.tail_cons, Tape.write_mk'] ** case intro.intro \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 f : \u0393 \u2192 \u03c3 \u2192 \u0393 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) v : \u03c3 L : ListBlank \u0393 a : \u0393 R : ListBlank \u0393 \u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)     (stepAux (trNormal dec q) v (trTape' enc0 L (ListBlank.cons (f a v) R)))     (trCfg enc enc0 (stepAux q v (Tape.mk' L (ListBlank.cons (f a v) R)))) ** apply IH ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a q)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.load a q) v (Tape.mk' L R))) ** simp only [trNormal, stepAux_read dec enc0 encdec] ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 a : \u0393 \u2192 \u03c3 \u2192 \u03c3 q : Stmt\u2081 IH :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M))     (stepAux (Stmt.load (fun x s => a (ListBlank.head R) s) (trNormal dec q)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.load a q) v (Tape.mk' L R))) ** apply IH ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R))) IH\u2082 :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch p q\u2081 q\u2082)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.branch p q\u2081 q\u2082) v (Tape.mk' L R))) ** simp only [trNormal, stepAux_read dec enc0 encdec, stepAux] ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 p : \u0393 \u2192 \u03c3 \u2192 Bool q\u2081 q\u2082 : Stmt\u2081 IH\u2081 :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R))) IH\u2082 :   \u2200 (v : \u03c3) (L R : ListBlank \u0393),     Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))       (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R))) v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M))     (bif p (ListBlank.head R) v then stepAux (trNormal dec q\u2081) v (trTape' enc0 L R)     else stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))     (trCfg enc enc0 (bif p (Tape.mk' L R).head v then stepAux q\u2081 v (Tape.mk' L R) else stepAux q\u2082 v (Tape.mk' L R))) ** cases p R.head v <;> [apply IH\u2082; apply IH\u2081] ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 l : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto l)) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux (Stmt.goto l) v (Tape.mk' L R))) ** simp only [trNormal, stepAux_read dec enc0 encdec, stepAux, trCfg, trTape_mk'] ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 l : \u0393 \u2192 \u03c3 \u2192 \u039b v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) { l := some (\u039b'.normal (l (ListBlank.head R) v)), var := v, Tape := trTape' enc0 L R }     { l := Option.map \u039b'.normal (some (l (Tape.mk' L R).head v)), var := v, Tape := trTape' enc0 L R } ** apply ReflTransGen.refl ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))     (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R))) ** simp only [trNormal, stepAux, trCfg, stepAux_move, trTape'_move_left enc0,\n  trTape'_move_right enc0, trTape_mk'] ** \u0393 : Type u_1 inst\u271d\u00b2 : Inhabited \u0393 \u039b : Type u_2 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_3 inst\u271d : Inhabited \u03c3 n : \u2115 enc : \u0393 \u2192 Vector Bool n dec : Vector Bool n \u2192 \u0393 enc0 : enc default = Vector.replicate n false M : \u039b \u2192 Stmt\u2081 encdec : \u2200 (a : \u0393), dec (enc a) = a enc\u2080 : enc default = Vector.replicate n false x\u271d : Cfg\u2081 v\u271d : \u03c3 L\u271d R\u271d : ListBlank \u0393 v : \u03c3 L R : ListBlank \u0393 \u22a2 Reaches (step (tr enc dec M)) { l := none, var := v, Tape := trTape' enc0 L R }     { l := Option.map \u039b'.normal none, var := v, Tape := trTape' enc0 L R } ** apply ReflTransGen.refl ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.sub_lt_self ** a b : Int h : 0 < b \u22a2 a + 0 = a ** rw [Int.add_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.isEmpty_coe_sort ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s\u271d s : Finset \u03b1 \u22a2 IsEmpty { x // x \u2208 s } \u2194 s = \u2205 ** simpa using @Set.isEmpty_coe_sort \u03b1 s ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsStoppingTime.measurableSet_stopping_time_le ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} ** exact measurableSet_le_stopping_time h\u03c4 h\u03c0 ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 this : MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} ** rw [measurableSet_min_iff h\u03c4 h\u03c0] at this ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 this : MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2227 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u22a2 MeasurableSet {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} ** exact this.2 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pairwise_disjoint_Ioo_int_cast ** \u03b1 : Type u_1 inst\u271d : OrderedRing \u03b1 a : \u03b1 \u22a2 Pairwise (Disjoint on fun n => Ioo (\u2191n) (\u2191n + 1)) ** simpa only [zero_add] using pairwise_disjoint_Ioo_add_int_cast (0 : \u03b1) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_zero ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C \u22a2 setToFun \u03bc T hT 0 = 0 ** erw [setToFun_eq hT (integrable_zero _ _ _), Integrable.toL1_zero, ContinuousLinearMap.map_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.sign_eq_div_abs ** a : Int az : a = 0 \u22a2 sign a = div a \u2191(natAbs a) ** simp [az] ** Qed",
        "informal": ""
    },
    {
        "formal": "Holor.cprankMax_mul ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 x : Holor \u03b1 [d] \u22a2 CPRankMax 0 (x \u2297 0) ** simp [mul_zero x, CPRankMax.zero] ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 n : \u2115 x : Holor \u03b1 [d] y\u2081 y\u2082 : Holor \u03b1 ds hy\u2081 : CPRankMax1 y\u2081 hy\u2082 : CPRankMax n y\u2082 \u22a2 CPRankMax (n + 1) (x \u2297 (y\u2081 + y\u2082)) ** rw [mul_left_distrib] ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 n : \u2115 x : Holor \u03b1 [d] y\u2081 y\u2082 : Holor \u03b1 ds hy\u2081 : CPRankMax1 y\u2081 hy\u2082 : CPRankMax n y\u2082 \u22a2 CPRankMax (n + 1) (x \u2297 y\u2081 + x \u2297 y\u2082) ** rw [Nat.add_comm] ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 n : \u2115 x : Holor \u03b1 [d] y\u2081 y\u2082 : Holor \u03b1 ds hy\u2081 : CPRankMax1 y\u2081 hy\u2082 : CPRankMax n y\u2082 \u22a2 CPRankMax (1 + n) (x \u2297 y\u2081 + x \u2297 y\u2082) ** apply cprankMax_add ** case a \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 n : \u2115 x : Holor \u03b1 [d] y\u2081 y\u2082 : Holor \u03b1 ds hy\u2081 : CPRankMax1 y\u2081 hy\u2082 : CPRankMax n y\u2082 \u22a2 CPRankMax 1 (x \u2297 y\u2081) ** exact cprankMax_1 (CPRankMax1.cons _ _ hy\u2081) ** case a \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Ring \u03b1 n : \u2115 x : Holor \u03b1 [d] y\u2081 y\u2082 : Holor \u03b1 ds hy\u2081 : CPRankMax1 y\u2081 hy\u2082 : CPRankMax n y\u2082 \u22a2 CPRankMax n (x \u2297 y\u2082) ** exact cprankMax_mul _ x y\u2082 hy\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_mul_laverage ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f\u271d g : \u03b1 \u2192 \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc ** cases' eq_or_ne \u03bc 0 with h\u03bc h\u03bc ** case inl \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f\u271d g : \u03b1 \u2192 \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : \u03bc = 0 \u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc ** rw [h\u03bc, lintegral_zero_measure, laverage_zero_measure, mul_zero] ** case inr \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f\u271d g : \u03b1 \u2192 \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e h\u03bc : \u03bc \u2260 0 \u22a2 \u2191\u2191\u03bc univ * \u2a0d\u207b (x : \u03b1), f x \u2202\u03bc = \u222b\u207b (x : \u03b1), f x \u2202\u03bc ** rw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 h\u03bc) (measure_ne_top _ _)] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.length_filterMap_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 Option \u03b2 l : List \u03b1 \u22a2 length (filterMap f l) \u2264 length l ** rw [\u2190 length_map _ some, map_filterMap_some_eq_filter_map_is_some, \u2190 length_map _ f] ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 Option \u03b2 l : List \u03b1 \u22a2 length (filter (fun b => Option.isSome b) (map f l)) \u2264 length (map f l) ** apply length_filter_le ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** let \u03b4 : \u211d\u22650 := \u27e8\u03b5 / 2, (half_pos \u03b5pos).le\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** have \u03b4pos : 0 < \u03b4 := half_pos \u03b5pos ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** let fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** have int_fp : Integrable (fun x => (fp x : \u211d)) \u03bc := hf.real_toNNReal ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** rcases exists_lt_lowerSemicontinuous_integral_gt_nnreal fp int_fp \u03b4pos with\n  \u27e8gp, fp_lt_gp, gpcont, gp_lt_top, gp_integrable, gpint\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** let fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** have int_fm : Integrable (fun x => (fm x : \u211d)) \u03bc := hf.neg.real_toNNReal ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** rcases exists_upperSemicontinuous_le_integral_le fm int_fm \u03b4pos with\n  \u27e8gm, gm_le_fm, gmcont, gm_integrable, gmint\u27e9 ** case intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** let g : \u03b1 \u2192 EReal := fun x => (gp x : EReal) - gm x ** case intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** have ae_g : \u2200\u1d50 x \u2202\u03bc, (g x).toReal = (gp x : EReal).toReal - (gm x : EReal).toReal := by\n  filter_upwards [gp_lt_top] with _ hx\n  rw [EReal.toReal_sub] <;> simp [hx.ne] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2203 g,     (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227       LowerSemicontinuous g \u2227         (Integrable fun x => EReal.toReal (g x)) \u2227           (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** refine' \u27e8g, ?lt, ?lsc, ?int, ?aelt, ?intlt\u27e9 ** case lt \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x  case lsc \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 LowerSemicontinuous g  case int \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 Integrable fun x => EReal.toReal (g x)  case aelt \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4  case intlt \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** case int =>\n  show Integrable (fun x => EReal.toReal (g x)) \u03bc\n  rw [integrable_congr ae_g]\n  convert gp_integrable.sub gm_integrable\n  simp ** case lt \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x  case lsc \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 LowerSemicontinuous g  case aelt \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4  case intlt \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** case intlt =>\n  show (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) < (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5;\n  exact\n    calc\n      (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) = \u222b x : \u03b1, EReal.toReal (gp x) - EReal.toReal (gm x) \u2202\u03bc :=\n        integral_congr_ae ae_g\n      _ = (\u222b x : \u03b1, EReal.toReal (gp x) \u2202\u03bc) - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n        simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]\n        exact integral_sub gp_integrable gm_integrable\n      _ < (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n        apply sub_lt_sub_right\n        convert gpint\n        simp only [EReal.toReal_coe_ennreal]\n      _ \u2264 (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - ((\u222b x : \u03b1, \u2191(fm x) \u2202\u03bc) - \u03b4) := (sub_le_sub_left gmint _)\n      _ = (\u222b x : \u03b1, f x \u2202\u03bc) + 2 * \u03b4 := by\n        simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring\n      _ = (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by congr 1; field_simp [mul_comm] ** case lt \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x  case lsc \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 LowerSemicontinuous g  case aelt \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 ** case aelt =>\n  show \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u22a4\n  filter_upwards [gp_lt_top] with ?_ hx\n  simp only [sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top,\n    lt_top_iff_ne_top.1 hx, EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff,\n    EReal.coe_ennreal_ne_bot] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) ** filter_upwards [gp_lt_top] with _ hx ** case h \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) a\u271d : \u03b1 hx : gp a\u271d < \u22a4 \u22a2 EReal.toReal (g a\u271d) = EReal.toReal \u2191(gp a\u271d) - EReal.toReal \u2191\u2191(gm a\u271d) ** rw [EReal.toReal_sub] <;> simp [hx.ne] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 Integrable fun x => EReal.toReal (g x) ** rw [integrable_congr ae_g] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 Integrable fun x => EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) ** convert gp_integrable.sub gm_integrable ** case h.e'_5.h \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x\u271d : \u03b1 \u22a2 EReal.toReal \u2191(gp x\u271d) - EReal.toReal \u2191\u2191(gm x\u271d) = ((fun x => ENNReal.toReal (gp x)) - fun x => \u2191(gm x)) x\u271d ** simp ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), EReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** exact\n  calc\n    (\u222b x : \u03b1, (g x).toReal \u2202\u03bc) = \u222b x : \u03b1, EReal.toReal (gp x) - EReal.toReal (gm x) \u2202\u03bc :=\n      integral_congr_ae ae_g\n    _ = (\u222b x : \u03b1, EReal.toReal (gp x) \u2202\u03bc) - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n      simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]\n      exact integral_sub gp_integrable gm_integrable\n    _ < (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - \u222b x : \u03b1, \u2191(gm x) \u2202\u03bc := by\n      apply sub_lt_sub_right\n      convert gpint\n      simp only [EReal.toReal_coe_ennreal]\n    _ \u2264 (\u222b x : \u03b1, \u2191(fp x) \u2202\u03bc) + \u2191\u03b4 - ((\u222b x : \u03b1, \u2191(fm x) \u2202\u03bc) - \u03b4) := (sub_le_sub_left gmint _)\n    _ = (\u222b x : \u03b1, f x \u2202\u03bc) + 2 * \u03b4 := by\n      simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring\n    _ = (\u222b x : \u03b1, f x \u2202\u03bc) + \u03b5 := by congr 1; field_simp [mul_comm] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u2202\u03bc =     \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc ** simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), ENNReal.toReal (gp x) - \u2191(gm x) \u2202\u03bc = \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc ** exact integral_sub gp_integrable gm_integrable ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4 - \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc ** apply sub_lt_sub_right ** case h \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), EReal.toReal \u2191(gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4 ** convert gpint ** case h.e'_3.h.e'_7.h \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x\u271d : \u03b1 \u22a2 EReal.toReal \u2191(gp x\u271d) = ENNReal.toReal (gp x\u271d) ** simp only [EReal.toReal_coe_ennreal] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + \u2191\u03b4 - (\u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - \u2191\u03b4) = \u222b (x : \u03b1), f x \u2202\u03bc + 2 * \u2191\u03b4 ** simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), \u2191(Real.toNNReal (f x)) \u2202\u03bc + \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } -       (\u222b (x : \u03b1), \u2191(Real.toNNReal (-f x)) \u2202\u03bc - \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) }) =     \u222b (x : \u03b1), \u2191(Real.toNNReal (f x)) \u2202\u03bc - \u222b (x : \u03b1), \u2191(Real.toNNReal (-f x)) \u2202\u03bc +       2 * \u2191{ val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } ** ring ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u222b (x : \u03b1), f x \u2202\u03bc + 2 * \u2191\u03b4 = \u222b (x : \u03b1), f x \u2202\u03bc + \u03b5 ** congr 1 ** case e_a \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 2 * \u2191\u03b4 = \u03b5 ** field_simp [mul_comm] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4 ** filter_upwards [gp_lt_top] with ?_ hx ** case h \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x\u271d : \u03b1 hx : gp x\u271d < \u22a4 \u22a2 g x\u271d < \u22a4 ** simp only [sub_eq_add_neg, Ne.def, (EReal.add_lt_top _ _).ne, lt_top_iff_ne_top,\n  lt_top_iff_ne_top.1 hx, EReal.coe_ennreal_eq_top_iff, not_false_iff, EReal.neg_eq_top_iff,\n  EReal.coe_ennreal_ne_bot] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2200 (x : \u03b1), \u2191(f x) < g x ** intro x ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 \u2191(f x) < g x ** rw [EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (f x)] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 \u2191\u2191(Real.toNNReal (f x)) - \u2191\u2191(Real.toNNReal (-f x)) < g x ** refine' EReal.sub_lt_sub_of_lt_of_le _ _ _ _ ** case refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 \u2191\u2191(Real.toNNReal (f x)) < \u2191(gp x) ** simp only [EReal.coe_ennreal_lt_coe_ennreal_iff] ** case refine'_1 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 \u2191(Real.toNNReal (f x)) < gp x ** exact fp_lt_gp x ** case refine'_2 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 \u2191\u2191(gm x) \u2264 \u2191\u2191(Real.toNNReal (-f x)) ** simp only [ENNReal.coe_le_coe, EReal.coe_ennreal_le_coe_ennreal_iff] ** case refine'_2 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 gm x \u2264 Real.toNNReal (-f x) ** exact gm_le_fm x ** case refine'_3 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 \u2191\u2191(gm x) \u2260 \u22a5 ** simp only [EReal.coe_ennreal_ne_bot, Ne.def, not_false_iff] ** case refine'_4 \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 \u2191\u2191(Real.toNNReal (-f x)) \u2260 \u22a4 ** simp only [EReal.coe_nnreal_ne_top, Ne.def, not_false_iff] ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 LowerSemicontinuous g ** apply LowerSemicontinuous.add' ** case hf \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 LowerSemicontinuous fun z => \u2191(gp z) ** exact continuous_coe_ennreal_ereal.comp_lowerSemicontinuous gpcont fun x y hxy =>\n    EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy ** case hg \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 LowerSemicontinuous fun z => -\u2191\u2191(gm z) ** apply continuous_neg.comp_upperSemicontinuous_antitone _ fun x y hxy =>\n    EReal.neg_le_neg_iff.2 hxy ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 UpperSemicontinuous fun z => \u2191\u2191(gm z) ** apply continuous_coe_ennreal_ereal.comp_upperSemicontinuous _ fun x y hxy =>\n    EReal.coe_ennreal_le_coe_ennreal_iff.2 hxy ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 UpperSemicontinuous fun z => \u2191(gm z) ** exact ENNReal.continuous_coe.comp_upperSemicontinuous gmcont fun x y hxy =>\n    ENNReal.coe_le_coe.2 hxy ** case hcont \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) \u22a2 \u2200 (x : \u03b1), ContinuousAt (fun p => p.1 + p.2) (\u2191(gp x), -\u2191\u2191(gm x)) ** intro x ** case hcont \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 ContinuousAt (fun p => p.1 + p.2) (\u2191(gp x), -\u2191\u2191(gm x)) ** exact EReal.continuousAt_add (by simp) (by simp) ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 (\u2191(gp x), -\u2191\u2191(gm x)).1 \u2260 \u22a4 \u2228 (\u2191(gp x), -\u2191\u2191(gm x)).2 \u2260 \u22a5 ** simp ** \u03b1 : Type u_1 inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : WeaklyRegular \u03bc inst\u271d : SigmaFinite \u03bc f : \u03b1 \u2192 \u211d hf : Integrable f \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u03b4 : \u211d\u22650 := { val := \u03b5 / 2, property := (_ : 0 \u2264 \u03b5 / 2) } \u03b4pos : 0 < \u03b4 fp : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (f x) int_fp : Integrable fun x => \u2191(fp x) gp : \u03b1 \u2192 \u211d\u22650\u221e fp_lt_gp : \u2200 (x : \u03b1), \u2191(fp x) < gp x gpcont : LowerSemicontinuous gp gp_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, gp x < \u22a4 gp_integrable : Integrable fun x => ENNReal.toReal (gp x) gpint : \u222b (x : \u03b1), ENNReal.toReal (gp x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(fp x) \u2202\u03bc + (fun a => \u2191a) \u03b4 fm : \u03b1 \u2192 \u211d\u22650 := fun x => Real.toNNReal (-f x) int_fm : Integrable fun x => \u2191(fm x) gm : \u03b1 \u2192 \u211d\u22650 gm_le_fm : \u2200 (x : \u03b1), gm x \u2264 fm x gmcont : UpperSemicontinuous gm gm_integrable : Integrable fun x => \u2191(gm x) gmint : \u222b (x : \u03b1), \u2191(fm x) \u2202\u03bc - (fun a => \u2191a) \u03b4 \u2264 \u222b (x : \u03b1), \u2191(gm x) \u2202\u03bc g : \u03b1 \u2192 EReal := fun x => \u2191(gp x) - \u2191\u2191(gm x) ae_g : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, EReal.toReal (g x) = EReal.toReal \u2191(gp x) - EReal.toReal \u2191\u2191(gm x) x : \u03b1 \u22a2 (\u2191(gp x), -\u2191\u2191(gm x)).1 \u2260 \u22a5 \u2228 (\u2191(gp x), -\u2191\u2191(gm x)).2 \u2260 \u22a4 ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurePreserving_finTwoArrow ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1\u271d : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : Fintype \u03b9 m\u271d\u00b9 : (i : \u03b9) \u2192 OuterMeasure (\u03b1\u271d i) m\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1\u271d i) \u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1\u271d i) inst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i) inst\u271d\u00b9 : Fintype \u03b9' \u03b1 : Type u m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc \u22a2 MeasurePreserving \u2191MeasurableEquiv.finTwoArrow ** simpa only [Matrix.vec_single_eq_const, Matrix.vecCons_const] using\n  measurePreserving_finTwoArrow_vec \u03bc \u03bc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.map_add ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd\u271d \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u03b2 hf : Measurable f \u22a2 map f (\u03bc + \u03bd) = map f \u03bc + map f \u03bd ** simp [\u2190 map\u2097_apply_of_measurable hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.elem_cons_self ** \u03b1 : Type u_1 as : List \u03b1 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : LawfulBEq \u03b1 a : \u03b1 \u22a2 elem a (a :: as) = true ** simp [elem_cons] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.integral_indicator_norm_ge_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 ** have htendsto :\n    \u2200\u1d50 x \u2202\u03bc, Tendsto (fun M : \u2115 => { x | (M : \u211d) \u2264 \u2016f x\u2016\u208a }.indicator f x) atTop (\ud835\udcdd 0) :=\n  univ_mem' (id fun x => tendsto_indicator_ge f x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) \u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 ** have hmeas : \u2200 M : \u2115, AEStronglyMeasurable ({ x | (M : \u211d) \u2264 \u2016f x\u2016\u208a }.indicator f) \u03bc := by\n  intro M\n  apply hf.1.indicator\n  apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const\n    hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc \u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 ** have hbound : HasFiniteIntegral (fun x => \u2016f x\u2016) \u03bc := by\n  rw [mem\u2112p_one_iff_integrable] at hf\n  exact hf.norm.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 this : Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) \u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 ** rw [ENNReal.tendsto_atTop_zero] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 this :   \u2200 (\u03b5 : \u211d\u22650\u221e),     \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5 \u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8M, hM\u27e9 := this (ENNReal.ofReal \u03b5) (ENNReal.ofReal_pos.2 h\u03b5) ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 this :   \u2200 (\u03b5 : \u211d\u22650\u221e),     \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5 M : \u2115 hM : \u2200 (n : \u2115), n \u2265 M \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 ** simp only [true_and_iff, ge_iff_le, zero_tsub, zero_le, sub_zero, zero_add, coe_nnnorm,\n  Set.mem_Icc] at hM ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 this :   \u2200 (\u03b5 : \u211d\u22650\u221e),     \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5 M : \u2115 hM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 M, \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e8M, _\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 this :   \u2200 (\u03b5 : \u211d\u22650\u221e),     \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5 M : \u2115 hM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 ** convert hM M le_rfl ** case h.e'_3.h.e'_4.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 this :   \u2200 (\u03b5 : \u211d\u22650\u221e),     \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5 M : \u2115 hM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 x\u271d : \u03b1 \u22a2 \u2191\u2016Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x\u271d\u2016\u208a = ENNReal.ofReal \u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016\u208a} f x\u271d\u2016 ** simp only [coe_nnnorm, ENNReal.ofReal_eq_coe_nnreal (norm_nonneg _)] ** case h.e'_3.h.e'_4.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 this :   \u2200 (\u03b5 : \u211d\u22650\u221e),     \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc \u2264 \u03b5 M : \u2115 hM : \u2200 (n : \u2115), M \u2264 n \u2192 \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2202\u03bc \u2264 ENNReal.ofReal \u03b5 x\u271d : \u03b1 \u22a2 \u2191\u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016} f x\u271d\u2016\u208a =     \u2191{ val := \u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016\u208a} f x\u271d\u2016, property := (_ : 0 \u2264 \u2016Set.indicator {x | \u2191M \u2264 \u2016f x\u2016\u208a} f x\u271d\u2016) } ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) \u22a2 \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc ** intro M ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) M : \u2115 \u22a2 AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc ** apply hf.1.indicator ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) M : \u2115 \u22a2 MeasurableSet {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} ** apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const\n  hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc \u22a2 HasFiniteIntegral fun x => \u2016f x\u2016 ** rw [mem\u2112p_one_iff_integrable] at hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Integrable f hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc \u22a2 HasFiniteIntegral fun x => \u2016f x\u2016 ** exact hf.norm.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a - 0\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) ** refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 \u22a2 \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016 ** refine' fun n => univ_mem' (id fun x => _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 n : \u2115 x : \u03b1 \u22a2 x \u2208 {x | (fun a => \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016) x} ** by_cases hx : (n : \u211d) \u2264 \u2016f x\u2016 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 n : \u2115 x : \u03b1 hx : \u2191n \u2264 \u2016f x\u2016 \u22a2 x \u2208 {x | (fun a => \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016) x} ** dsimp ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 n : \u2115 x : \u03b1 hx : \u2191n \u2264 \u2016f x\u2016 \u22a2 \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f x\u2016 \u2264 \u2016f x\u2016 ** rwa [Set.indicator_of_mem] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 n : \u2115 x : \u03b1 hx : \u00ac\u2191n \u2264 \u2016f x\u2016 \u22a2 x \u2208 {x | (fun a => \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f a\u2016 \u2264 \u2016f a\u2016) x} ** dsimp ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 n : \u2115 x : \u03b1 hx : \u00ac\u2191n \u2264 \u2016f x\u2016 \u22a2 \u2016Set.indicator {x | \u2191n \u2264 \u2016f x\u2016\u208a} f x\u2016 \u2264 \u2016f x\u2016 ** rw [Set.indicator_of_not_mem, norm_zero] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 n : \u2115 x : \u03b1 hx : \u00ac\u2191n \u2264 \u2016f x\u2016 \u22a2 0 \u2264 \u2016f x\u2016 ** exact norm_nonneg _ ** case neg.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hf : Mem\u2112p f 1 hmeas\u271d : StronglyMeasurable f \u03b5 : \u211d h\u03b5 : 0 < \u03b5 htendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun M => Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f x) atTop (\ud835\udcdd 0) hmeas : \u2200 (M : \u2115), AEStronglyMeasurable (Set.indicator {x | \u2191M \u2264 \u2191\u2016f x\u2016\u208a} f) \u03bc hbound : HasFiniteIntegral fun x => \u2016f x\u2016 n : \u2115 x : \u03b1 hx : \u00ac\u2191n \u2264 \u2016f x\u2016 \u22a2 \u00acx \u2208 {x | \u2191n \u2264 \u2016f x\u2016\u208a} ** assumption ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.natAbs_div ** a b : Int w\u271d\u00b9 w\u271d : Nat \u22a2 natAbs (div (\u2191w\u271d\u00b9) (-\u2191w\u271d)) = Nat.div (natAbs \u2191w\u271d\u00b9) (natAbs (-\u2191w\u271d)) ** rw [Int.div_neg, natAbs_neg, natAbs_neg] ** a b : Int w\u271d\u00b9 w\u271d : Nat \u22a2 natAbs (div \u2191w\u271d\u00b9 \u2191w\u271d) = Nat.div (natAbs \u2191w\u271d\u00b9) (natAbs \u2191w\u271d) ** rfl ** a b : Int w\u271d\u00b9 w\u271d : Nat \u22a2 natAbs (div (-\u2191w\u271d\u00b9) \u2191w\u271d) = Nat.div (natAbs (-\u2191w\u271d\u00b9)) (natAbs \u2191w\u271d) ** rw [Int.neg_div, natAbs_neg, natAbs_neg] ** a b : Int w\u271d\u00b9 w\u271d : Nat \u22a2 natAbs (div (-\u2191w\u271d\u00b9) (-\u2191w\u271d)) = Nat.div (natAbs (-\u2191w\u271d\u00b9)) (natAbs (-\u2191w\u271d)) ** rw [Int.neg_div_neg, natAbs_neg, natAbs_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.variance_le_expectation_sq ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 \u22a2 variance X \u2119 \u2264 \u222b (a : \u03a9), (X ^ 2) a ** by_cases hX : Mem\u2112p X 2 ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 \u22a2 variance X \u2119 \u2264 \u222b (a : \u03a9), (X ^ 2) a ** rw [variance, evariance_eq_lintegral_ofReal, \u2190 integral_eq_lintegral_of_nonneg_ae] ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 \u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a  case neg.hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 \u22a2 0 \u2264\u1d50[\u2119] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2  case neg.hfm \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 \u22a2 AEStronglyMeasurable (fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2) \u2119 ** by_cases hint : Integrable X ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : Integrable X \u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a  case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : \u00acIntegrable X \u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a  case neg.hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 \u22a2 0 \u2264\u1d50[\u2119] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2  case neg.hfm \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 \u22a2 AEStronglyMeasurable (fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2) \u2119 ** swap ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : Mem\u2112p X 2 \u22a2 variance X \u2119 \u2264 \u222b (a : \u03a9), (X ^ 2) a ** rw [variance_def' hX] ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : Mem\u2112p X 2 \u22a2 (\u222b (a : \u03a9), (X ^ 2) a) - (\u222b (a : \u03a9), X a) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a ** simp only [sq_nonneg, sub_le_self_iff] ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : \u00acIntegrable X \u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a ** simp only [integral_undef hint, Pi.pow_apply, Pi.sub_apply, sub_zero] ** case neg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : \u00acIntegrable X \u22a2 \u222b (a : \u03a9), X a ^ 2 \u2264 \u222b (a : \u03a9), X a ^ 2 ** exact le_rfl ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : Integrable X \u22a2 \u222b (a : \u03a9), (X a - \u222b (x : \u03a9), X x) ^ 2 \u2264 \u222b (a : \u03a9), (X ^ 2) a ** rw [integral_undef] ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : Integrable X \u22a2 0 \u2264 \u222b (a : \u03a9), (X ^ 2) a ** exact integral_nonneg fun a => sq_nonneg _ ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : Integrable X \u22a2 \u00acIntegrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2 ** intro h ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : Integrable X h : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2 \u22a2 False ** have A : Mem\u2112p (X - fun \u03c9 : \u03a9 => \ud835\udd3c[X]) 2 \u2119 :=\n  (mem\u2112p_two_iff_integrable_sq (hint.aestronglyMeasurable.sub aestronglyMeasurable_const)).2 h ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : Integrable X h : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2 A : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2 \u22a2 False ** have B : Mem\u2112p (fun _ : \u03a9 => \ud835\udd3c[X]) 2 \u2119 := mem\u2112p_const _ ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : Integrable X h : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2 A : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2 B : Mem\u2112p (fun x => \u222b (a : \u03a9), X a) 2 \u22a2 False ** apply hX ** case pos \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : Integrable X h : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2 A : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2 B : Mem\u2112p (fun x => \u222b (a : \u03a9), X a) 2 \u22a2 Mem\u2112p X 2 ** convert A.add B ** case h.e'_5 \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 hint : Integrable X h : Integrable fun a => (X a - \u222b (x : \u03a9), X x) ^ 2 A : Mem\u2112p (X - fun \u03c9 => \u222b (a : \u03a9), X a) 2 B : Mem\u2112p (fun x => \u222b (a : \u03a9), X a) 2 \u22a2 X = (X - fun \u03c9 => \u222b (a : \u03a9), X a) + fun x => \u222b (a : \u03a9), X a ** simp ** case neg.hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 \u22a2 0 \u2264\u1d50[\u2119] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2 ** exact @ae_of_all _ (_) _ _ fun x => sq_nonneg _ ** case neg.hfm \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u03a9 \u2192 \u211d hm : AEStronglyMeasurable X \u2119 hX : \u00acMem\u2112p X 2 \u22a2 AEStronglyMeasurable (fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x) ^ 2) \u2119 ** exact (AEMeasurable.pow_const (hm.aemeasurable.sub_const _) _).aestronglyMeasurable ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant ** G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U \u22a2 0 < innerContent \u03bc U ** have : (interior (U : Set G)).Nonempty ** case this G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U \u22a2 Set.Nonempty (interior \u2191U)  G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U this : Set.Nonempty (interior \u2191U) \u22a2 0 < innerContent \u03bc U ** rwa [U.isOpen.interior_eq] ** G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U this : Set.Nonempty (interior \u2191U) \u22a2 0 < innerContent \u03bc U ** rcases compact_covered_by_mul_left_translates K.2 this with \u27e8s, hs\u27e9 ** case intro G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U this : Set.Nonempty (interior \u2191U) s : Finset G hs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U \u22a2 0 < innerContent \u03bc U ** suffices \u03bc K \u2264 s.card * \u03bc.innerContent U by\n  exact (ENNReal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2 ** case intro G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U this : Set.Nonempty (interior \u2191U) s : Finset G hs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U \u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Finset.card s) * innerContent \u03bc U ** have : (K : Set G) \u2286 \u2191(\u2a06 g \u2208 s, Opens.comap (Homeomorph.mulLeft g).toContinuousMap U) := by\n  simpa only [Opens.iSup_def, Opens.coe_comap, Subtype.coe_mk] ** case intro G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U this\u271d : Set.Nonempty (interior \u2191U) s : Finset G hs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U this : \u2191K \u2286 \u2191(\u2a06 g \u2208 s, \u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft g))) U) \u22a2 (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Finset.card s) * innerContent \u03bc U ** refine' (\u03bc.le_innerContent _ _ this).trans _ ** case intro G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U this\u271d : Set.Nonempty (interior \u2191U) s : Finset G hs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U this : \u2191K \u2286 \u2191(\u2a06 g \u2208 s, \u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft g))) U) \u22a2 (Finset.sum s fun d => innerContent \u03bc (\u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft d))) U)) \u2264     \u2191(Finset.card s) * innerContent \u03bc U ** simp only [\u03bc.is_mul_left_invariant_innerContent h3, Finset.sum_const, nsmul_eq_mul, le_refl] ** G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U this\u271d : Set.Nonempty (interior \u2191U) s : Finset G hs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U this : (fun s => \u2191(toFun \u03bc s)) K \u2264 \u2191(Finset.card s) * innerContent \u03bc U \u22a2 0 < innerContent \u03bc U ** exact (ENNReal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2 ** G : Type w inst\u271d\u00b3 : TopologicalSpace G \u03bc : Content G inst\u271d\u00b2 : T2Space G inst\u271d\u00b9 : Group G inst\u271d : TopologicalGroup G h3 :   \u2200 (g : G) {K : Compacts G},     (fun s => \u2191(toFun \u03bc s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =       (fun s => \u2191(toFun \u03bc s)) K K : Compacts G hK : (fun s => \u2191(toFun \u03bc s)) K \u2260 0 U : Opens G hU : Set.Nonempty \u2191U this : Set.Nonempty (interior \u2191U) s : Finset G hs : K.carrier \u2286 \u22c3 g \u2208 s, (fun h => g * h) \u207b\u00b9' \u2191U \u22a2 \u2191K \u2286 \u2191(\u2a06 g \u2208 s, \u2191(Opens.comap (Homeomorph.toContinuousMap (Homeomorph.mulLeft g))) U) ** simpa only [Opens.iSup_def, Opens.coe_comap, Subtype.coe_mk] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.le_log2 ** n k : Nat h : n \u2260 0 \u22a2 0 \u2264 log2 n \u2194 2 ^ 0 \u2264 n ** simp [show 1 \u2264 n from Nat.pos_of_ne_zero h] ** n k\u271d : Nat h : n \u2260 0 k : Nat \u22a2 k + 1 \u2264 log2 n \u2194 2 ^ (k + 1) \u2264 n ** rw [log2] ** n k\u271d : Nat h : n \u2260 0 k : Nat \u22a2 (k + 1 \u2264 if n \u2265 2 then log2 (n / 2) + 1 else 0) \u2194 2 ^ (k + 1) \u2264 n ** split ** case inl n k\u271d : Nat h : n \u2260 0 k : Nat h\u271d : n \u2265 2 \u22a2 k + 1 \u2264 log2 (n / 2) + 1 \u2194 2 ^ (k + 1) \u2264 n ** have n0 : 0 < n / 2 := (Nat.le_div_iff_mul_le (by decide)).2 \u2039_\u203a ** case inl n k\u271d : Nat h : n \u2260 0 k : Nat h\u271d : n \u2265 2 n0 : 0 < n / 2 \u22a2 k + 1 \u2264 log2 (n / 2) + 1 \u2194 2 ^ (k + 1) \u2264 n ** simp [Nat.add_le_add_iff_right, le_log2 (Nat.ne_of_gt n0), le_div_iff_mul_le, Nat.pow_succ] ** n k\u271d : Nat h : n \u2260 0 k : Nat h\u271d : n \u2265 2 \u22a2 0 < 2 ** decide ** case inr n k\u271d : Nat h : n \u2260 0 k : Nat h\u271d : \u00acn \u2265 2 \u22a2 k + 1 \u2264 0 \u2194 2 ^ (k + 1) \u2264 n ** simp only [le_zero_eq, succ_ne_zero, false_iff] ** case inr n k\u271d : Nat h : n \u2260 0 k : Nat h\u271d : \u00acn \u2265 2 \u22a2 \u00ac2 ^ (k + 1) \u2264 n ** refine mt (Nat.le_trans ?_) \u2039_\u203a ** case inr n k\u271d : Nat h : n \u2260 0 k : Nat h\u271d : \u00acn \u2265 2 \u22a2 2 \u2264 2 ^ (k + 1) ** exact Nat.pow_le_pow_of_le_right (Nat.succ_pos 1) (Nat.le_add_left 1 k) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image\u2082_insert_left ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 \u03b6 : Type u_11 \u03b6' : Type u_12 \u03bd : Type u_13 inst\u271d\u2078 : DecidableEq \u03b1' inst\u271d\u2077 : DecidableEq \u03b2' inst\u271d\u2076 : DecidableEq \u03b3 inst\u271d\u2075 : DecidableEq \u03b3' inst\u271d\u2074 : DecidableEq \u03b4 inst\u271d\u00b3 : DecidableEq \u03b4' inst\u271d\u00b2 : DecidableEq \u03b5 inst\u271d\u00b9 : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 inst\u271d : DecidableEq \u03b1 \u22a2 \u2191(image\u2082 f (insert a s) t) = \u2191(image (fun b => f a b) t \u222a image\u2082 f s t) ** push_cast ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 \u03b6 : Type u_11 \u03b6' : Type u_12 \u03bd : Type u_13 inst\u271d\u2078 : DecidableEq \u03b1' inst\u271d\u2077 : DecidableEq \u03b2' inst\u271d\u2076 : DecidableEq \u03b3 inst\u271d\u2075 : DecidableEq \u03b3' inst\u271d\u2074 : DecidableEq \u03b4 inst\u271d\u00b3 : DecidableEq \u03b4' inst\u271d\u00b2 : DecidableEq \u03b5 inst\u271d\u00b9 : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 inst\u271d : DecidableEq \u03b1 \u22a2 image2 f (insert a \u2191s) \u2191t = (fun b => f a b) '' \u2191t \u222a image2 f \u2191s \u2191t ** exact image2_insert_left ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Finite.pi ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 t : (d : \u03b4) \u2192 Set (\u03ba d) ht : \u2200 (d : \u03b4), Set.Finite (t d) \u22a2 Set.Finite (Set.pi univ t) ** cases _root_.nonempty_fintype \u03b4 ** case intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 t : (d : \u03b4) \u2192 Set (\u03ba d) ht : \u2200 (d : \u03b4), Set.Finite (t d) val\u271d : Fintype \u03b4 \u22a2 Set.Finite (Set.pi univ t) ** lift t to \u2200 d, Finset (\u03ba d) using ht ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 val\u271d : Fintype \u03b4 t : (i : \u03b4) \u2192 Finset (\u03ba i) \u22a2 Set.Finite (Set.pi univ fun i => \u2191(t i)) ** classical\n  rw [\u2190 Fintype.coe_piFinset]\n  apply Finset.finite_toSet ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 val\u271d : Fintype \u03b4 t : (i : \u03b4) \u2192 Finset (\u03ba i) \u22a2 Set.Finite (Set.pi univ fun i => \u2191(t i)) ** rw [\u2190 Fintype.coe_piFinset] ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 val\u271d : Fintype \u03b4 t : (i : \u03b4) \u2192 Finset (\u03ba i) \u22a2 Set.Finite \u2191(Fintype.piFinset fun i => t i) ** apply Finset.finite_toSet ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.toMeasure_apply_eq_zero_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : MeasurableSpace \u03b1 p : PMF \u03b1 s t : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(toMeasure p) s = 0 \u2194 Disjoint (support p) s ** rw [toMeasure_apply_eq_toOuterMeasure_apply p s hs, toOuterMeasure_apply_eq_zero_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_congr_ae ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** by_cases hm : m \u2264 m0 ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : m \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : m \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm) ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** exact (condexp_ae_eq_condexpL1 hm f).trans\n  (Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])\n    (condexp_ae_eq_condexpL1 hm g).symm) ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** simp_rw [condexp_of_not_le hm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : \u00acm \u2264 m0 \u22a2 0 =\u1d50[\u03bc] 0 ** rfl ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 0 =\u1d50[\u03bc] 0 ** rfl ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 h : f =\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u2191\u2191(condexpL1 hm \u03bc f) =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc g) ** rw [condexpL1_congr_ae hm h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_restrict_uIoc_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d : LinearOrder \u03b1 a b : \u03b1 P : \u03b1 \u2192 Prop \u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (\u0399 a b), P x) \u2194     (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (Ioc a b), P x) \u2227 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (Ioc b a), P x ** rw [ae_restrict_uIoc_eq, eventually_sup] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.ae_eq_image_of_ae_eq_comap ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : s =\u1da0[ae (comap f \u03bc)] t \u22a2 f '' s =\u1da0[ae \u03bc] f '' t ** rw [EventuallyEq, ae_iff] at hst \u22a2 ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 \u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0 ** have h_eq_\u03b1 : { a : \u03b1 | \u00acs a = t a } = s \\ t \u222a t \\ s := by\n  ext1 x\n  simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]\n  tauto ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 h_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s \u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0 ** have h_eq_\u03b2 : { a : \u03b2 | \u00ac(f '' s) a = (f '' t) a } = f '' s \\ f '' t \u222a f '' t \\ f '' s := by\n  ext1 x\n  simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff]\n  tauto ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 h_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s h_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' s \\ f '' t \u222a f '' t \\ f '' s \u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0 ** rw [\u2190 Set.image_diff hfi, \u2190 Set.image_diff hfi, \u2190 Set.image_union] at h_eq_\u03b2 ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 h_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s h_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s) \u22a2 \u2191\u2191\u03bc {a | \u00ac(f '' s) a = (f '' t) a} = 0 ** rw [h_eq_\u03b2] ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 h_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s h_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s) \u22a2 \u2191\u2191\u03bc (f '' (s \\ t \u222a t \\ s)) = 0 ** rw [h_eq_\u03b1] at hst ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) (s \\ t \u222a t \\ s) = 0 h_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s h_eq_\u03b2 : {a | \u00ac(f '' s) a = (f '' t) a} = f '' (s \\ t \u222a t \\ s) \u22a2 \u2191\u2191\u03bc (f '' (s \\ t \u222a t \\ s)) = 0 ** exact measure_image_eq_zero_of_comap_eq_zero f \u03bc hfi hf hst ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 \u22a2 {a | \u00acs a = t a} = s \\ t \u222a t \\ s ** ext1 x ** case h \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 x : \u03b1 \u22a2 x \u2208 {a | \u00acs a = t a} \u2194 x \u2208 s \\ t \u222a t \\ s ** simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff] ** case h \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 x : \u03b1 \u22a2 \u00ac(s x \u2194 t x) \u2194 x \u2208 s \u2227 \u00acx \u2208 t \u2228 x \u2208 t \u2227 \u00acx \u2208 s ** tauto ** \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 h_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s \u22a2 {a | \u00ac(f '' s) a = (f '' t) a} = f '' s \\ f '' t \u222a f '' t \\ f '' s ** ext1 x ** case h \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 h_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s x : \u03b2 \u22a2 x \u2208 {a | \u00ac(f '' s) a = (f '' t) a} \u2194 x \u2208 f '' s \\ f '' t \u222a f '' t \\ f '' s ** simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff] ** case h \u03b1 : Type u_1 \u03b2\u271d : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2\u271d inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 \u03b2 : Type u_8 inst\u271d : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 \u03bc : Measure \u03b2 hfi : Injective f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 NullMeasurableSet (f '' s) s t : Set \u03b1 hst : \u2191\u2191(comap f \u03bc) {a | \u00acs a = t a} = 0 h_eq_\u03b1 : {a | \u00acs a = t a} = s \\ t \u222a t \\ s x : \u03b2 \u22a2 \u00ac((f '' s) x \u2194 (f '' t) x) \u2194 x \u2208 f '' s \u2227 \u00acx \u2208 f '' t \u2228 x \u2208 f '' t \u2227 \u00acx \u2208 f '' s ** tauto ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_subordinate_pairwise_disjoint ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 h : \u2200 (i : \u03b9), NullMeasurableSet (s i) hd : Pairwise (AEDisjoint \u03bc on s) \u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t) ** choose t ht_sub htm ht_eq using fun i => exists_measurable_subset_ae_eq (h i) ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t\u271d : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 h : \u2200 (i : \u03b9), NullMeasurableSet (s i) hd : Pairwise (AEDisjoint \u03bc on s) t : \u03b9 \u2192 Set \u03b1 ht_sub : \u2200 (i : \u03b9), t i \u2286 s i htm : \u2200 (i : \u03b9), MeasurableSet (t i) ht_eq : \u2200 (i : \u03b9), t i =\u1d50[\u03bc] s i \u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t) ** rcases exists_null_pairwise_disjoint_diff hd with \u27e8u, hum, hu\u2080, hud\u27e9 ** case intro.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t\u271d : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 h : \u2200 (i : \u03b9), NullMeasurableSet (s i) hd : Pairwise (AEDisjoint \u03bc on s) t : \u03b9 \u2192 Set \u03b1 ht_sub : \u2200 (i : \u03b9), t i \u2286 s i htm : \u2200 (i : \u03b9), MeasurableSet (t i) ht_eq : \u2200 (i : \u03b9), t i =\u1d50[\u03bc] s i u : \u03b9 \u2192 Set \u03b1 hum : \u2200 (i : \u03b9), MeasurableSet (u i) hu\u2080 : \u2200 (i : \u03b9), \u2191\u2191\u03bc (u i) = 0 hud : Pairwise (Disjoint on fun i => s i \\ u i) \u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t) ** exact\n  \u27e8fun i => t i \\ u i, fun i => (diff_subset _ _).trans (ht_sub _), fun i =>\n    (ht_eq _).symm.trans (diff_null_ae_eq_self (hu\u2080 i)).symm, fun i => (htm i).diff (hum i),\n    hud.mono fun i j h =>\n      h.mono (diff_subset_diff_left (ht_sub i)) (diff_subset_diff_left (ht_sub j))\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ProgMeasurable.finset_prod ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m inst\u271d\u00b2 : MeasurableSpace \u03b9 \u03b3 : Type u_4 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : ContinuousMul \u03b2 U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Finset \u03b3 h : \u2200 (c : \u03b3), c \u2208 s \u2192 ProgMeasurable f (U c) \u22a2 ProgMeasurable f fun i a => \u220f c in s, U c i a ** convert ProgMeasurable.finset_prod' h using 1 ** case h.e'_9 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m inst\u271d\u00b2 : MeasurableSpace \u03b9 \u03b3 : Type u_4 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : ContinuousMul \u03b2 U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Finset \u03b3 h : \u2200 (c : \u03b3), c \u2208 s \u2192 ProgMeasurable f (U c) \u22a2 (fun i a => \u220f c in s, U c i a) = \u220f c in s, U c ** ext (i a) ** case h.e'_9.h.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : Preorder \u03b9 u v : \u03b9 \u2192 \u03a9 \u2192 \u03b2 f : Filtration \u03b9 m inst\u271d\u00b2 : MeasurableSpace \u03b9 \u03b3 : Type u_4 inst\u271d\u00b9 : CommMonoid \u03b2 inst\u271d : ContinuousMul \u03b2 U : \u03b3 \u2192 \u03b9 \u2192 \u03a9 \u2192 \u03b2 s : Finset \u03b3 h : \u2200 (c : \u03b3), c \u2208 s \u2192 ProgMeasurable f (U c) i : \u03b9 a : \u03a9 \u22a2 \u220f c in s, U c i a = Finset.prod s (fun c => U c) i a ** simp only [Finset.prod_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_add_adjacent_intervals ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hbc : IntervalIntegrable f \u03bc b c \u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc + \u222b (x : \u211d) in b..c, f x \u2202\u03bc = \u222b (x : \u211d) in a..c, f x \u2202\u03bc ** rw [\u2190 add_neg_eq_zero, \u2190 integral_symm, integral_add_adjacent_intervals_cancel hab hbc] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.norm_integral_eq_norm_integral_Ioc ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d f : \u211d \u2192 E \u22a2 \u2016\u222b (x : \u211d) in a..b, f x \u2202\u03bc\u2016 = \u2016\u222b (x : \u211d) in \u0399 a b, f x \u2202\u03bc\u2016 ** rw [\u2190 norm_integral_min_max, integral_of_le min_le_max, uIoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.Cofix.bisim\u2082 ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y) \u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR (fun {i} => RelLast \u03b1 r) (dest x) (dest y) ** intros ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y) x\u271d y\u271d : Cofix F \u03b1 a\u271d : r x\u271d y\u271d \u22a2 LiftR (fun {i} => RelLast \u03b1 r) (dest x\u271d) (dest y\u271d) ** rw [\u2190 LiftR_RelLast_iff] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y) x\u271d y\u271d : Cofix F \u03b1 a\u271d : r x\u271d y\u271d \u22a2 LiftR' (RelLast' \u03b1 r) (dest x\u271d) (dest y\u271d) ** apply h ** case a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 LiftR' (RelLast' \u03b1 r) (dest x) (dest y) x\u271d y\u271d : Cofix F \u03b1 a\u271d : r x\u271d y\u271d \u22a2 r x\u271d y\u271d ** assumption ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.DominatedFinMeasAdditive.zero ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hC : 0 \u2264 C \u22a2 DominatedFinMeasAdditive \u03bc 0 C ** refine' \u27e8FinMeasAdditive.zero, fun s _ _ => _\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hC : 0 \u2264 C s : Set \u03b1 x\u271d\u00b9 : MeasurableSet s x\u271d : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2016OfNat.ofNat 0 s\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s) ** rw [Pi.zero_apply, norm_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hC : 0 \u2264 C s : Set \u03b1 x\u271d\u00b9 : MeasurableSet s x\u271d : \u2191\u2191\u03bc s < \u22a4 \u22a2 0 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s) ** exact mul_nonneg hC toReal_nonneg ** Qed",
        "informal": ""
    },
    {
        "formal": "surjOn_Ioi_of_monotone_surjective ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : PartialOrder \u03b2 f : \u03b1 \u2192 \u03b2 h_mono : Monotone f h_surj : Surjective f a : \u03b1 \u22a2 SurjOn f (Ioi a) (Ioi (f a)) ** rw [\u2190 compl_Iic, \u2190 compl_compl (Ioi (f a))] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : PartialOrder \u03b2 f : \u03b1 \u2192 \u03b2 h_mono : Monotone f h_surj : Surjective f a : \u03b1 \u22a2 SurjOn f (Iic a)\u1d9c (Ioi (f a))\u1d9c\u1d9c ** refine' MapsTo.surjOn_compl _ h_surj ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : PartialOrder \u03b2 f : \u03b1 \u2192 \u03b2 h_mono : Monotone f h_surj : Surjective f a : \u03b1 \u22a2 MapsTo f (Iic a) (Ioi (f a))\u1d9c ** exact fun x hx => (h_mono hx).not_lt ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.ext_iff' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b2 } \u22a2 \u03ba = \u03b7 \u2194 \u2200 (a : \u03b1) (s : Set \u03b2), MeasurableSet s \u2192 \u2191\u2191(\u2191\u03ba a) s = \u2191\u2191(\u2191\u03b7 a) s ** simp_rw [ext_iff, Measure.ext_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.noncommProd_insert_of_not_mem ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b) ha : \u00aca \u2208 s \u22a2 Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute ** convert noncommProd_lemma _ f comm using 3 ** case h.e'_2.h.e'_2.h.a F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b) ha : \u00aca \u2208 s x\u271d : \u03b2 \u22a2 x\u271d \u2208 f a ::\u2098 Multiset.map f s.val \u2194 x\u271d \u2208 Multiset.map f (insert a s).val ** simp [@eq_comm _ (f a)] ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b2 inst\u271d\u00b9 : Monoid \u03b3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 comm : Set.Pairwise \u2191(insert a s) fun a b => Commute (f a) (f b) ha : \u00aca \u2208 s \u22a2 Multiset.noncommProd (f a ::\u2098 Multiset.map f s.val)       (_ : Set.Pairwise {x | x \u2208 f a ::\u2098 Multiset.map f s.val} Commute) =     f a * noncommProd s f (_ : Set.Pairwise \u2191s fun a b => Commute (f a) (f b)) ** rw [Multiset.noncommProd_cons, noncommProd] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.martingalePart_bdd_difference ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131\u271d : Filtration \u2115 m0 n : \u2115 R : \u211d\u22650 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u2131 : Filtration \u2115 m0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |martingalePart f \u2131 \u03bc (i + 1) \u03c9 - martingalePart f \u2131 \u03bc i \u03c9| \u2264 \u2191(2 * R) ** filter_upwards [hbdd, predictablePart_bdd_difference \u2131 hbdd] with \u03c9 h\u03c9\u2081 h\u03c9\u2082 i ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131\u271d : Filtration \u2115 m0 n : \u2115 R : \u211d\u22650 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u2131 : Filtration \u2115 m0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R h\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R i : \u2115 \u22a2 |martingalePart f \u2131 \u03bc (i + 1) \u03c9 - martingalePart f \u2131 \u03bc i \u03c9| \u2264 \u2191(2 * R) ** simp only [two_mul, martingalePart, Pi.sub_apply] ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131\u271d : Filtration \u2115 m0 n : \u2115 R : \u211d\u22650 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u2131 : Filtration \u2115 m0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R h\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R i : \u2115 \u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R) ** have : |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| := by\n  ring_nf ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131\u271d : Filtration \u2115 m0 n : \u2115 R : \u211d\u22650 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u2131 : Filtration \u2115 m0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R h\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R i : \u2115 this :   |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =     |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R) ** rw [this] ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131\u271d : Filtration \u2115 m0 n : \u2115 R : \u211d\u22650 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u2131 : Filtration \u2115 m0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R h\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R i : \u2115 this :   |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =     |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u22a2 |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R) ** exact (abs_sub _ _).trans (add_le_add (h\u03c9\u2081 i) (h\u03c9\u2082 i)) ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131\u271d : Filtration \u2115 m0 n : \u2115 R : \u211d\u22650 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u2131 : Filtration \u2115 m0 hbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R \u03c9 : \u03a9 h\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R h\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R i : \u2115 \u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =     |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| ** ring_nf ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.map_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 p : PMF \u03b1 b : \u03b2 \u22a2 map (Function.const \u03b1 b) p = pure b ** simp only [map, Function.comp, bind_const, Function.const] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pairwise_disjoint_Ico_int_cast ** \u03b1 : Type u_1 inst\u271d : OrderedRing \u03b1 a : \u03b1 \u22a2 Pairwise (Disjoint on fun n => Ico (\u2191n) (\u2191n + 1)) ** simpa only [zero_add] using pairwise_disjoint_Ico_add_int_cast (0 : \u03b1) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.induction_stronglyMeasurable ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} \u22a2 \u2200 (f : { x // x \u2208 Lp F p }), AEStronglyMeasurable' m (\u2191\u2191f) \u03bc \u2192 P f ** intro f hf ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc \u22a2 P f ** suffices h_add_ae :\n  \u2200 \u2983f g\u2984, \u2200 hf : Mem\u2112p f p \u03bc, \u2200 hg : Mem\u2112p g p \u03bc, \u2200 _ : AEStronglyMeasurable' m f \u03bc,\n    \u2200 _ : AEStronglyMeasurable' m g \u03bc, Disjoint (Function.support f) (Function.support g) \u2192\n      P (hf.toLp f) \u2192 P (hg.toLp g) \u2192 P (hf.toLp f + hg.toLp g) ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc h_add_ae :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     AEStronglyMeasurable' m f \u03bc \u2192       AEStronglyMeasurable' m g \u03bc \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) \u22a2 P f  case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc \u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     AEStronglyMeasurable' m f \u03bc \u2192       AEStronglyMeasurable' m g \u03bc \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** exact Lp.induction_stronglyMeasurable_aux hm hp_ne_top h_ind h_add_ae h_closed f hf ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f : { x // x \u2208 Lp F p } hf : AEStronglyMeasurable' m (\u2191\u2191f) \u03bc \u22a2 \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     AEStronglyMeasurable' m f \u03bc \u2192       AEStronglyMeasurable' m g \u03bc \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** intro f g hf hg hfm hgm h_disj hPf hPg ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** let s_f : Set \u03b1 := Function.support (hfm.mk f) ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hs_f : MeasurableSet[m] s_f := hfm.stronglyMeasurable_mk.measurableSet_support ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hs_f_eq : s_f =\u1d50[\u03bc] Function.support f := hfm.ae_eq_mk.symm.support ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** let s_g : Set \u03b1 := Function.support (hgm.mk g) ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hs_g : MeasurableSet[m] s_g := hgm.stronglyMeasurable_mk.measurableSet_support ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hs_g_eq : s_g =\u1d50[\u03bc] Function.support g := hgm.ae_eq_mk.symm.support ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have h_inter_empty : (s_f \u2229 s_g : Set \u03b1) =\u1d50[\u03bc] (\u2205 : Set \u03b1) := by\n  refine' (hs_f_eq.inter hs_g_eq).trans _\n  suffices Function.support f \u2229 Function.support g = \u2205 by rw [this]\n  exact Set.disjoint_iff_inter_eq_empty.mp h_disj ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** let f' := (s_f \\ s_g).indicator (hfm.mk f) ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hff' : f =\u1d50[\u03bc] f' := by\n  have : s_f \\ s_g =\u1d50[\u03bc] s_f := by\n    rw [\u2190 Set.diff_inter_self_eq_diff, Set.inter_comm]\n    refine' ((ae_eq_refl s_f).diff h_inter_empty).trans _\n    rw [Set.diff_empty]\n  refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm\n  rw [Set.indicator_support]\n  exact hfm.ae_eq_mk.symm ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hf'_meas : StronglyMeasurable[m] f' := hfm.stronglyMeasurable_mk.indicator (hs_f.diff hs_g) ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hf'_Lp : Mem\u2112p f' p \u03bc := hf.ae_eq hff' ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** let g' := (s_g \\ s_f).indicator (hgm.mk g) ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hgg' : g =\u1d50[\u03bc] g' := by\n  have : s_g \\ s_f =\u1d50[\u03bc] s_g := by\n    rw [\u2190 Set.diff_inter_self_eq_diff]\n    refine' ((ae_eq_refl s_g).diff h_inter_empty).trans _\n    rw [Set.diff_empty]\n  refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm\n  rw [Set.indicator_support]\n  exact hgm.ae_eq_mk.symm ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) hgg' : g =\u1d50[\u03bc] g' \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hg'_meas : StronglyMeasurable[m] g' := hgm.stronglyMeasurable_mk.indicator (hs_g.diff hs_f) ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) hgg' : g =\u1d50[\u03bc] g' hg'_meas : StronglyMeasurable g' \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have hg'_Lp : Mem\u2112p g' p \u03bc := hg.ae_eq hgg' ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) hgg' : g =\u1d50[\u03bc] g' hg'_meas : StronglyMeasurable g' hg'_Lp : Mem\u2112p g' p \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** have h_disj : Disjoint (Function.support f') (Function.support g') :=\n  haveI : Disjoint (s_f \\ s_g) (s_g \\ s_f) := disjoint_sdiff_sdiff\n  this.mono Set.support_indicator_subset Set.support_indicator_subset ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj\u271d : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) hgg' : g =\u1d50[\u03bc] g' hg'_meas : StronglyMeasurable g' hg'_Lp : Mem\u2112p g' p h_disj : Disjoint (Function.support f') (Function.support g') \u22a2 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** rw [\u2190 Mem\u2112p.toLp_congr hf'_Lp hf hff'.symm] at hPf \u22a2 ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj\u271d : Disjoint (Function.support f) (Function.support g) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p hPf : P (Mem\u2112p.toLp f' hf'_Lp) g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) hgg' : g =\u1d50[\u03bc] g' hg'_meas : StronglyMeasurable g' hg'_Lp : Mem\u2112p g' p h_disj : Disjoint (Function.support f') (Function.support g') \u22a2 P (Mem\u2112p.toLp f' hf'_Lp + Mem\u2112p.toLp g hg) ** rw [\u2190 Mem\u2112p.toLp_congr hg'_Lp hg hgg'.symm] at hPg \u22a2 ** case h_add_ae \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj\u271d : Disjoint (Function.support f) (Function.support g) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p hPf : P (Mem\u2112p.toLp f' hf'_Lp) g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) hgg' : g =\u1d50[\u03bc] g' hg'_meas : StronglyMeasurable g' hg'_Lp : Mem\u2112p g' p hPg : P (Mem\u2112p.toLp g' hg'_Lp) h_disj : Disjoint (Function.support f') (Function.support g') \u22a2 P (Mem\u2112p.toLp f' hf'_Lp + Mem\u2112p.toLp g' hg'_Lp) ** exact h_add hf'_Lp hg'_Lp hf'_meas hg'_meas h_disj hPf hPg ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g \u22a2 s_f \u2229 s_g =\u1d50[\u03bc] \u2205 ** refine' (hs_f_eq.inter hs_g_eq).trans _ ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g \u22a2 Function.support f \u2229 Function.support g =\u1d50[\u03bc] \u2205 ** suffices Function.support f \u2229 Function.support g = \u2205 by rw [this] ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g \u22a2 Function.support f \u2229 Function.support g = \u2205 ** exact Set.disjoint_iff_inter_eq_empty.mp h_disj ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g this : Function.support f \u2229 Function.support g = \u2205 \u22a2 Function.support f \u2229 Function.support g =\u1d50[\u03bc] \u2205 ** rw [this] ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) \u22a2 f =\u1d50[\u03bc] f' ** have : s_f \\ s_g =\u1d50[\u03bc] s_f := by\n  rw [\u2190 Set.diff_inter_self_eq_diff, Set.inter_comm]\n  refine' ((ae_eq_refl s_f).diff h_inter_empty).trans _\n  rw [Set.diff_empty] ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) this : s_f \\ s_g =\u1d50[\u03bc] s_f \u22a2 f =\u1d50[\u03bc] f' ** refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) this : s_f \\ s_g =\u1d50[\u03bc] s_f \u22a2 Set.indicator s_f (AEStronglyMeasurable'.mk f hfm) =\u1d50[\u03bc] f ** rw [Set.indicator_support] ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) this : s_f \\ s_g =\u1d50[\u03bc] s_f \u22a2 AEStronglyMeasurable'.mk f hfm =\u1d50[\u03bc] f ** exact hfm.ae_eq_mk.symm ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) \u22a2 s_f \\ s_g =\u1d50[\u03bc] s_f ** rw [\u2190 Set.diff_inter_self_eq_diff, Set.inter_comm] ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) \u22a2 s_f \\ (s_f \u2229 s_g) =\u1d50[\u03bc] s_f ** refine' ((ae_eq_refl s_f).diff h_inter_empty).trans _ ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) \u22a2 s_f \\ \u2205 =\u1d50[\u03bc] s_f ** rw [Set.diff_empty] ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) \u22a2 g =\u1d50[\u03bc] g' ** have : s_g \\ s_f =\u1d50[\u03bc] s_g := by\n  rw [\u2190 Set.diff_inter_self_eq_diff]\n  refine' ((ae_eq_refl s_g).diff h_inter_empty).trans _\n  rw [Set.diff_empty] ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) this : s_g \\ s_f =\u1d50[\u03bc] s_g \u22a2 g =\u1d50[\u03bc] g' ** refine' ((indicator_ae_eq_of_ae_eq_set this).trans _).symm ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) this : s_g \\ s_f =\u1d50[\u03bc] s_g \u22a2 Set.indicator s_g (AEStronglyMeasurable'.mk g hgm) =\u1d50[\u03bc] g ** rw [Set.indicator_support] ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) this : s_g \\ s_f =\u1d50[\u03bc] s_g \u22a2 AEStronglyMeasurable'.mk g hgm =\u1d50[\u03bc] g ** exact hgm.ae_eq_mk.symm ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) \u22a2 s_g \\ s_f =\u1d50[\u03bc] s_g ** rw [\u2190 Set.diff_inter_self_eq_diff] ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) \u22a2 s_g \\ (s_f \u2229 s_g) =\u1d50[\u03bc] s_g ** refine' ((ae_eq_refl s_g).diff h_inter_empty).trans _ ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2079 : CompleteSpace E' inst\u271d\u2078 : NormedSpace \u211d E' inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \ud835\udd5c F inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \ud835\udd5c F' inst\u271d\u00b3 : NormedSpace \u211d F' inst\u271d\u00b2 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : Fact (1 \u2264 p) inst\u271d : NormedSpace \u211d F hm : m \u2264 m0 hp_ne_top : p \u2260 \u22a4 P : { x // x \u2208 Lp F p } \u2192 Prop h_ind :   \u2200 (c : F) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     P \u2191(simpleFunc.indicatorConst p (_ : MeasurableSet s) (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) h_add :   \u2200 \u2983f g : \u03b1 \u2192 F\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),     StronglyMeasurable f \u2192       StronglyMeasurable g \u2192         Disjoint (Function.support f) (Function.support g) \u2192           P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) h_closed : IsClosed {f | P \u2191f} f\u271d : { x // x \u2208 Lp F p } hf\u271d : AEStronglyMeasurable' m (\u2191\u2191f\u271d) \u03bc f g : \u03b1 \u2192 F hf : Mem\u2112p f p hg : Mem\u2112p g p hfm : AEStronglyMeasurable' m f \u03bc hgm : AEStronglyMeasurable' m g \u03bc h_disj : Disjoint (Function.support f) (Function.support g) hPf : P (Mem\u2112p.toLp f hf) hPg : P (Mem\u2112p.toLp g hg) s_f : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk f hfm) hs_f : MeasurableSet s_f hs_f_eq : s_f =\u1d50[\u03bc] Function.support f s_g : Set \u03b1 := Function.support (AEStronglyMeasurable'.mk g hgm) hs_g : MeasurableSet s_g hs_g_eq : s_g =\u1d50[\u03bc] Function.support g h_inter_empty : s_f \u2229 s_g =\u1d50[\u03bc] \u2205 f' : \u03b1 \u2192 F := Set.indicator (s_f \\ s_g) (AEStronglyMeasurable'.mk f hfm) hff' : f =\u1d50[\u03bc] f' hf'_meas : StronglyMeasurable f' hf'_Lp : Mem\u2112p f' p g' : \u03b1 \u2192 F := Set.indicator (s_g \\ s_f) (AEStronglyMeasurable'.mk g hgm) \u22a2 s_g \\ \u2205 =\u1d50[\u03bc] s_g ** rw [Set.diff_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.back_eq ** s : String \u22a2 back s = List.getLastD s.data default ** match s, s.1.eq_nil_or_concat with\n| \u27e8_\u27e9, .inl rfl => rfl\n| \u27e8_\u27e9, .inr \u27e8cs, c, rfl\u27e9 => simp [back, prev_of_valid, get_of_valid] ** s : String \u22a2 back { data := [] } = List.getLastD { data := [] }.data default ** rfl ** s : String cs : List Char c : Char \u22a2 back { data := cs ++ [c] } = List.getLastD { data := cs ++ [c] }.data default ** simp [back, prev_of_valid, get_of_valid] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.unifIntegrable_of' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 UnifIntegrable f p \u03bc ** have hpzero := (lt_of_lt_of_le zero_lt_one hp).ne.symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 \u22a2 UnifIntegrable f p \u03bc ** by_cases h\u03bc : \u03bc Set.univ = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u22a2 UnifIntegrable f p \u03bc ** intro \u03b5 h\u03b5 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 \u03b4 x,     \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8C, hCpos, hC\u27e9 := h (\u03b5 / 2) (half_pos h\u03b5) ** case neg.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) \u22a2 \u2203 \u03b4 x,     \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e8(\u03b5 / (2 * C)) ^ ENNReal.toReal p,\n  Real.rpow_pos_of_pos (div_pos h\u03b5 (mul_pos two_pos (NNReal.coe_pos.2 hCpos))) _,\n  fun i s hs h\u03bcs => _\u27e9 ** case neg.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) \u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** by_cases h\u03bcs' : \u03bc s = 0 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u2191\u2191\u03bc univ = 0 \u22a2 UnifIntegrable f p \u03bc ** rw [Measure.measure_univ_eq_zero] at h\u03bc ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u03bc = 0 \u22a2 UnifIntegrable f p \u03bc ** exact h\u03bc.symm \u25b8 unifIntegrable_zero_meas ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u2191\u2191\u03bc s = 0 \u22a2 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** rw [(snorm_eq_zero_iff ((hf i).indicator hs).aestronglyMeasurable hpzero).2\n    (indicator_meas_zero h\u03bcs')] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u2191\u2191\u03bc s = 0 \u22a2 0 \u2264 ENNReal.ofReal \u03b5 ** norm_num ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 snorm (indicator s (f i)) p \u03bc \u2264     snorm (indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i)) p \u03bc + snorm (indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc ** refine' le_trans (Eq.le _) (snorm_add_le\n  (StronglyMeasurable.aestronglyMeasurable\n    ((hf i).indicator (hs.inter (stronglyMeasurable_const.measurableSet_le (hf i).nnnorm))))\n  (StronglyMeasurable.aestronglyMeasurable\n    ((hf i).indicator (hs.inter ((hf i).nnnorm.measurableSet_lt stronglyMeasurable_const))))\n  hp) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 snorm (indicator s (f i)) p \u03bc =     snorm (indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i) + indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc ** congr ** case e_f \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 indicator s (f i) = indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i) + indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i) ** change _ = fun x => (s \u2229 { x : \u03b1 | C \u2264 \u2016f i x\u2016\u208a }).indicator (f i) x +\n  (s \u2229 { x : \u03b1 | \u2016f i x\u2016\u208a < C }).indicator (f i) x ** case e_f \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 indicator s (f i) = fun x => indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i) x + indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i) x ** rw [\u2190 Set.indicator_union_of_disjoint] ** case e_f \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 indicator s (f i) = indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a} \u222a s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i) ** rw [\u2190 Set.inter_union_distrib_left, (by ext; simp [le_or_lt] :\n    { x : \u03b1 | C \u2264 \u2016f i x\u2016\u208a } \u222a { x : \u03b1 | \u2016f i x\u2016\u208a < C } = Set.univ),\n  Set.inter_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 {x | C \u2264 \u2016f i x\u2016\u208a} \u222a {x | \u2016f i x\u2016\u208a < C} = univ ** ext ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 x\u271d : \u03b1 \u22a2 x\u271d \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} \u222a {x | \u2016f i x\u2016\u208a < C} \u2194 x\u271d \u2208 univ ** simp [le_or_lt] ** case e_f.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 Disjoint (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (s \u2229 {x | \u2016f i x\u2016\u208a < C}) ** refine' (Disjoint.inf_right' _ _).inf_left' _ ** case e_f.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 Disjoint {x | C \u2264 \u2016f i x\u2016\u208a} {x | \u2016f i x\u2016\u208a < C} ** rw [disjoint_iff_inf_le] ** case e_f.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 {x | C \u2264 \u2016f i x\u2016\u208a} \u2293 {x | \u2016f i x\u2016\u208a < C} \u2264 \u22a5 ** rintro x \u27e8hx\u2081, hx\u2082\u27e9 ** case e_f.h.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 x : \u03b1 hx\u2081 : x \u2208 {x | C \u2264 \u2016f i x\u2016\u208a} hx\u2082 : x \u2208 {x | \u2016f i x\u2016\u208a < C} \u22a2 x \u2208 \u22a5 ** rw [Set.mem_setOf_eq] at hx\u2081 hx\u2082 ** case e_f.h.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 x : \u03b1 hx\u2081 : C \u2264 \u2016f i x\u2016\u208a hx\u2082 : \u2016f i x\u2016\u208a < C \u22a2 x \u2208 \u22a5 ** exact False.elim (hx\u2082.ne (eq_of_le_of_not_lt hx\u2081 (not_lt.2 hx\u2082.le)).symm) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 snorm (indicator (s \u2229 {x | C \u2264 \u2016f i x\u2016\u208a}) (f i)) p \u03bc + snorm (indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc \u2264     snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc + \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** refine' add_le_add\n  (snorm_mono fun x => norm_indicator_le_of_subset (Set.inter_subset_right _ _) _ _) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 snorm (indicator (s \u2229 {x | \u2016f i x\u2016\u208a < C}) (f i)) p \u03bc \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** rw [\u2190 Set.indicator_indicator] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 snorm (indicator s (indicator {x | \u2016f i x\u2016\u208a < C} (f i))) p \u03bc \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** rw [snorm_indicator_eq_snorm_restrict hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p (Measure.restrict \u03bc s) \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** have : \u2200\u1d50 x \u2202\u03bc.restrict s, \u2016{ x : \u03b1 | \u2016f i x\u2016\u208a < C }.indicator (f i) x\u2016 \u2264 C := by\n  refine' ae_of_all _ _\n  simp_rw [norm_indicator_eq_indicator_norm]\n  exact Set.indicator_le' (fun x (hx : _ < _) => hx.le) fun _ _ => NNReal.coe_nonneg _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 this : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016 \u2264 \u2191C \u22a2 snorm (indicator {x | \u2016f i x\u2016\u208a < C} (f i)) p (Measure.restrict \u03bc s) \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** refine' le_trans (snorm_le_of_ae_bound this) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 this : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016 \u2264 \u2191C \u22a2 \u2191\u2191(Measure.restrict \u03bc s) univ ^ (ENNReal.toReal p)\u207b\u00b9 * ENNReal.ofReal \u2191C \u2264 \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) ** rw [mul_comm, Measure.restrict_apply' hs, Set.univ_inter, ENNReal.ofReal_coe_nnreal, one_div] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) x\u2016 \u2264 \u2191C ** refine' ae_of_all _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 \u2200 (a : \u03b1), \u2016indicator {x | \u2016f i x\u2016\u208a < C} (f i) a\u2016 \u2264 \u2191C ** simp_rw [norm_indicator_eq_indicator_norm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 \u2200 (a : \u03b1), indicator {x | \u2016f i x\u2016\u208a < C} (fun a => \u2016f i a\u2016) a \u2264 \u2191C ** exact Set.indicator_le' (fun x (hx : _ < _) => hx.le) fun _ _ => NNReal.coe_nonneg _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc + \u2191C * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) \u2264     ENNReal.ofReal (\u03b5 / 2) + \u2191C * ENNReal.ofReal (\u03b5 / (2 * \u2191C)) ** refine' add_le_add (hC i) (mul_le_mul_left' _ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p) \u2264 ENNReal.ofReal (\u03b5 / (2 * \u2191C)) ** rwa [ENNReal.rpow_one_div_le_iff (ENNReal.toReal_pos hpzero hp'),\n  ENNReal.ofReal_rpow_of_pos (div_pos h\u03b5 (mul_pos two_pos (NNReal.coe_pos.2 hCpos)))] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 ENNReal.ofReal (\u03b5 / 2) + \u2191C * ENNReal.ofReal (\u03b5 / (2 * \u2191C)) \u2264 ENNReal.ofReal (\u03b5 / 2) + ENNReal.ofReal (\u03b5 / 2) ** refine' add_le_add_left _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 \u2191C * ENNReal.ofReal (\u03b5 / (2 * \u2191C)) \u2264 ENNReal.ofReal (\u03b5 / 2) ** rw [\u2190 ENNReal.ofReal_coe_nnreal, \u2190 ENNReal.ofReal_mul (NNReal.coe_nonneg _), \u2190 div_div,\n  mul_div_cancel' _ (NNReal.coe_pos.2 hCpos).ne.symm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 hf : \u2200 (i : \u03b9), StronglyMeasurable (f i) h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 C, 0 < C \u2227 \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5 hpzero : p \u2260 0 h\u03bc : \u00ac\u2191\u2191\u03bc univ = 0 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 C : \u211d\u22650 hCpos : 0 < C hC : \u2200 (i : \u03b9), snorm (indicator {x | C \u2264 \u2016f i x\u2016\u208a} (f i)) p \u03bc \u2264 ENNReal.ofReal (\u03b5 / 2) i : \u03b9 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal ((\u03b5 / (2 * \u2191C)) ^ ENNReal.toReal p) h\u03bcs' : \u00ac\u2191\u2191\u03bc s = 0 \u22a2 ENNReal.ofReal (\u03b5 / 2) + ENNReal.ofReal (\u03b5 / 2) \u2264 ENNReal.ofReal \u03b5 ** rw [\u2190 ENNReal.ofReal_add (half_pos h\u03b5).le (half_pos h\u03b5).le, add_halves] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Egorov.exists_notConvergentSeq_lt ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : Countable \u03b9 h\u03b5 : 0 < \u03b5 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : \u2191\u2191\u03bc s \u2260 \u22a4 hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) n : \u2115 \u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n) ** have \u27e8N, hN\u27e9 := (ENNReal.tendsto_atTop ENNReal.zero_ne_top).1\n  (measure_notConvergentSeq_tendsto_zero hf hg hsm hs hfg n) (ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (by\n    rw [gt_iff_lt, ENNReal.ofReal_pos]\n    exact mul_pos h\u03b5 (pow_pos (by norm_num) n)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : Countable \u03b9 h\u03b5 : 0 < \u03b5 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : \u2191\u2191\u03bc s \u2260 \u22a4 hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) n : \u2115 N : \u03b9 hN :   \u2200 (n_1 : \u03b9),     n_1 \u2265 N \u2192       \u2191\u2191\u03bc (s \u2229 notConvergentSeq (fun n => f n) g n n_1) \u2208         Icc (0 - ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (0 + ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) \u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n) ** rw [zero_add] at hN ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : Countable \u03b9 h\u03b5 : 0 < \u03b5 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : \u2191\u2191\u03bc s \u2260 \u22a4 hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) n : \u2115 N : \u03b9 hN :   \u2200 (n_1 : \u03b9),     n_1 \u2265 N \u2192       \u2191\u2191\u03bc (s \u2229 notConvergentSeq (fun n => f n) g n n_1) \u2208         Icc (0 - ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) (ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n)) \u22a2 \u2203 j, \u2191\u2191\u03bc (s \u2229 notConvergentSeq f g n j) \u2264 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n) ** exact \u27e8N, (hN N le_rfl).2\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : Countable \u03b9 h\u03b5 : 0 < \u03b5 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : \u2191\u2191\u03bc s \u2260 \u22a4 hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) n : \u2115 \u22a2 ENNReal.ofReal (\u03b5 * 2\u207b\u00b9 ^ n) > 0 ** rw [gt_iff_lt, ENNReal.ofReal_pos] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : Countable \u03b9 h\u03b5 : 0 < \u03b5 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : \u2191\u2191\u03bc s \u2260 \u22a4 hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) n : \u2115 \u22a2 0 < \u03b5 * 2\u207b\u00b9 ^ n ** exact mul_pos h\u03b5 (pow_pos (by norm_num) n) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u00b3 : MetricSpace \u03b2 \u03bc : Measure \u03b1 n\u271d : \u2115 i j : \u03b9 s : Set \u03b1 \u03b5 : \u211d f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : SemilatticeSup \u03b9 inst\u271d\u00b9 : Nonempty \u03b9 inst\u271d : Countable \u03b9 h\u03b5 : 0 < \u03b5 hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : \u2191\u2191\u03bc s \u2260 \u22a4 hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) n : \u2115 \u22a2 0 < 2\u207b\u00b9 ** norm_num ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.finStronglyMeasurable_of_stronglyMeasurable ** \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 \u22a2 FinStronglyMeasurable f \u03bc ** borelize G ** \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G \u22a2 FinStronglyMeasurable f \u03bc ** haveI : SeparableSpace (Set.range f \u222a {0} : Set G) :=\n  hf_meas.separableSpace_range_union_singleton ** \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(Set.range f \u222a {0}) \u22a2 FinStronglyMeasurable f \u03bc ** let fs := SimpleFunc.approxOn f hf_meas.measurable (Set.range f \u222a {0}) 0 (by simp) ** \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(Set.range f \u222a {0}) fs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) \u22a2 FinStronglyMeasurable f \u03bc ** refine' \u27e8fs, _, _\u27e9 ** \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(Set.range f \u222a {0}) \u22a2 0 \u2208 Set.range f \u222a {0} ** simp ** case refine'_1 \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(Set.range f \u222a {0}) fs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) \u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 ** have h_fs_Lp : \u2200 n, Mem\u2112p (fs n) p \u03bc :=\n  SimpleFunc.mem\u2112p_approxOn_range hf_meas.measurable hf ** case refine'_1 \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(Set.range f \u222a {0}) fs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) h_fs_Lp : \u2200 (n : \u2115), Mem\u2112p (\u2191(fs n)) p \u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (support \u2191(fs n)) < \u22a4 ** exact fun n => (fs n).measure_support_lt_top_of_mem\u2112p (h_fs_Lp n) hp_ne_zero hp_ne_top ** case refine'_2 \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(Set.range f \u222a {0}) fs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) \u22a2 \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) ** intro x ** case refine'_2 \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(Set.range f \u222a {0}) fs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) x : \u03b1 \u22a2 Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x)) ** apply SimpleFunc.tendsto_approxOn ** case refine'_2.hx \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(Set.range f \u222a {0}) fs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) x : \u03b1 \u22a2 f x \u2208 closure (Set.range f \u222a {0}) ** apply subset_closure ** case refine'_2.hx.a \u03b1 : Type u_1 G : Type u_2 p : \u211d\u22650\u221e m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 G hf : Mem\u2112p f p hf_meas : StronglyMeasurable f hp_ne_zero : p \u2260 0 hp_ne_top : p \u2260 \u22a4 this\u271d\u00b9 : MeasurableSpace G := borel G this\u271d : BorelSpace G this : SeparableSpace \u2191(Set.range f \u222a {0}) fs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f (_ : Measurable f) (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) x : \u03b1 \u22a2 f x \u2208 Set.range f \u222a {0} ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_smul_eq_self ** \ud835\udd5c : Type u_1 M : Type u_2 \u03b1 : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst\u271d\u2079 : MeasurableSpace G inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E inst\u271d\u2075 : NormedAddCommGroup F \u03bc\u271d : Measure G f\u271d : G \u2192 E g\u271d : G inst\u271d\u2074 : Group G inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSMul G \u03b1 \u03bc : Measure \u03b1 inst\u271d : SMulInvariantMeasure G \u03b1 \u03bc f : \u03b1 \u2192 E g : G \u22a2 \u222b (x : \u03b1), f (g \u2022 x) \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc ** have h : MeasurableEmbedding fun x : \u03b1 => g \u2022 x := (MeasurableEquiv.smul g).measurableEmbedding ** \ud835\udd5c : Type u_1 M : Type u_2 \u03b1 : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst\u271d\u2079 : MeasurableSpace G inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : CompleteSpace E inst\u271d\u2075 : NormedAddCommGroup F \u03bc\u271d : Measure G f\u271d : G \u2192 E g\u271d : G inst\u271d\u2074 : Group G inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MulAction G \u03b1 inst\u271d\u00b9 : MeasurableSMul G \u03b1 \u03bc : Measure \u03b1 inst\u271d : SMulInvariantMeasure G \u03b1 \u03bc f : \u03b1 \u2192 E g : G h : MeasurableEmbedding fun x => g \u2022 x \u22a2 \u222b (x : \u03b1), f (g \u2022 x) \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc ** rw [\u2190 h.integral_map, map_smul] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.val_add_of_le ** n : \u2115 inst\u271d : NeZero n a b : ZMod n h : n \u2264 val a + val b \u22a2 val (a + b) = val a + val b - n ** rw [val_add_val_of_le h] ** n : \u2115 inst\u271d : NeZero n a b : ZMod n h : n \u2264 val a + val b \u22a2 val (a + b) = val (a + b) + n - n ** exact eq_tsub_of_add_eq rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "List.modifyNth_eq_set_get ** \u03b1 : Type u_1 f : \u03b1 \u2192 \u03b1 n : Nat l : List \u03b1 h : n < length l \u22a2 modifyNth f n l = set l n (f (get l { val := n, isLt := h })) ** rw [modifyNth_eq_set_get?, get?_eq_get h] ** \u03b1 : Type u_1 f : \u03b1 \u2192 \u03b1 n : Nat l : List \u03b1 h : n < length l \u22a2 Option.getD ((fun a => set l n (f a)) <$> some (get l { val := n, isLt := h })) l =     set l n (f (get l { val := n, isLt := h })) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.martingalePart_eq_sum ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 \u22a2 martingalePart f \u2131 \u03bc = fun n => f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) ** unfold martingalePart predictablePart ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n : \u2115 \u22a2 (fun n => f n - \u2211 i in Finset.range n, \u03bc[f (i + 1) - f i|\u2191\u2131 i]) = fun n =>     f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) ** ext1 n ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d n : \u2115 \u22a2 f n - \u2211 i in Finset.range n, \u03bc[f (i + 1) - f i|\u2191\u2131 i] =     f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) ** rw [Finset.eq_sum_range_sub f n, \u2190 add_sub, \u2190 Finset.sum_sub_distrib] ** Qed",
        "informal": ""
    },
    {
        "formal": "Orientation.measure_orthonormalBasis ** \u03b9 : Type u_1 F : Type u_2 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : InnerProductSpace \u211d F inst\u271d\u00b2 : FiniteDimensional \u211d F inst\u271d\u00b9 : MeasurableSpace F inst\u271d : BorelSpace F m n : \u2115 _i : Fact (finrank \u211d F = n) o : Orientation \u211d F (Fin n) b : OrthonormalBasis \u03b9 \u211d F \u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1 ** have e : \u03b9 \u2243 Fin n := by\n  refine' Fintype.equivFinOfCardEq _\n  rw [\u2190 _i.out, finrank_eq_card_basis b.toBasis] ** \u03b9 : Type u_1 F : Type u_2 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : InnerProductSpace \u211d F inst\u271d\u00b2 : FiniteDimensional \u211d F inst\u271d\u00b9 : MeasurableSpace F inst\u271d : BorelSpace F m n : \u2115 _i : Fact (finrank \u211d F = n) o : Orientation \u211d F (Fin n) b : OrthonormalBasis \u03b9 \u211d F e : \u03b9 \u2243 Fin n \u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1 ** have A : \u21d1b = b.reindex e \u2218 e := by\n  ext x\n  simp only [OrthonormalBasis.coe_reindex, Function.comp_apply, Equiv.symm_apply_apply] ** \u03b9 : Type u_1 F : Type u_2 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : InnerProductSpace \u211d F inst\u271d\u00b2 : FiniteDimensional \u211d F inst\u271d\u00b9 : MeasurableSpace F inst\u271d : BorelSpace F m n : \u2115 _i : Fact (finrank \u211d F = n) o : Orientation \u211d F (Fin n) b : OrthonormalBasis \u03b9 \u211d F e : \u03b9 \u2243 Fin n A : \u2191b = \u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e \u22a2 \u2191\u2191(AlternatingMap.measure (volumeForm o)) (parallelepiped \u2191b) = 1 ** rw [A, parallelepiped_comp_equiv, AlternatingMap.measure_parallelepiped,\n  o.abs_volumeForm_apply_of_orthonormal, ENNReal.ofReal_one] ** \u03b9 : Type u_1 F : Type u_2 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : InnerProductSpace \u211d F inst\u271d\u00b2 : FiniteDimensional \u211d F inst\u271d\u00b9 : MeasurableSpace F inst\u271d : BorelSpace F m n : \u2115 _i : Fact (finrank \u211d F = n) o : Orientation \u211d F (Fin n) b : OrthonormalBasis \u03b9 \u211d F \u22a2 \u03b9 \u2243 Fin n ** refine' Fintype.equivFinOfCardEq _ ** \u03b9 : Type u_1 F : Type u_2 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : InnerProductSpace \u211d F inst\u271d\u00b2 : FiniteDimensional \u211d F inst\u271d\u00b9 : MeasurableSpace F inst\u271d : BorelSpace F m n : \u2115 _i : Fact (finrank \u211d F = n) o : Orientation \u211d F (Fin n) b : OrthonormalBasis \u03b9 \u211d F \u22a2 Fintype.card \u03b9 = n ** rw [\u2190 _i.out, finrank_eq_card_basis b.toBasis] ** \u03b9 : Type u_1 F : Type u_2 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : InnerProductSpace \u211d F inst\u271d\u00b2 : FiniteDimensional \u211d F inst\u271d\u00b9 : MeasurableSpace F inst\u271d : BorelSpace F m n : \u2115 _i : Fact (finrank \u211d F = n) o : Orientation \u211d F (Fin n) b : OrthonormalBasis \u03b9 \u211d F e : \u03b9 \u2243 Fin n \u22a2 \u2191b = \u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e ** ext x ** case h \u03b9 : Type u_1 F : Type u_2 inst\u271d\u2075 : Fintype \u03b9 inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : InnerProductSpace \u211d F inst\u271d\u00b2 : FiniteDimensional \u211d F inst\u271d\u00b9 : MeasurableSpace F inst\u271d : BorelSpace F m n : \u2115 _i : Fact (finrank \u211d F = n) o : Orientation \u211d F (Fin n) b : OrthonormalBasis \u03b9 \u211d F e : \u03b9 \u2243 Fin n x : \u03b9 \u22a2 \u2191b x = (\u2191(OrthonormalBasis.reindex b e) \u2218 \u2191e) x ** simp only [OrthonormalBasis.coe_reindex, Function.comp_apply, Equiv.symm_apply_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_piecewise ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : DecidablePred fun x => x \u2208 s hs : MeasurableSet s hf : IntegrableOn f s hg : IntegrableOn g s\u1d9c \u22a2 \u222b (x : \u03b1), piecewise s f g x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc + \u222b (x : \u03b1) in s\u1d9c, g x \u2202\u03bc ** rw [\u2190 Set.indicator_add_compl_eq_piecewise,\n  integral_add' (hf.integrable_indicator hs) (hg.integrable_indicator hs.compl),\n  integral_indicator hs, integral_indicator hs.compl] ** Qed",
        "informal": ""
    },
    {
        "formal": "aemeasurable_restrict_iff_comap_subtype ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f\u271d g : \u03b1 \u2192 \u03b2 \u03bc\u271d \u03bd : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u03b2 \u22a2 AEMeasurable f \u2194 AEMeasurable (f \u2218 Subtype.val) ** rw [\u2190 map_comap_subtype_coe hs, (MeasurableEmbedding.subtype_coe hs).aemeasurable_map_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.compProd_apply_eq_compProdFun ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = compProdFun \u03ba \u03b7 a s ** rw [compProd, dif_pos] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191{                 val := fun a =>                   Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)                     (_ :                       \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),                         (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192                           Pairwise (Disjoint on f) \u2192                             compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),                 property :=                   (_ :                     Measurable fun a =>                       Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)                         (_ :                           \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),                             (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192                               Pairwise (Disjoint on f) \u2192                                 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }             a)       s =     compProdFun \u03ba \u03b7 a s  case hc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7 ** swap ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191{                 val := fun a =>                   Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)                     (_ :                       \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),                         (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192                           Pairwise (Disjoint on f) \u2192                             compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),                 property :=                   (_ :                     Measurable fun a =>                       Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)                         (_ :                           \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),                             (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192                               Pairwise (Disjoint on f) \u2192                                 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }             a)       s =     compProdFun \u03ba \u03b7 a s ** change\n  Measure.ofMeasurable (fun s _ => compProdFun \u03ba \u03b7 a s) (compProdFun_empty \u03ba \u03b7 a)\n      (compProdFun_iUnion \u03ba \u03b7 a) s =\n    \u222b\u207b b, \u03b7 (a, b) {c | (b, c) \u2208 s} \u2202\u03ba a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)             (_ :               \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),                 (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192                   Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)))       s =     \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a ** rw [Measure.ofMeasurable_apply _ hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 compProdFun \u03ba \u03b7 a s = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a ** rfl ** case hc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a : \u03b1 hs : MeasurableSet s \u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7 ** constructor <;> infer_instance ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.VectorMeasure.MutuallySingular.smul_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 L : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u2078 : AddCommMonoid L inst\u271d\u2077 : TopologicalSpace L inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : AddCommMonoid N inst\u271d\u00b3 : TopologicalSpace N v v\u2081 v\u2082 : VectorMeasure \u03b1 M w w\u2081 w\u2082 : VectorMeasure \u03b1 N R : Type u_6 inst\u271d\u00b2 : Semiring R inst\u271d\u00b9 : DistribMulAction R N inst\u271d : ContinuousConstSMul R N r : R h : v \u27c2\u1d65 w s : Set \u03b1 hmeas : MeasurableSet s hs\u2081 : \u2200 (t : Set \u03b1), t \u2286 s \u2192 \u2191v t = 0 hs\u2082 : \u2200 (t : Set \u03b1), t \u2286 s\u1d9c \u2192 \u2191w t = 0 t : Set \u03b1 ht : t \u2286 s\u1d9c \u22a2 \u2191(r \u2022 w) t = 0 ** simp only [coe_smul, Pi.smul_apply, hs\u2082 t ht, smul_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_one_le_of_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** by_cases hr : r = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** by_cases h\u03bc : IsFiniteMeasure \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : IsFiniteMeasure \u03bc \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r  case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : \u00acIsFiniteMeasure \u03bc \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** swap ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : IsFiniteMeasure \u03bc \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** haveI := h\u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** rw [integral_eq_integral_pos_part_sub_integral_neg_part hfint, sub_nonneg] at hfint' ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** have hposbdd : \u222b \u03c9, max (f \u03c9) 0 \u2202\u03bc \u2264 (\u03bc Set.univ).toReal \u2022 (r : \u211d) := by\n  rw [\u2190 integral_const]\n  refine' integral_mono_ae hfint.real_toNNReal (integrable_const (r : \u211d)) _\n  filter_upwards [hf] with \u03c9 h\u03c9 using Real.toNNReal_le_iff_le_coe.2 h\u03c9 ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc hposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** rw [Mem\u2112p.snorm_eq_integral_rpow_norm one_ne_zero ENNReal.one_ne_top\n    (mem\u2112p_one_iff_integrable.2 hfint),\n  ENNReal.ofReal_le_iff_le_toReal\n    (ENNReal.mul_ne_top (ENNReal.mul_ne_top ENNReal.two_ne_top <| @measure_ne_top _ _ _ h\u03bc _)\n      ENNReal.coe_ne_top)] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc hposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r \u22a2 (\u222b (a : \u03b1), \u2016f a\u2016 ^ ENNReal.toReal 1 \u2202\u03bc) ^ (ENNReal.toReal 1)\u207b\u00b9 \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r) ** simp_rw [ENNReal.one_toReal, _root_.inv_one, Real.rpow_one, Real.norm_eq_abs, \u2190\n  max_zero_add_max_neg_zero_eq_abs_self, \u2190 Real.coe_toNNReal'] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc hposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r \u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) + \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r) ** rw [integral_add hfint.real_toNNReal] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : r = 0 \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** suffices f =\u1d50[\u03bc] 0 by\n  rw [snorm_congr_ae this, snorm_zero, hr, ENNReal.coe_zero, mul_zero] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : r = 0 \u22a2 f =\u1d50[\u03bc] 0 ** rw [hr] at hf ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u21910 hr : r = 0 \u22a2 f =\u1d50[\u03bc] 0 ** norm_cast at hf ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hr : r = 0 hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0 \u22a2 f =\u1d50[\u03bc] 0 ** have hnegf : \u222b x, -f x \u2202\u03bc = 0 := by\n  rw [integral_neg, neg_eq_zero]\n  exact le_antisymm (integral_nonpos_of_ae hf) hfint' ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hr : r = 0 hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0 hnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0 \u22a2 f =\u1d50[\u03bc] 0 ** have := (integral_eq_zero_iff_of_nonneg_ae ?_ hfint.neg).1 hnegf ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : r = 0 this : f =\u1d50[\u03bc] 0 \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** rw [snorm_congr_ae this, snorm_zero, hr, ENNReal.coe_zero, mul_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hr : r = 0 hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0 \u22a2 \u222b (x : \u03b1), -f x \u2202\u03bc = 0 ** rw [integral_neg, neg_eq_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hr : r = 0 hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0 \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = 0 ** exact le_antisymm (integral_nonpos_of_ae hf) hfint' ** case pos.refine_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hr : r = 0 hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0 hnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0 this : -f =\u1d50[\u03bc] 0 \u22a2 f =\u1d50[\u03bc] 0 ** filter_upwards [this] with \u03c9 h\u03c9 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hr : r = 0 hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0 hnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0 this : -f =\u1d50[\u03bc] 0 \u03c9 : \u03b1 h\u03c9 : (-f) \u03c9 = OfNat.ofNat 0 \u03c9 \u22a2 f \u03c9 = OfNat.ofNat 0 \u03c9 ** rwa [Pi.neg_apply, Pi.zero_apply, neg_eq_zero] at h\u03c9 ** case pos.refine_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hr : r = 0 hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0 hnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0 \u22a2 0 \u2264\u1d50[\u03bc] -f ** filter_upwards [hf] with \u03c9 h\u03c9 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hr : r = 0 hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 0 hnegf : \u222b (x : \u03b1), -f x \u2202\u03bc = 0 \u03c9 : \u03b1 h\u03c9 : f \u03c9 \u2264 0 \u22a2 OfNat.ofNat 0 \u03c9 \u2264 (-f) \u03c9 ** rwa [Pi.zero_apply, Pi.neg_apply, Right.nonneg_neg_iff] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : \u00acIsFiniteMeasure \u03bc \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** have : \u03bc Set.univ = \u221e := by\n  by_contra h\u03bc'\n  exact h\u03bc (IsFiniteMeasure.mk <| lt_top_iff_ne_top.2 h\u03bc') ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : \u00acIsFiniteMeasure \u03bc this : \u2191\u2191\u03bc univ = \u22a4 \u22a2 snorm f 1 \u03bc \u2264 2 * \u2191\u2191\u03bc univ * \u2191r ** rw [this, ENNReal.mul_top', if_neg, ENNReal.top_mul', if_neg] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : \u00acIsFiniteMeasure \u03bc \u22a2 \u2191\u2191\u03bc univ = \u22a4 ** by_contra h\u03bc' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : \u00acIsFiniteMeasure \u03bc h\u03bc' : \u00ac\u2191\u2191\u03bc univ = \u22a4 \u22a2 False ** exact h\u03bc (IsFiniteMeasure.mk <| lt_top_iff_ne_top.2 h\u03bc') ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : \u00acIsFiniteMeasure \u03bc this : \u2191\u2191\u03bc univ = \u22a4 \u22a2 snorm f 1 \u03bc \u2264 \u22a4 ** exact le_top ** case neg.hnc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : \u00acIsFiniteMeasure \u03bc this : \u2191\u2191\u03bc univ = \u22a4 \u22a2 \u00ac\u2191r = 0 ** simp [hr] ** case neg.hnc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : 0 \u2264 \u222b (x : \u03b1), f x \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc : \u00acIsFiniteMeasure \u03bc this : \u2191\u2191\u03bc univ = \u22a4 \u22a2 \u00ac2 = 0 ** norm_num ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc \u22a2 \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r ** rw [\u2190 integral_const] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc \u22a2 \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 \u222b (x : \u03b1), \u2191r \u2202\u03bc ** refine' integral_mono_ae hfint.real_toNNReal (integrable_const (r : \u211d)) _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc \u22a2 (fun \u03c9 => max (f \u03c9) 0) \u2264\u1d50[\u03bc] fun x => \u2191r ** filter_upwards [hf] with \u03c9 h\u03c9 using Real.toNNReal_le_iff_le_coe.2 h\u03c9 ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc hposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r \u22a2 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc + \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 ENNReal.toReal (2 * \u2191\u2191\u03bc univ * \u2191r) ** simp only [Real.coe_toNNReal', ENNReal.toReal_mul, ENNReal.one_toReal, ENNReal.coe_toReal,\n  ge_iff_le, Left.nonneg_neg_iff, Left.neg_nonpos_iff, toReal_ofNat] at hfint' \u22a2 ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc hposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r \u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r ** refine' (add_le_add_left hfint' _).trans _ ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), max (-f a) 0 \u2202\u03bc \u2264 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc hposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r \u22a2 \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc + \u222b (a : \u03b1), max (f a) 0 \u2202\u03bc \u2264 2 * ENNReal.toReal (\u2191\u2191\u03bc univ) * \u2191r ** rwa [\u2190 two_mul, mul_assoc, mul_le_mul_left (two_pos : (0 : \u211d) < 2)] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 r : \u211d\u22650 f : \u03b1 \u2192 \u211d hfint : Integrable f hfint' : \u222b (a : \u03b1), \u2191(Real.toNNReal (-f a)) \u2202\u03bc \u2264 \u222b (a : \u03b1), \u2191(Real.toNNReal (f a)) \u2202\u03bc hf : \u2200\u1d50 (\u03c9 : \u03b1) \u2202\u03bc, f \u03c9 \u2264 \u2191r hr : \u00acr = 0 h\u03bc this : IsFiniteMeasure \u03bc hposbdd : \u222b (\u03c9 : \u03b1), max (f \u03c9) 0 \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2191r \u22a2 Integrable fun a => \u2191(Real.toNNReal (-f a)) ** exact hfint.neg.sup (integrable_zero _ _ \u03bc) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condCdfRat_le_one ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a \u22a2 condCdfRat \u03c1 a r \u2264 1 ** unfold condCdfRat ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a \u22a2 ite (a \u2208 condCdfSet \u03c1) (fun r => ENNReal.toReal (preCdf \u03c1 r a)) (fun r => if r < 0 then 0 else 1) r \u2264 1 ** split_ifs with h ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h : \u00aca \u2208 condCdfSet \u03c1 \u22a2 (fun r => if r < 0 then 0 else 1) r \u2264 1 ** dsimp only ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h : \u00aca \u2208 condCdfSet \u03c1 \u22a2 (if r < 0 then 0 else 1) \u2264 1 ** split_ifs ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h : \u00aca \u2208 condCdfSet \u03c1 h\u271d : r < 0 \u22a2 0 \u2264 1  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h : \u00aca \u2208 condCdfSet \u03c1 h\u271d : \u00acr < 0 \u22a2 1 \u2264 1 ** exacts [zero_le_one, le_rfl] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h : a \u2208 condCdfSet \u03c1 \u22a2 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r \u2264 1 ** refine' ENNReal.toReal_le_of_le_ofReal zero_le_one _ ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h : a \u2208 condCdfSet \u03c1 \u22a2 preCdf \u03c1 r a \u2264 ENNReal.ofReal 1 ** rw [ENNReal.ofReal_one] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h : a \u2208 condCdfSet \u03c1 \u22a2 preCdf \u03c1 r a \u2264 1 ** exact (hasCondCdf_of_mem_condCdfSet h).le_one r ** Qed",
        "informal": ""
    },
    {
        "formal": "ack_add_one_sq_lt_ack_add_three ** n : \u2115 \u22a2 (ack 0 n + 1) ^ 2 \u2264 ack (0 + 3) n ** simpa using sq_le_two_pow_add_one_minus_three (n + 2) ** m : \u2115 \u22a2 (ack (m + 1) 0 + 1) ^ 2 \u2264 ack (m + 1 + 3) 0 ** rw [ack_succ_zero, ack_succ_zero] ** m : \u2115 \u22a2 (ack m 1 + 1) ^ 2 \u2264 ack (m + 3) 1 ** apply ack_add_one_sq_lt_ack_add_three ** m n : \u2115 \u22a2 (ack (m + 1) (n + 1) + 1) ^ 2 \u2264 ack (m + 1 + 3) (n + 1) ** rw [ack_succ_succ, ack_succ_succ] ** m n : \u2115 \u22a2 (ack m (ack (m + 1) n) + 1) ^ 2 \u2264 ack (m + 3) (ack (m + 3 + 1) n) ** apply (ack_add_one_sq_lt_ack_add_three _ _).trans (ack_mono_right _ <| ack_mono_left _ _) ** m n : \u2115 \u22a2 m + 1 \u2264 m + 3 + 1 ** linarith ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.card_uIcc ** a b : \u2124 \u22a2 ofNat (card (range (toNat (a \u2294 b + 1 - a \u2293 b)))) = ofNat (natAbs (b - a) + 1) ** change ((\u2191) : \u2115 \u2192 \u2124) _ = ((\u2191) : \u2115 \u2192 \u2124) _ ** a b : \u2124 \u22a2 \u2191(card (range (toNat (a \u2294 b + 1 - a \u2293 b)))) = \u2191(natAbs (b - a) + 1) ** rw [card_range, sup_eq_max, inf_eq_min,\n  Int.toNat_of_nonneg (sub_nonneg_of_le <| le_add_one min_le_max), Int.ofNat_add,\n  Int.coe_natAbs, add_comm, add_sub_assoc, max_sub_min_eq_abs, add_comm, Int.ofNat_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.setToL1_add_left ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' f : { x // x \u2208 Lp E 1 } \u22a2 \u2191(setToL1 (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f = \u2191(setToL1 hT) f + \u2191(setToL1 hT') f ** suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by\n  rw [this, ContinuousLinearMap.add_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' f : { x // x \u2208 Lp E 1 } \u22a2 setToL1 (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) = setToL1 hT + setToL1 hT' ** refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc (hT.add hT')) _ _ _ _ _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' f : { x // x \u2208 Lp E 1 } \u22a2 ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d) =     setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) ** ext1 f ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f =     \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f ** suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM \u03b1 E \u03bc (hT.add hT') f by\n  rw [\u2190 this]; rfl ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f ** rw [setToL1_eq_setToL1SCLM, setToL1_eq_setToL1SCLM, setToL1SCLM_add_left hT hT'] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' f : { x // x \u2208 Lp E 1 } this : setToL1 (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C')) = setToL1 hT + setToL1 hT' \u22a2 \u2191(setToL1 (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f = \u2191(setToL1 hT) f + \u2191(setToL1 hT') f ** rw [this, ContinuousLinearMap.add_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } this : \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f =     \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f ** rw [\u2190 this] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' f\u271d : { x // x \u2208 Lp E 1 } f : { x // x \u2208 simpleFunc E 1 \u03bc } this : \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc (_ : DominatedFinMeasAdditive \u03bc (T + T') (C + C'))) f \u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT + setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT) \u2191f + \u2191(setToL1 hT') \u2191f ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurePreserving_mul_prod_inv ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd \u22a2 MeasurePreserving fun z => (z.2 * z.1, z.1\u207b\u00b9) ** convert (measurePreserving_prod_inv_mul_swap \u03bd \u03bc).comp (measurePreserving_prod_mul_swap \u03bc \u03bd)\n  using 1 ** case h.e'_5 G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd \u22a2 (fun z => (z.2 * z.1, z.1\u207b\u00b9)) = (fun z => (z.2, z.2\u207b\u00b9 * z.1)) \u2218 fun z => (z.2, z.2 * z.1) ** ext1 \u27e8x, y\u27e9 ** case h.e'_5.h.mk G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd x y : G \u22a2 ((x, y).2 * (x, y).1, (x, y).1\u207b\u00b9) = ((fun z => (z.2, z.2\u207b\u00b9 * z.1)) \u2218 fun z => (z.2, z.2 * z.1)) (x, y) ** simp_rw [Function.comp_apply, mul_inv_rev, inv_mul_cancel_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.snd_apply' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 f : \u03b2 \u2192 \u03b3 g : \u03b3 \u2192 \u03b1 \u03ba : { x // x \u2208 kernel \u03b1 (\u03b2 \u00d7 \u03b3) } a : \u03b1 s : Set \u03b3 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(snd \u03ba) a) s = \u2191\u2191(\u2191\u03ba a) {p | p.2 \u2208 s} ** rw [snd_apply, Measure.map_apply measurable_snd hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 f : \u03b2 \u2192 \u03b3 g : \u03b3 \u2192 \u03b1 \u03ba : { x // x \u2208 kernel \u03b1 (\u03b2 \u00d7 \u03b3) } a : \u03b1 s : Set \u03b3 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191\u03ba a) (Prod.snd \u207b\u00b9' s) = \u2191\u2191(\u2191\u03ba a) {p | p.2 \u2208 s} ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.pure_bindOnSupport ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 a : \u03b1 f : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2 \u22a2 bindOnSupport (pure a) f = f a (_ : a \u2208 support (pure a)) ** refine' PMF.ext fun b => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 a : \u03b1 f : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2 b : \u03b2 \u22a2 \u2191(bindOnSupport (pure a) f) b = \u2191(f a (_ : a \u2208 support (pure a))) b ** simp only [bindOnSupport_apply, pure_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 a : \u03b1 f : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2 b : \u03b2 \u22a2 (\u2211' (a_1 : \u03b1),       (if a_1 = a then 1 else 0) *         if h : (if a_1 = a then 1 else 0) = 0 then 0 else \u2191(f a_1 (_ : \u00ac\u2191(pure a) a_1 = 0)) b) =     \u2191(f a (_ : a \u2208 support (pure a))) b ** refine' _root_.trans (tsum_congr fun a' => _) (tsum_ite_eq a _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 f\u271d : (a : \u03b1) \u2192 a \u2208 support p \u2192 PMF \u03b2 a : \u03b1 f : (a' : \u03b1) \u2192 a' \u2208 support (pure a) \u2192 PMF \u03b2 b : \u03b2 a' : \u03b1 \u22a2 ((if a' = a then 1 else 0) * if h : (if a' = a then 1 else 0) = 0 then 0 else \u2191(f a' (_ : \u00ac\u2191(pure a) a' = 0)) b) =     if a' = a then \u2191(f a (_ : a \u2208 support (pure a))) b else 0 ** by_cases h : a' = a <;> simp [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lpMeasSubgroupToLpTrim_add ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 lpMeasSubgroupToLpTrim F p \u03bc hm (f + g) = lpMeasSubgroupToLpTrim F p \u03bc hm f + lpMeasSubgroupToLpTrim F p \u03bc hm g ** ext1 ** case h \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[Measure.trim \u03bc hm]     \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f + lpMeasSubgroupToLpTrim F p \u03bc hm g) ** refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm ** case h \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[Measure.trim \u03bc hm]     \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g) ** refine' ae_eq_trim_of_stronglyMeasurable hm (Lp.stronglyMeasurable _) _ _ ** case h.refine'_2 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm (f + g)) =\u1d50[\u03bc]     \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g) ** refine' (lpMeasSubgroupToLpTrim_ae_eq hm _).trans _ ** case h.refine'_2 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191\u2191(f + g) =\u1d50[\u03bc] \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g) ** refine'\n  EventuallyEq.trans _\n    (EventuallyEq.add (lpMeasSubgroupToLpTrim_ae_eq hm f).symm\n      (lpMeasSubgroupToLpTrim_ae_eq hm g).symm) ** case h.refine'_2 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191\u2191(f + g) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x ** refine' (Lp.coeFn_add _ _).trans _ ** case h.refine'_2 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 \u2191\u2191\u2191f + \u2191\u2191\u2191g =\u1d50[\u03bc] fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x ** exact eventually_of_forall fun x => by rfl ** case h.refine'_1 \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } \u22a2 StronglyMeasurable (\u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm f) + \u2191\u2191(lpMeasSubgroupToLpTrim F p \u03bc hm g)) ** exact (Lp.stronglyMeasurable _).add (Lp.stronglyMeasurable _) ** \u03b1 : Type u_1 E' : Type u_2 F : Type u_3 F' : Type u_4 \ud835\udd5c : Type u_5 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' \u03b9 : Type u_6 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 f g : { x // x \u2208 lpMeasSubgroup F m p \u03bc } x : \u03b1 \u22a2 (\u2191\u2191\u2191f + \u2191\u2191\u2191g) x = (fun x => \u2191\u2191\u2191f x + \u2191\u2191\u2191g x) x ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 \u03c6,     (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227       \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5 ** rw [lintegral_eq_nnreal] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 \u03c6,     (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227       \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5 ** have := ENNReal.lt_add_right h h\u03b5 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this :   \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <     (\u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc) + \u03b5 \u22a2 \u2203 \u03c6,     (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227       \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5 ** erw [ENNReal.biSup_add] at this <;> [skip; exact \u27e80, fun x => zero_le _\u27e9] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this :   \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc <     \u2a06 i \u2208 fun \u03c6 => \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x, SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5 \u22a2 \u2203 \u03c6,     (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227       \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5 ** simp_rw [lt_iSup_iff, iSup_lt_iff, iSup_le_iff] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this :   \u2203 i h b,     b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc + \u03b5 \u2227       \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b \u22a2 \u2203 \u03c6,     (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227       \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5 ** rcases this with \u27e8\u03c6, hle : \u2200 x, \u2191(\u03c6 x) \u2264 f x, b, hb\u03c6, hb\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 hle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x b : \u211d\u22650\u221e hb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5 hb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b \u22a2 \u2203 \u03c6,     (\u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x) \u2227       \u2200 (\u03c8 : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5 ** refine' \u27e8\u03c6, hle, fun \u03c8 h\u03c8 => _\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 hle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x b : \u211d\u22650\u221e hb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5 hb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5 ** have : (map (\u2191) \u03c6).lintegral \u03bc \u2260 \u221e := ne_top_of_le_ne_top h (by exact le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) \u03c6 hle) ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 hle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x b : \u211d\u22650\u221e hb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5 hb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x this : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c8 - \u03c6)) \u03bc < \u03b5 ** rw [\u2190 ENNReal.add_lt_add_iff_left this, \u2190 add_lintegral, \u2190 SimpleFunc.map_add @ENNReal.coe_add] ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 hle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x b : \u211d\u22650\u221e hb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5 hb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x this : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (\u03c6 + (\u03c8 - \u03c6))) \u03bc <     SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5 ** refine' (hb _ fun x => le_trans _ (max_le (hle x) (h\u03c8 x))).trans_lt hb\u03c6 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 hle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x b : \u211d\u22650\u221e hb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5 hb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x this : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 x : \u03b1 \u22a2 \u2191(\u2191(\u03c6 + (\u03c8 - \u03c6)) x) \u2264 max \u2191(\u2191\u03c6 x) \u2191(\u2191\u03c8 x) ** norm_cast ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 hle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x b : \u211d\u22650\u221e hb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5 hb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x this : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 x : \u03b1 \u22a2 \u2191(\u03c6 + (\u03c8 - \u03c6)) x \u2264 max (\u2191\u03c6 x) (\u2191\u03c8 x) ** simp only [add_apply, sub_apply, add_tsub_eq_max] ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 hle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x b : \u211d\u22650\u221e hb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5 hb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x this : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 x : \u03b1 \u22a2 max (\u2191\u03c6 x) (\u2191\u03c8 x) \u2264 max (\u2191\u03c6 x) (\u2191\u03c8 x) ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e h : \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u03c6 : \u03b1 \u2192\u209b \u211d\u22650 hle : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x b : \u211d\u22650\u221e hb\u03c6 : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc + \u03b5 hb : \u2200 (i : \u03b1 \u2192\u209b \u211d\u22650), (\u2200 (x : \u03b1), \u2191(\u2191i x) \u2264 f x) \u2192 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) \u03bc \u2264 b \u03c8 : \u03b1 \u2192\u209b \u211d\u22650 h\u03c8 : \u2200 (x : \u03b1), \u2191(\u2191\u03c8 x) \u2264 f x \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2264     \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc ** exact le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) \u03c6 hle ** Qed",
        "informal": ""
    },
    {
        "formal": "List.eraseP_map ** \u03b2 : Type u_1 \u03b1 : Type u_2 p : \u03b1 \u2192 Bool f : \u03b2 \u2192 \u03b1 b : \u03b2 l : List \u03b2 \u22a2 eraseP p (map f (b :: l)) = map f (eraseP (p \u2218 f) (b :: l)) ** by_cases h : p (f b) <;> simp [h, eraseP_map f l, eraseP_cons_of_pos] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.preimage_mul_right_one' ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : Group \u03b1 s t : Finset \u03b1 a b : \u03b1 \u22a2 preimage 1 (fun x => x * b\u207b\u00b9) (_ : Set.InjOn (fun x => x * b\u207b\u00b9) ((fun x => x * b\u207b\u00b9) \u207b\u00b9' \u21911)) = {b} ** rw [preimage_mul_right_one, inv_inv] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.val_mul ** n : \u2115 a b : ZMod n \u22a2 val (a * b) = val a * val b % n ** cases n ** case zero a b : ZMod Nat.zero \u22a2 val (a * b) = val a * val b % Nat.zero ** rw [Nat.mod_zero] ** case zero a b : ZMod Nat.zero \u22a2 val (a * b) = val a * val b ** apply Int.natAbs_mul ** case succ n\u271d : \u2115 a b : ZMod (Nat.succ n\u271d) \u22a2 val (a * b) = val a * val b % Nat.succ n\u271d ** apply Fin.val_mul ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.induction ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) f : \u03b1 \u2192\u209b \u03b3 \u22a2 P f ** generalize h : f.range \\ {0} = s ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) f : \u03b1 \u2192\u209b \u03b3 s : Finset \u03b3 h : SimpleFunc.range f \\ {0} = s \u22a2 P f ** rw [\u2190 Finset.coe_inj, Finset.coe_sdiff, Finset.coe_singleton, SimpleFunc.coe_range] at h ** case empty \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191\u2205 \u22a2 P f ** rw [Finset.coe_empty, diff_eq_empty, range_subset_singleton] at h ** case empty \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) f : \u03b1 \u2192\u209b \u03b3 h : \u2191f = Function.const \u03b1 0 \u22a2 P f ** convert h_ind 0 MeasurableSet.univ ** case h.e'_1 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) f : \u03b1 \u2192\u209b \u03b3 h : \u2191f = Function.const \u03b1 0 \u22a2 f = piecewise univ (_ : MeasurableSet univ) (const \u03b1 0) (const \u03b1 0) ** ext x ** case h.e'_1.H \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) f : \u03b1 \u2192\u209b \u03b3 h : \u2191f = Function.const \u03b1 0 x : \u03b1 \u22a2 \u2191f x = \u2191(piecewise univ (_ : MeasurableSet univ) (const \u03b1 0) (const \u03b1 0)) x ** simp [h] ** case insert \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) \u22a2 P f ** have mx := f.measurableSet_preimage {x} ** case insert \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) \u22a2 P f ** let g := SimpleFunc.piecewise (f \u207b\u00b9' {x}) mx 0 f ** case insert \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g \u22a2 P f ** convert h_add _ Pg (h_ind x mx) ** case insert \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g \u22a2 Disjoint (support \u2191g) (support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0))) ** rw [disjoint_iff_inf_le] ** case insert \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g \u22a2 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2264 \u22a5 ** rintro y ** case insert \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g y : \u03b1 \u22a2 y \u2208 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2192 y \u2208 \u22a5 ** by_cases hy : y \u2208 f \u207b\u00b9' {x} ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f \u22a2 P g ** apply ih ** case h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f \u22a2 range \u2191g \\ {0} = \u2191s ** simp only [SimpleFunc.coe_piecewise, range_piecewise] ** case h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f \u22a2 (\u21910 '' (\u2191f \u207b\u00b9' {x}) \u222a \u2191f '' (\u2191f \u207b\u00b9' {x})\u1d9c) \\ {0} = \u2191s ** rw [image_compl_preimage, union_diff_distrib, diff_diff_comm, h, Finset.coe_insert,\n  insert_diff_self_of_not_mem, diff_eq_empty.mpr, Set.empty_union] ** case h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f \u22a2 \u21910 '' (\u2191f \u207b\u00b9' {x}) \u2286 {0} ** rw [Set.image_subset_iff] ** case h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f \u22a2 \u2191f \u207b\u00b9' {x} \u2286 \u21910 \u207b\u00b9' {0} ** convert Set.subset_univ _ ** case h.e'_4 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f \u22a2 \u21910 \u207b\u00b9' {0} = univ ** exact preimage_const_of_mem (mem_singleton _) ** case h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f \u22a2 \u00acx \u2208 \u2191s ** rwa [Finset.mem_coe] ** case h.e'_1 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g \u22a2 f = g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0) ** ext1 y ** case h.e'_1.H \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g y : \u03b1 \u22a2 \u2191f y = \u2191(g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) y ** by_cases hy : y \u2208 f \u207b\u00b9' {x} ** case pos \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g y : \u03b1 hy : y \u2208 \u2191f \u207b\u00b9' {x} \u22a2 \u2191f y = \u2191(g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) y ** simpa [piecewise_eq_of_mem _ _ _ hy, -piecewise_eq_indicator] ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g y : \u03b1 hy : \u00acy \u2208 \u2191f \u207b\u00b9' {x} \u22a2 \u2191f y = \u2191(g + piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) y ** simp [piecewise_eq_of_not_mem _ _ _ hy, -piecewise_eq_indicator] ** case pos \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g y : \u03b1 hy : y \u2208 \u2191f \u207b\u00b9' {x} \u22a2 y \u2208 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2192 y \u2208 \u22a5 ** simp [piecewise_eq_of_mem _ _ _ hy, -piecewise_eq_indicator] ** case neg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3\u271d : Type u_3 \u03b4 : Type u_4 \u03b1 : Type u_5 \u03b3 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : AddMonoid \u03b3 P : (\u03b1 \u2192\u209b \u03b3) \u2192 Prop h_ind : \u2200 (c : \u03b3) {s : Set \u03b1} (hs : MeasurableSet s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0)) h_add : \u2200 \u2983f g : \u03b1 \u2192\u209b \u03b3\u2984, Disjoint (support \u2191f) (support \u2191g) \u2192 P f \u2192 P g \u2192 P (f + g) x : \u03b3 s : Finset \u03b3 hxs : \u00acx \u2208 s ih : \u2200 (f : \u03b1 \u2192\u209b \u03b3), range \u2191f \\ {0} = \u2191s \u2192 P f f : \u03b1 \u2192\u209b \u03b3 h : range \u2191f \\ {0} = \u2191(insert x s) mx : MeasurableSet (\u2191f \u207b\u00b9' {x}) g : \u03b1 \u2192\u209b \u03b3 := piecewise (\u2191f \u207b\u00b9' {x}) mx 0 f Pg : P g y : \u03b1 hy : \u00acy \u2208 \u2191f \u207b\u00b9' {x} \u22a2 y \u2208 support \u2191g \u2293 support \u2191(piecewise (\u2191f \u207b\u00b9' {x}) mx (const \u03b1 x) (const \u03b1 0)) \u2192 y \u2208 \u22a5 ** simp [piecewise_eq_of_not_mem _ _ _ hy, -piecewise_eq_indicator] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.inter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u_6 s t u : Set \u03b1 m : MeasurableSpace \u03b1 s\u2081 s\u2082 : Set \u03b1 h\u2081 : MeasurableSet s\u2081 h\u2082 : MeasurableSet s\u2082 \u22a2 MeasurableSet (s\u2081 \u2229 s\u2082) ** rw [inter_eq_compl_compl_union_compl] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u_6 s t u : Set \u03b1 m : MeasurableSpace \u03b1 s\u2081 s\u2082 : Set \u03b1 h\u2081 : MeasurableSet s\u2081 h\u2082 : MeasurableSet s\u2082 \u22a2 MeasurableSet (s\u2081\u1d9c \u222a s\u2082\u1d9c)\u1d9c ** exact (h\u2081.compl.union h\u2082.compl).compl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.dist_indicatorConstLp_eq_norm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 dist (indicatorConstLp p hs h\u03bcs c) (indicatorConstLp p ht h\u03bct c) =     \u2016indicatorConstLp p (_ : MeasurableSet (s \u2206 t)) (_ : \u2191\u2191\u03bc (s \u2206 t) \u2260 \u22a4) c\u2016 ** rw [Lp.dist_edist, edist_indicatorConstLp_eq_nnnorm, ENNReal.coe_toReal, Lp.coe_nnnorm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.measure_inv ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : MeasurableSpace H inst\u271d\u00b2 : InvolutiveInv G inst\u271d\u00b9 : MeasurableInv G \u03bc : Measure G inst\u271d : IsInvInvariant \u03bc A : Set G \u22a2 \u2191\u2191\u03bc A\u207b\u00b9 = \u2191\u2191\u03bc A ** rw [\u2190 inv_apply, inv_eq_self] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.variance_eq ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : Mem\u2112p X 2 \u22a2 variance X \u03bc = \u222b (x : \u03a9), ((X - fun x => \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) x \u2202\u03bc ** rw [variance, evariance_eq_lintegral_ofReal, \u2190 ofReal_integral_eq_lintegral_ofReal,\n  ENNReal.toReal_ofReal] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : Mem\u2112p X 2 \u22a2 \u222b (x : \u03a9), (X x - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 \u2202\u03bc = \u222b (x : \u03a9), ((X - fun x => \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2) x \u2202\u03bc ** rfl ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : Mem\u2112p X 2 \u22a2 0 \u2264 \u222b (x : \u03a9), (X x - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 \u2202\u03bc ** exact integral_nonneg fun \u03c9 => pow_two_nonneg _ ** case hfi \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : Mem\u2112p X 2 \u22a2 Integrable fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 ** convert (hX.sub <| mem\u2112p_const (\u03bc [X])).integrable_norm_rpow two_ne_zero ENNReal.two_ne_top\n  with \u03c9 ** case h.e'_5.h \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : Mem\u2112p X 2 \u03c9 : \u03a9 \u22a2 (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 = \u2016(X - fun x => \u222b (x : \u03a9), X x \u2202\u03bc) \u03c9\u2016 ^ ENNReal.toReal 2 ** simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toReal,\n  Real.rpow_two, sq_abs, abs_pow] ** case f_nn \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d : IsFiniteMeasure \u03bc hX : Mem\u2112p X 2 \u22a2 0 \u2264\u1d50[\u03bc] fun \u03c9 => (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc) ^ 2 ** exact ae_of_all _ fun \u03c9 => pow_two_nonneg _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_ball_mul_of_pos ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d : Set E x : E r : \u211d hr : 0 < r s : \u211d \u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s) ** have : ball (0 : E) (r * s) = r \u2022 ball (0 : E) s := by\n  simp only [_root_.smul_ball hr.ne' (0 : E) s, Real.norm_eq_abs, abs_of_nonneg hr.le, smul_zero] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d : Set E x : E r : \u211d hr : 0 < r s : \u211d this : ball 0 (r * s) = r \u2022 ball 0 s \u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s) ** simp only [this, addHaar_smul, abs_of_nonneg hr.le, addHaar_ball_center, abs_pow] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d : Set E x : E r : \u211d hr : 0 < r s : \u211d \u22a2 ball 0 (r * s) = r \u2022 ball 0 s ** simp only [_root_.smul_ball hr.ne' (0 : E) s, Real.norm_eq_abs, abs_of_nonneg hr.le, smul_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.eventually_div_right_iff ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : MeasurableSpace H inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MeasurableMul G \u03bc : Measure G inst\u271d : IsMulRightInvariant \u03bc t : G p : G \u2192 Prop \u22a2 (\u2200\u1d50 (x : G) \u2202\u03bc, p (x / t)) \u2194 \u2200\u1d50 (x : G) \u2202\u03bc, p x ** conv_rhs => rw [Filter.Eventually, \u2190 map_div_right_ae \u03bc t]; rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.eq_of_mem_ordConnectedSection_of_uIcc_subset ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x y z : \u03b1 hx : x \u2208 ordConnectedSection s hy : y \u2208 ordConnectedSection s h : [[x, y]] \u2286 s \u22a2 x = y ** rcases hx with \u27e8x, rfl\u27e9 ** case intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 y z : \u03b1 hy : y \u2208 ordConnectedSection s x : \u2191s h : [[ordConnectedProj s x, y]] \u2286 s \u22a2 ordConnectedProj s x = y ** rcases hy with \u27e8y, rfl\u27e9 ** case intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 z : \u03b1 x y : \u2191s h : [[ordConnectedProj s x, ordConnectedProj s y]] \u2286 s \u22a2 ordConnectedProj s x = ordConnectedProj s y ** exact\n  ordConnectedProj_eq.2\n    (mem_ordConnectedComponent_trans\n      (mem_ordConnectedComponent_trans (ordConnectedProj_mem_ordConnectedComponent _ _) h)\n      (mem_ordConnectedComponent_ordConnectedProj _ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.powersetCard_sup ** \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u \u22a2 sup (powersetCard (Nat.succ n) u) id = u ** apply le_antisymm ** case a \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u \u22a2 sup (powersetCard (Nat.succ n) u) id \u2264 u ** simp_rw [Finset.sup_le_iff, mem_powersetCard] ** case a \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u \u22a2 \u2200 (b : Finset \u03b1), b \u2286 u \u2227 card b = Nat.succ n \u2192 id b \u2264 u ** rintro x \u27e8h, -\u27e9 ** case a.intro \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u x : Finset \u03b1 h : x \u2286 u \u22a2 id x \u2264 u ** exact h ** case a \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u \u22a2 u \u2264 sup (powersetCard (Nat.succ n) u) id ** rw [sup_eq_biUnion, le_iff_subset, subset_iff] ** case a \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u \u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id ** cases' (Nat.succ_le_of_lt hn).eq_or_lt with h' h' ** case a.inl \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u h' : Nat.succ n = card u \u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id ** simp [h'] ** case a.inr \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u h' : Nat.succ n < card u \u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id ** intro x hx ** case a.inr \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u h' : Nat.succ n < card u x : \u03b1 hx : x \u2208 u \u22a2 x \u2208 Finset.biUnion (powersetCard (Nat.succ n) u) id ** simp only [mem_biUnion, exists_prop, id.def] ** case a.inr \u03b1 : Type u_1 s t : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u h' : Nat.succ n < card u x : \u03b1 hx : x \u2208 u \u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a ** obtain \u27e8t, ht\u27e9 : \u2203 t, t \u2208 powersetCard n (u.erase x) := powersetCard_nonempty\n  (le_trans (Nat.le_pred_of_lt hn) pred_card_le_card_erase) ** case a.inr.intro \u03b1 : Type u_1 s t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u h' : Nat.succ n < card u x : \u03b1 hx : x \u2208 u t : Finset \u03b1 ht : t \u2208 powersetCard n (erase u x) \u22a2 \u2203 a, a \u2208 powersetCard (Nat.succ n) u \u2227 x \u2208 a ** refine' \u27e8insert x t, _, mem_insert_self _ _\u27e9 ** case a.inr.intro \u03b1 : Type u_1 s t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u h' : Nat.succ n < card u x : \u03b1 hx : x \u2208 u t : Finset \u03b1 ht : t \u2208 powersetCard n (erase u x) \u22a2 insert x t \u2208 powersetCard (Nat.succ n) u ** rw [\u2190 insert_erase hx, powersetCard_succ_insert (not_mem_erase _ _)] ** case a.inr.intro \u03b1 : Type u_1 s t\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 u : Finset \u03b1 n : \u2115 hn : n < card u h' : Nat.succ n < card u x : \u03b1 hx : x \u2208 u t : Finset \u03b1 ht : t \u2208 powersetCard n (erase u x) \u22a2 insert x t \u2208 powersetCard (Nat.succ n) (erase u x) \u222a image (insert x) (powersetCard n (erase u x)) ** exact mem_union_right _ (mem_image_of_mem _ ht) ** Qed",
        "informal": ""
    },
    {
        "formal": "le_radius_cauchyPowerSeries ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 \u22a2 \u2191R \u2264 FormalMultilinearSeries.radius (cauchyPowerSeries f c \u2191R) ** refine'\n  (cauchyPowerSeries f c R).le_radius_of_bound\n    ((2 * \u03c0)\u207b\u00b9 * \u222b \u03b8 : \u211d in (0)..2 * \u03c0, \u2016f (circleMap c R \u03b8)\u2016) fun n => _ ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 n : \u2115 \u22a2 \u2016cauchyPowerSeries f c (\u2191R) n\u2016 * \u2191R ^ n \u2264 (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016 ** refine' (mul_le_mul_of_nonneg_right (norm_cauchyPowerSeries_le _ _ _ _)\n  (pow_nonneg R.coe_nonneg _)).trans _ ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 n : \u2115 \u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * |\u2191R|\u207b\u00b9 ^ n * \u2191R ^ n \u2264     (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016 ** rw [_root_.abs_of_nonneg R.coe_nonneg] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 n : \u2115 \u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R)\u207b\u00b9 ^ n * \u2191R ^ n \u2264     (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016 ** cases' eq_or_ne (R ^ n : \u211d) 0 with hR hR ** case inl E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 n : \u2115 hR : \u2191(R ^ n) = 0 \u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R)\u207b\u00b9 ^ n * \u2191R ^ n \u2264     (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016 ** rw_mod_cast [hR, mul_zero] ** case inl E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 n : \u2115 hR : R ^ n = 0 \u22a2 0 \u2264 (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016 ** exact mul_nonneg (inv_nonneg.2 Real.two_pi_pos.le)\n  (intervalIntegral.integral_nonneg Real.two_pi_pos.le fun _ _ => norm_nonneg _) ** case inr E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 n : \u2115 hR : \u2191(R ^ n) \u2260 0 \u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R)\u207b\u00b9 ^ n * \u2191R ^ n \u2264     (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016 ** rw [inv_pow] ** case inr E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 n : \u2115 hR : \u2191(R ^ n) \u2260 0 \u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R ^ n)\u207b\u00b9 * \u2191R ^ n \u2264     (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016 ** have : (R:\u211d) ^ n \u2260 0 := by norm_cast at hR \u22a2 ** case inr E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 n : \u2115 hR : \u2191(R ^ n) \u2260 0 this : \u2191R ^ n \u2260 0 \u22a2 ((2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016) * (\u2191R ^ n)\u207b\u00b9 * \u2191R ^ n \u2264     (2 * \u03c0)\u207b\u00b9 * \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016f (circleMap c (\u2191R) \u03b8)\u2016 ** rw [inv_mul_cancel_right\u2080 this] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R : \u211d\u22650 n : \u2115 hR : \u2191(R ^ n) \u2260 0 \u22a2 \u2191R ^ n \u2260 0 ** norm_cast at hR \u22a2 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.measurableSet_iSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9\u271d : Sort u_6 s\u271d t u : Set \u03b1 \u03b9 : Sort u_7 m : \u03b9 \u2192 MeasurableSpace \u03b1 s : Set \u03b1 \u22a2 MeasurableSet s \u2194 GenerateMeasurable {s | \u2203 i, MeasurableSet s} s ** simp only [iSup, measurableSet_sSup, exists_range_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.DynkinSystem.generateFrom_eq ** \u03b1 : Type u_1 d : DynkinSystem \u03b1 s : Set (Set \u03b1) hs : IsPiSystem s \u22a2 ofMeasurableSpace       (toMeasurableSpace (generate s)         (_ : \u2200 (t\u2081 t\u2082 : Set \u03b1), Has (generate s) t\u2081 \u2192 Has (generate s) t\u2082 \u2192 Has (generate s) (t\u2081 \u2229 t\u2082))) \u2264     ofMeasurableSpace (generateFrom s) ** rw [ofMeasurableSpace_toMeasurableSpace] ** \u03b1 : Type u_1 d : DynkinSystem \u03b1 s : Set (Set \u03b1) hs : IsPiSystem s \u22a2 generate s \u2264 ofMeasurableSpace (generateFrom s) ** exact generate_le _ fun t ht => measurableSet_generateFrom ht ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.eval\u2082_pow ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 p : MvPolynomial \u03c3 R \u22a2 eval\u2082 f g (p ^ 0) = eval\u2082 f g p ^ 0 ** rw [pow_zero, pow_zero] ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 p : MvPolynomial \u03c3 R \u22a2 eval\u2082 f g 1 = 1 ** exact eval\u2082_one _ _ ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n\u271d m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p\u271d q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 p : MvPolynomial \u03c3 R n : \u2115 \u22a2 eval\u2082 f g (p ^ (n + 1)) = eval\u2082 f g p ^ (n + 1) ** rw [pow_add, pow_one, pow_add, pow_one, eval\u2082_mul, eval\u2082_pow] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.tendsto_approxOn ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : PseudoEMetricSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 f\u271d f : \u03b2 \u2192 \u03b1 hf : Measurable f s : Set \u03b1 y\u2080 : \u03b1 h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s x : \u03b2 hx : f x \u2208 closure s \u22a2 Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) x) atTop (\ud835\udcdd (f x)) ** haveI : Nonempty s := \u27e8\u27e8y\u2080, h\u2080\u27e9\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : PseudoEMetricSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 f\u271d f : \u03b2 \u2192 \u03b1 hf : Measurable f s : Set \u03b1 y\u2080 : \u03b1 h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s x : \u03b2 hx : f x \u2208 closure s this : Nonempty \u2191s \u22a2 Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) x) atTop (\ud835\udcdd (f x)) ** rw [\u2190 @Subtype.range_coe _ s, \u2190 image_univ, \u2190 (denseRange_denseSeq s).closure_eq] at hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : PseudoEMetricSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 f\u271d f : \u03b2 \u2192 \u03b1 hf : Measurable f s : Set \u03b1 y\u2080 : \u03b1 h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s x : \u03b2 this : Nonempty \u2191s hx : f x \u2208 closure (Subtype.val '' closure (Set.range (denseSeq \u2191s))) \u22a2 Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) x) atTop (\ud835\udcdd (f x)) ** simp (config := { iota := false }) only [approxOn, coe_comp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : PseudoEMetricSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 f\u271d f : \u03b2 \u2192 \u03b1 hf : Measurable f s : Set \u03b1 y\u2080 : \u03b1 h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s x : \u03b2 this : Nonempty \u2191s hx : f x \u2208 closure (Subtype.val '' closure (Set.range (denseSeq \u2191s))) \u22a2 Tendsto (fun n => (\u2191(nearestPt (fun k => Nat.casesOn k y\u2080 (Subtype.val \u2218 denseSeq \u2191s)) n) \u2218 f) x) atTop (\ud835\udcdd (f x)) ** refine' tendsto_nearestPt (closure_minimal _ isClosed_closure hx) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : PseudoEMetricSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 f\u271d f : \u03b2 \u2192 \u03b1 hf : Measurable f s : Set \u03b1 y\u2080 : \u03b1 h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s x : \u03b2 this : Nonempty \u2191s hx : f x \u2208 closure (Subtype.val '' closure (Set.range (denseSeq \u2191s))) \u22a2 Subtype.val '' closure (Set.range (denseSeq \u2191s)) \u2286     closure (Set.range fun k => Nat.casesOn k y\u2080 (Subtype.val \u2218 denseSeq \u2191s)) ** simp (config := { iota := false }) only [Nat.range_casesOn, closure_union, range_comp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : PseudoEMetricSpace \u03b1 inst\u271d\u00b2 : OpensMeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 f\u271d f : \u03b2 \u2192 \u03b1 hf : Measurable f s : Set \u03b1 y\u2080 : \u03b1 h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s x : \u03b2 this : Nonempty \u2191s hx : f x \u2208 closure (Subtype.val '' closure (Set.range (denseSeq \u2191s))) \u22a2 Subtype.val '' closure (Set.range (denseSeq \u2191s)) \u2286 closure {y\u2080} \u222a closure (Subtype.val '' Set.range (denseSeq \u2191s)) ** exact\n  Subset.trans (image_closure_subset_closure_image continuous_subtype_val)\n    (subset_union_right _ _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.AssocList.forIn_eq ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b4 : Type u_1 inst\u271d : Monad m l : AssocList \u03b1 \u03b2 init : \u03b4 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4) \u22a2 forIn l init f = forIn (toList l) init f ** simp [forIn, List.forIn] ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b4 : Type u_1 inst\u271d : Monad m l : AssocList \u03b1 \u03b2 init : \u03b4 f : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4) \u22a2 AssocList.forIn l init f = List.forIn.loop f (toList l) init ** induction l generalizing init <;> simp [AssocList.forIn, List.forIn.loop] ** case cons m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b4 : Type u_1 inst\u271d : Monad m f : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4) key\u271d : \u03b1 value\u271d : \u03b2 tail\u271d : AssocList \u03b1 \u03b2 tail_ih\u271d : \u2200 (init : \u03b4), AssocList.forIn tail\u271d init f = List.forIn.loop f (toList tail\u271d) init init : \u03b4 \u22a2 (do       let __do_lift \u2190 f (key\u271d, value\u271d) init       match __do_lift with         | ForInStep.done d => pure d         | ForInStep.yield d => AssocList.forIn tail\u271d d f) =     do     let __do_lift \u2190 f (key\u271d, value\u271d) init     match __do_lift with       | ForInStep.done d => pure d       | ForInStep.yield b => List.forIn.loop f (toList tail\u271d) b ** congr ** case cons.e_a m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b4 : Type u_1 inst\u271d : Monad m f : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4) key\u271d : \u03b1 value\u271d : \u03b2 tail\u271d : AssocList \u03b1 \u03b2 tail_ih\u271d : \u2200 (init : \u03b4), AssocList.forIn tail\u271d init f = List.forIn.loop f (toList tail\u271d) init init : \u03b4 \u22a2 (fun __do_lift =>       match __do_lift with       | ForInStep.done d => pure d       | ForInStep.yield d => AssocList.forIn tail\u271d d f) =     fun __do_lift =>     match __do_lift with     | ForInStep.done d => pure d     | ForInStep.yield b => List.forIn.loop f (toList tail\u271d) b ** funext a ** case cons.e_a.h m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b4 : Type u_1 inst\u271d : Monad m f : \u03b1 \u00d7 \u03b2 \u2192 \u03b4 \u2192 m (ForInStep \u03b4) key\u271d : \u03b1 value\u271d : \u03b2 tail\u271d : AssocList \u03b1 \u03b2 tail_ih\u271d : \u2200 (init : \u03b4), AssocList.forIn tail\u271d init f = List.forIn.loop f (toList tail\u271d) init init : \u03b4 a : ForInStep \u03b4 \u22a2 (match a with     | ForInStep.done d => pure d     | ForInStep.yield d => AssocList.forIn tail\u271d d f) =     match a with     | ForInStep.done d => pure d     | ForInStep.yield b => List.forIn.loop f (toList tail\u271d) b ** split <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** have integrand_eq : \u2200 \u03c9,\n    ENNReal.ofReal (\u222b t in (0)..f \u03c9, g t) = (\u222b\u207b t in Ioc 0 (f \u03c9), ENNReal.ofReal (g t)) := by\n  intro \u03c9\n  have g_ae_nn : 0 \u2264\u1d50[volume.restrict (Ioc 0 (f \u03c9))] g := by\n    filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f \u03c9)))]\n      with x hx using g_nn x hx.1\n  rw [\u2190 ofReal_integral_eq_lintegral_ofReal (g_intble' (f \u03c9) (f_nn \u03c9)).1 g_ae_nn]\n  congr\n  exact intervalIntegral.integral_of_le (f_nn \u03c9) ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) \u22a2 \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** rw [lintegral_congr integrand_eq] ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) \u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (t : \u211d) in Ioc 0 (f a), ENNReal.ofReal (g t) \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** simp_rw [\u2190 lintegral_indicator (fun t => ENNReal.ofReal (g t)) measurableSet_Ioc] ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) \u22a2 \u222b\u207b (a : \u03b1), \u222b\u207b (a_1 : \u211d), indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) a_1 \u2202\u03bc =     \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t) ** rw [\u2190 lintegral_indicator _ measurableSet_Ioi, lintegral_lintegral_swap] ** case hf \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) aux\u2082 :   (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2) \u22a2 AEMeasurable (Function.uncurry fun a a_1 => indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) a_1) ** rw [aux\u2082] ** case hf \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) aux\u2082 :   (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2) \u22a2 AEMeasurable (indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2)) ** have mble\u2080 : MeasurableSet {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := by\n  simpa only [mem_univ, Pi.zero_apply, gt_iff_lt, not_lt, ge_iff_le, true_and] using\n    measurableSet_region_between_oc measurable_zero f_mble  MeasurableSet.univ ** case hf \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) aux\u2082 :   (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2) mble\u2080 : MeasurableSet {p | p.2 \u2208 Ioc 0 (f p.1)} \u22a2 AEMeasurable (indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2)) ** exact (ENNReal.measurable_ofReal.comp (g_mble.comp measurable_snd)).aemeasurable.indicator\u2080\n  mble\u2080.nullMeasurableSet ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t \u22a2 \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t ** intro t ht ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t t : \u211d ht : 0 \u2264 t \u22a2 IntervalIntegrable g volume 0 t ** cases' eq_or_lt_of_le ht with h h ** case inl \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t t : \u211d ht : 0 \u2264 t h : 0 = t \u22a2 IntervalIntegrable g volume 0 t ** simp [\u2190 h] ** case inr \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t t : \u211d ht : 0 \u2264 t h : 0 < t \u22a2 IntervalIntegrable g volume 0 t ** exact g_intble t h ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t \u22a2 \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) ** intro \u03c9 ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t \u03c9 : \u03b1 \u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) ** have g_ae_nn : 0 \u2264\u1d50[volume.restrict (Ioc 0 (f \u03c9))] g := by\n  filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f \u03c9)))]\n    with x hx using g_nn x hx.1 ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t \u03c9 : \u03b1 g_ae_nn : 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g \u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) ** rw [\u2190 ofReal_integral_eq_lintegral_ofReal (g_intble' (f \u03c9) (f_nn \u03c9)).1 g_ae_nn] ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t \u03c9 : \u03b1 g_ae_nn : 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g \u22a2 ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = ENNReal.ofReal (\u222b (x : \u211d) in Ioc 0 (f \u03c9), g x) ** congr ** case e_r \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t \u03c9 : \u03b1 g_ae_nn : 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g \u22a2 \u222b (t : \u211d) in 0 ..f \u03c9, g t = \u222b (x : \u211d) in Ioc 0 (f \u03c9), g x ** exact intervalIntegral.integral_of_le (f_nn \u03c9) ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t \u03c9 : \u03b1 \u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioc 0 (f \u03c9)))] g ** filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f \u03c9)))]\n  with x hx using g_nn x hx.1 ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) \u22a2 \u222b\u207b (y : \u211d), \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y \u2202\u03bc =     \u222b\u207b (a : \u211d), indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) a ** apply congr_arg ** case h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) \u22a2 (fun y => \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y \u2202\u03bc) = fun a =>     indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) a ** funext s ** case h.h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 \u222b\u207b (x : \u03b1), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s \u2202\u03bc =     indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s ** simp_rw [aux\u2081] ** case h.h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 \u222b\u207b (x : \u03b1), ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u2202\u03bc =     indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s ** rw [lintegral_const_mul'] ** case h.h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator (Ici s) (fun x => 1) (f a) \u2202\u03bc =     indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s  case h.h.hr \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4 ** swap ** case h.h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator (Ici s) (fun x => 1) (f a) \u2202\u03bc =     indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s ** simp_rw [show\n    (fun a => (Ici s).indicator (fun _ : \u211d => (1 : \u211d\u22650\u221e)) (f a)) = fun a =>\n      {a : \u03b1 | s \u2264 f a}.indicator (fun _ => 1) a\n    by funext a; by_cases s \u2264 f a <;> simp [h]] ** case h.h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1), indicator {a | s \u2264 f a} (fun x => 1) a \u2202\u03bc =     indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s ** rw [lintegral_indicator\u2080] ** case h.h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1) in {a | s \u2264 f a}, 1 \u2202\u03bc =     indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s  case h.h.hs \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 NullMeasurableSet {a | s \u2264 f a} ** swap ** case h.h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * \u222b\u207b (a : \u03b1) in {a | s \u2264 f a}, 1 \u2202\u03bc =     indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a} * ENNReal.ofReal (g t)) s ** rw [lintegral_one, Measure.restrict_apply MeasurableSet.univ, univ_inter, indicator_mul_left,\n  mul_assoc,\n  show\n    (Ioi 0).indicator (fun _x : \u211d => (1 : \u211d\u22650\u221e)) s * \u03bc {a : \u03b1 | s \u2264 f a} =\n      (Ioi 0).indicator (fun _x : \u211d => 1 * \u03bc {a : \u03b1 | s \u2264 f a}) s\n    by by_cases 0 < s <;> simp [h]] ** case h.h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s =     indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a}) s * ENNReal.ofReal (g s) ** simp_rw [mul_comm _ (ENNReal.ofReal _), one_mul] ** case h.h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => \u2191\u2191\u03bc {a | s \u2264 f a}) s =     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun t => \u2191\u2191\u03bc {a | t \u2264 f a}) s ** rfl ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d \u22a2 (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) ** funext a ** case h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d a : \u03b1 \u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a) ** by_cases s \u2208 Ioc (0 : \u211d) (f a) ** case pos \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d a : \u03b1 h : s \u2208 Ioc 0 (f a) \u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a) ** simp only [h, show s \u2208 Ioi (0 : \u211d) from h.1, show f a \u2208 Ici s from h.2, indicator_of_mem,\n  mul_one] ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d a : \u03b1 h : \u00acs \u2208 Ioc 0 (f a) \u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a) ** have h_copy := h ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d a : \u03b1 h h_copy : \u00acs \u2208 Ioc 0 (f a) \u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a) ** simp only [mem_Ioc, not_and, not_le] at h ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d a : \u03b1 h_copy : \u00acs \u2208 Ioc 0 (f a) h : 0 < s \u2192 f a < s \u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a) ** by_cases h' : 0 < s ** case pos \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d a : \u03b1 h_copy : \u00acs \u2208 Ioc 0 (f a) h : 0 < s \u2192 f a < s h' : 0 < s \u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a) ** simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le, mul_zero] ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d a : \u03b1 h_copy : \u00acs \u2208 Ioc 0 (f a) h : 0 < s \u2192 f a < s h' : \u00ac0 < s \u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a) ** have : s \u2209 Ioi (0 : \u211d) := h' ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d a : \u03b1 h_copy : \u00acs \u2208 Ioc 0 (f a) h : 0 < s \u2192 f a < s h' : \u00ac0 < s this : \u00acs \u2208 Ioi 0 \u22a2 indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a) ** simp only [this, h', indicator_of_not_mem, not_false_iff, mul_zero,\n  zero_mul, mem_Ioc, false_and_iff] ** case h.h.hr \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4 ** apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) h : \u00ac0 < s \u22a2 indicator (Ioi 0) (fun x => 1) s \u2260 \u22a4 ** simp [h] ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 (fun a => indicator (Ici s) (fun x => 1) (f a)) = fun a => indicator {a | s \u2264 f a} (fun x => 1) a ** funext a ** case h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) a : \u03b1 \u22a2 indicator (Ici s) (fun x => 1) (f a) = indicator {a | s \u2264 f a} (fun x => 1) a ** by_cases s \u2264 f a <;> simp [h] ** case h.h.hs \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 NullMeasurableSet {a | s \u2264 f a} ** exact f_mble.nullMeasurable measurableSet_Ici ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s\u271d : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) s : \u211d aux\u2081 :   (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>     ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) \u22a2 indicator (Ioi 0) (fun _x => 1) s * \u2191\u2191\u03bc {a | s \u2264 f a} = indicator (Ioi 0) (fun _x => 1 * \u2191\u2191\u03bc {a | s \u2264 f a}) s ** by_cases 0 < s <;> simp [h] ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) \u22a2 (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2) ** funext p ** case h \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) p : \u03b1 \u00d7 \u211d \u22a2 Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) p =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) p ** cases p with | mk p_fst p_snd => ?_ ** case h.mk \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) p_fst : \u03b1 p_snd : \u211d \u22a2 Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) (p_fst, p_snd) =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd) ** rw [Function.uncurry_apply_pair] ** case h.mk \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) p_fst : \u03b1 p_snd : \u211d \u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd) ** by_cases p_snd \u2208 Ioc 0 (f p_fst) ** case pos \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) p_fst : \u03b1 p_snd : \u211d h : p_snd \u2208 Ioc 0 (f p_fst) \u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd) ** have h' : (p_fst, p_snd) \u2208 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := h ** case pos \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) p_fst : \u03b1 p_snd : \u211d h : p_snd \u2208 Ioc 0 (f p_fst) h' : (p_fst, p_snd) \u2208 {p | p.2 \u2208 Ioc 0 (f p.1)} \u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd) ** rw [Set.indicator_of_mem h', Set.indicator_of_mem h] ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) p_fst : \u03b1 p_snd : \u211d h : \u00acp_snd \u2208 Ioc 0 (f p_fst) \u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd) ** have h' : (p_fst, p_snd) \u2209 {p : \u03b1 \u00d7 \u211d | p.snd \u2208 Ioc 0 (f p.fst)} := h ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) p_fst : \u03b1 p_snd : \u211d h : \u00acp_snd \u2208 Ioc 0 (f p_fst) h' : \u00ac(p_fst, p_snd) \u2208 {p | p.2 \u2208 Ioc 0 (f p.1)} \u22a2 indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} (fun p => ENNReal.ofReal (g p.2)) (p_fst, p_snd) ** rw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h] ** \u03b1 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03b1 f : \u03b1 \u2192 \u211d g : \u211d \u2192 \u211d s : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc f_nn : 0 \u2264 f f_mble : Measurable f g_intble : \u2200 (t : \u211d), t > 0 \u2192 IntervalIntegrable g volume 0 t g_mble : Measurable g g_nn : \u2200 (t : \u211d), t > 0 \u2192 0 \u2264 g t g_intble' : \u2200 (t : \u211d), 0 \u2264 t \u2192 IntervalIntegrable g volume 0 t integrand_eq : \u2200 (\u03c9 : \u03b1), ENNReal.ofReal (\u222b (t : \u211d) in 0 ..f \u03c9, g t) = \u222b\u207b (t : \u211d) in Ioc 0 (f \u03c9), ENNReal.ofReal (g t) aux\u2082 :   (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =     indicator {p | p.2 \u2208 Ioc 0 (f p.1)} fun p => ENNReal.ofReal (g p.2) \u22a2 MeasurableSet {p | p.2 \u2208 Ioc 0 (f p.1)} ** simpa only [mem_univ, Pi.zero_apply, gt_iff_lt, not_lt, ge_iff_le, true_and] using\n  measurableSet_region_between_oc measurable_zero f_mble  MeasurableSet.univ ** Qed",
        "informal": ""
    },
    {
        "formal": "circleIntegral.norm_integral_le_of_norm_le_const ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 \u2264 R hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C this : |R| = R \u22a2 \u2200 (z : \u2102), z \u2208 sphere c |R| \u2192 \u2016f z\u2016 \u2264 C ** rwa [this] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 \u2264 R hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C this : |R| = R \u22a2 2 * \u03c0 * |R| * C = 2 * \u03c0 * R * C ** rw [this] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.leftpad_eq_leftpadTR ** \u22a2 @leftpad = @leftpadTR ** funext \u03b1 n a l ** case h.h.h.h \u03b1 : Type u_1 n : Nat a : \u03b1 l : List \u03b1 \u22a2 leftpad n a l = leftpadTR n a l ** simp [leftpad, leftpadTR, replicateTR_loop_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.restrict_ofFunction ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s : Set \u03b1 hm : Monotone m \u22a2 (fun t => m (t \u2229 s)) \u2205 = 0 ** simp ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s : Set \u03b1 hm : Monotone m \u22a2 m \u2205 = 0 ** simp [m_empty] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s : Set \u03b1 hm : Monotone m \u22a2 \u2191(restrict s) (OuterMeasure.ofFunction m m_empty) =     OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0) ** rw [restrict] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s : Set \u03b1 hm : Monotone m \u22a2 \u2191(LinearMap.comp (map Subtype.val) (comap Subtype.val)) (OuterMeasure.ofFunction m m_empty) =     OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0) ** simp only [LinearMap.comp_apply] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s : Set \u03b1 hm : Monotone m \u22a2 \u2191(map Subtype.val) (\u2191(comap Subtype.val) (OuterMeasure.ofFunction m m_empty)) =     OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0) ** rw [comap_ofFunction _ (Or.inl hm)] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s : Set \u03b1 hm : Monotone m \u22a2 \u2191(map Subtype.val)       (OuterMeasure.ofFunction (fun s_1 => m (Subtype.val '' s_1)) (_ : (fun s_1 => m (Subtype.val '' s_1)) \u2205 = 0)) =     OuterMeasure.ofFunction (fun t => m (t \u2229 s)) (_ : (fun t => m (t \u2229 s)) \u2205 = 0) ** simp only [map_ofFunction Subtype.coe_injective, Subtype.image_preimage_coe] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_zero_left ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc 0 C \u22a2 setToFun \u03bc 0 hT f = 0 ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc 0 C hf : Integrable f \u22a2 setToFun \u03bc 0 hT f = 0 ** rw [setToFun_eq hT hf] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc 0 C hf : Integrable f \u22a2 \u2191(L1.setToL1 hT) (Integrable.toL1 f hf) = 0 ** exact L1.setToL1_zero_left hT _ ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc 0 C hf : \u00acIntegrable f \u22a2 setToFun \u03bc 0 hT f = 0 ** exact setToFun_undef hT hf ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.eq_of_mem_uIcc_of_mem_uIcc' ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : DistribLattice \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 \u22a2 b \u2208 [[a, c]] \u2192 c \u2208 [[a, b]] \u2192 b = c ** simp_rw [mem_uIcc] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : DistribLattice \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 \u22a2 a \u2293 c \u2264 b \u2227 b \u2264 a \u2294 c \u2192 a \u2293 b \u2264 c \u2227 c \u2264 a \u2294 b \u2192 b = c ** exact Set.eq_of_mem_uIcc_of_mem_uIcc' ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.univ_eq_true_false ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1 x : Prop \u22a2 x \u2208 {True, False} ** rw [mem_insert_iff, mem_singleton_iff] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1 x : Prop \u22a2 x = True \u2228 x = False ** exact Classical.propComplete x ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.integrable_condCdf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d \u22a2 Integrable fun a => \u2191(condCdf \u03c1 a) x ** refine' integrable_of_forall_fin_meas_le _ (measure_lt_top \u03c1.fst univ) _ fun t _ _ => _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d \u22a2 AEStronglyMeasurable (fun a => \u2191(condCdf \u03c1 a) x) (Measure.fst \u03c1) ** exact (stronglyMeasurable_condCdf \u03c1 _).aestronglyMeasurable ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d t : Set \u03b1 x\u271d\u00b9 : MeasurableSet t x\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4 \u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191(condCdf \u03c1 x_1) x\u2016\u208a \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ ** have : \u2200 y, (\u2016condCdf \u03c1 y x\u2016\u208a : \u211d\u22650\u221e) \u2264 1 := by\n  intro y\n  rw [Real.nnnorm_of_nonneg (condCdf_nonneg _ _ _)]\n  simp only [ENNReal.coe_le_one_iff]\n  exact condCdf_le_one _ _ _ ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d t : Set \u03b1 x\u271d\u00b9 : MeasurableSet t x\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4 this : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1 \u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191(condCdf \u03c1 x_1) x\u2016\u208a \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ ** refine'\n  (set_lintegral_mono (measurable_condCdf _ _).ennnorm measurable_one fun y _ => this y).trans\n    _ ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d t : Set \u03b1 x\u271d\u00b9 : MeasurableSet t x\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4 this : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1 \u22a2 \u222b\u207b (x : \u03b1) in t, OfNat.ofNat 1 x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.fst \u03c1) univ ** simp only [Pi.one_apply, lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d t : Set \u03b1 x\u271d\u00b9 : MeasurableSet t x\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4 this : \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1 \u22a2 \u2191\u2191(Measure.fst \u03c1) t \u2264 \u2191\u2191(Measure.fst \u03c1) univ ** exact measure_mono (subset_univ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d t : Set \u03b1 x\u271d\u00b9 : MeasurableSet t x\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4 \u22a2 \u2200 (y : \u03b1), \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1 ** intro y ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d t : Set \u03b1 x\u271d\u00b9 : MeasurableSet t x\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4 y : \u03b1 \u22a2 \u2191\u2016\u2191(condCdf \u03c1 y) x\u2016\u208a \u2264 1 ** rw [Real.nnnorm_of_nonneg (condCdf_nonneg _ _ _)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d t : Set \u03b1 x\u271d\u00b9 : MeasurableSet t x\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4 y : \u03b1 \u22a2 \u2191{ val := \u2191(condCdf \u03c1 y) x, property := (_ : 0 \u2264 \u2191(condCdf \u03c1 y) x) } \u2264 1 ** simp only [ENNReal.coe_le_one_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d t : Set \u03b1 x\u271d\u00b9 : MeasurableSet t x\u271d : \u2191\u2191(Measure.fst \u03c1) t \u2260 \u22a4 y : \u03b1 \u22a2 { val := \u2191(condCdf \u03c1 y) x, property := (_ : 0 \u2264 \u2191(condCdf \u03c1 y) x) } \u2264 1 ** exact condCdf_le_one _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.setToSimpleFunc_indicator ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F \u22a2 setToSimpleFunc T (piecewise s hs (const \u03b1 x) (const \u03b1 0)) = \u2191(T s) x ** obtain rfl | hs_empty := s.eq_empty_or_nonempty ** case inr \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s \u22a2 setToSimpleFunc T (piecewise s hs (const \u03b1 x) (const \u03b1 0)) = \u2191(T s) x ** simp_rw [setToSimpleFunc] ** case inr \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s \u22a2 \u2211 x_1 in SimpleFunc.range (piecewise s hs (const \u03b1 x) (const \u03b1 0)),       \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =     \u2191(T s) x ** obtain rfl | hs_univ := eq_or_ne s univ ** case inr.inr \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ \u22a2 \u2211 x_1 in SimpleFunc.range (piecewise s hs (const \u03b1 x) (const \u03b1 0)),       \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =     \u2191(T s) x ** rw [range_indicator hs hs_empty hs_univ] ** case inr.inr \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ \u22a2 \u2211 x_1 in {x, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 = \u2191(T s) x ** by_cases hx0 : x = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u2211 x_1 in {x, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 = \u2191(T s) x ** rw [sum_insert] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x})) x +       \u2211 x_1 in {0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =     \u2191(T s) x  case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u00acx \u2208 {0} ** swap ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x})) x +       \u2211 x_1 in {0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =     \u2191(T s) x ** rw [sum_singleton, (T _).map_zero, add_zero] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x})) x = \u2191(T s) x ** congr ** case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x} = s ** simp only [coe_piecewise, piecewise_eq_indicator, coe_const, Pi.const_zero,\n  piecewise_eq_indicator] ** case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 indicator s (Function.const \u03b1 x) \u207b\u00b9' {x} = s ** rw [indicator_preimage, \u2190 Function.const_def, preimage_const_of_mem] ** case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 Set.ite s Set.univ (0 \u207b\u00b9' {x}) = s  case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 x \u2208 {x} ** swap ** case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 Set.ite s Set.univ (0 \u207b\u00b9' {x}) = s ** rw [\u2190 Pi.const_zero, \u2190 Function.const_def, preimage_const_of_not_mem] ** case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 Set.ite s Set.univ \u2205 = s  case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u00ac0 \u2208 {x} ** swap ** case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 Set.ite s Set.univ \u2205 = s ** simp ** case inl \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 x : F hs : MeasurableSet \u2205 \u22a2 setToSimpleFunc T (piecewise \u2205 hs (const \u03b1 x) (const \u03b1 0)) = \u2191(T \u2205) x ** simp only [hT_empty, ContinuousLinearMap.zero_apply, piecewise_empty, const_zero,\n  setToSimpleFunc_zero_apply] ** case inr.inl \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 x : F hs : MeasurableSet Set.univ hs_empty : Set.Nonempty Set.univ \u22a2 \u2211 x_1 in SimpleFunc.range (piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)),       \u2191(T (\u2191(piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =     \u2191(T Set.univ) x ** haveI h\u03b1 := hs_empty.to_type ** case inr.inl \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 x : F hs : MeasurableSet Set.univ hs_empty : Set.Nonempty Set.univ h\u03b1 : Nonempty \u03b1 \u22a2 \u2211 x_1 in SimpleFunc.range (piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)),       \u2191(T (\u2191(piecewise Set.univ hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 =     \u2191(T Set.univ) x ** simp [\u2190 Function.const_def] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : x = 0 \u22a2 \u2211 x_1 in {x, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 x) (const \u03b1 0)) \u207b\u00b9' {x_1})) x_1 = \u2191(T s) x ** simp_rw [hx0] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : x = 0 \u22a2 \u2211 x in {0, 0}, \u2191(T (\u2191(piecewise s hs (const \u03b1 0) (const \u03b1 0)) \u207b\u00b9' {x})) x = \u2191(T s) 0 ** simp ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u00acx \u2208 {0} ** rw [Finset.mem_singleton] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u00acx = 0 ** exact hx0 ** case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 x \u2208 {x} ** exact Set.mem_singleton x ** case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u00ac0 \u2208 {x} ** rw [Set.mem_singleton_iff] ** case neg.e_a.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' hT_empty : T \u2205 = 0 m : MeasurableSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s x : F hs_empty : Set.Nonempty s hs_univ : s \u2260 Set.univ hx0 : \u00acx = 0 \u22a2 \u00ac0 = x ** exact Ne.symm hx0 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.erase_inter_comm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t\u271d u v : Finset \u03b1 a\u271d b : \u03b1 s t : Finset \u03b1 a : \u03b1 \u22a2 erase s a \u2229 t = s \u2229 erase t a ** rw [erase_inter, inter_erase] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.compQuasiMeasurePreserving_eq_mk ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b3 inst\u271d\u00b9 : TopologicalSpace \u03b4 inst\u271d : MeasurableSpace \u03b2 \u03bd : Measure \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192\u2098[\u03bd] \u03b3 hf : QuasiMeasurePreserving f \u22a2 compQuasiMeasurePreserving g f hf = mk (\u2191g \u2218 f) (_ : AEStronglyMeasurable (\u2191g \u2218 f) \u03bc) ** rw [\u2190 compQuasiMeasurePreserving_mk g.aestronglyMeasurable hf, mk_coeFn] ** Qed",
        "informal": ""
    },
    {
        "formal": "Complex.volume_preserving_equiv_pi ** \u22a2 MeasurePreserving \u2191measurableEquivPi ** convert (measurableEquivPi.symm.measurable.measurePreserving volume).symm ** case h.e'_6  \u22a2 volume = Measure.map (\u2191(MeasurableEquiv.symm measurableEquivPi)) volume ** rw [\u2190 addHaarMeasure_eq_volume_pi, \u2190 Basis.parallelepiped_basisFun, \u2190 Basis.addHaar,\n  measurableEquivPi, Homeomorph.toMeasurableEquiv_symm_coe,\n  ContinuousLinearEquiv.symm_toHomeomorph, ContinuousLinearEquiv.coe_toHomeomorph,\n  Basis.map_addHaar, eq_comm] ** case h.e'_6  \u22a2 Basis.addHaar       (Basis.map (Pi.basisFun \u211d (Fin 2))         (ContinuousLinearEquiv.symm (LinearEquiv.toContinuousLinearEquiv (Basis.equivFun basisOneI))).toLinearEquiv) =     volume ** exact (Basis.addHaar_eq_iff _ _).mpr Complex.orthonormalBasisOneI.volume_parallelepiped ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.submartingale_of_expected_stoppedValue_mono ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf :   \u2200 (\u03c4 \u03c0 : \u03a9 \u2192 \u2115),     IsStoppingTime \ud835\udca2 \u03c4 \u2192       IsStoppingTime \ud835\udca2 \u03c0 \u2192         \u03c4 \u2264 \u03c0 \u2192 (\u2203 N, \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N) \u2192 \u222b (x : \u03a9), stoppedValue f \u03c4 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f \u03c0 x \u2202\u03bc \u22a2 Submartingale f \ud835\udca2 \u03bc ** refine' submartingale_of_set_integral_le hadp hint fun i j hij s hs => _ ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf :   \u2200 (\u03c4 \u03c0 : \u03a9 \u2192 \u2115),     IsStoppingTime \ud835\udca2 \u03c4 \u2192       IsStoppingTime \ud835\udca2 \u03c0 \u2192         \u03c4 \u2264 \u03c0 \u2192 (\u2203 N, \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N) \u2192 \u222b (x : \u03a9), stoppedValue f \u03c4 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f \u03c0 x \u2202\u03bc i j : \u2115 hij : i \u2264 j s : Set \u03a9 hs : MeasurableSet s \u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc ** classical\nspecialize hf (s.piecewise (fun _ => i) fun _ => j) _ (isStoppingTime_piecewise_const hij hs)\n  (isStoppingTime_const \ud835\udca2 j) (fun x => (ite_le_sup _ _ (x \u2208 s)).trans (max_eq_right hij).le)\n  \u27e8j, fun _ => le_rfl\u27e9\nrwa [stoppedValue_const, stoppedValue_piecewise_const,\n  integral_piecewise (\ud835\udca2.le _ _ hs) (hint _).integrableOn (hint _).integrableOn, \u2190\n  integral_add_compl (\ud835\udca2.le _ _ hs) (hint j), add_le_add_iff_right] at hf ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf :   \u2200 (\u03c4 \u03c0 : \u03a9 \u2192 \u2115),     IsStoppingTime \ud835\udca2 \u03c4 \u2192       IsStoppingTime \ud835\udca2 \u03c0 \u2192         \u03c4 \u2264 \u03c0 \u2192 (\u2203 N, \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 N) \u2192 \u222b (x : \u03a9), stoppedValue f \u03c4 x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f \u03c0 x \u2202\u03bc i j : \u2115 hij : i \u2264 j s : Set \u03a9 hs : MeasurableSet s \u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc ** specialize hf (s.piecewise (fun _ => i) fun _ => j) _ (isStoppingTime_piecewise_const hij hs)\n  (isStoppingTime_const \ud835\udca2 j) (fun x => (ite_le_sup _ _ (x \u2208 s)).trans (max_eq_right hij).le)\n  \u27e8j, fun _ => le_rfl\u27e9 ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) i j : \u2115 hij : i \u2264 j s : Set \u03a9 hs : MeasurableSet s hf :   \u222b (x : \u03a9), stoppedValue f (Set.piecewise s (fun x => i) fun x => j) x \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun x => j) x \u2202\u03bc \u22a2 \u222b (\u03c9 : \u03a9) in s, f i \u03c9 \u2202\u03bc \u2264 \u222b (\u03c9 : \u03a9) in s, f j \u03c9 \u2202\u03bc ** rwa [stoppedValue_const, stoppedValue_piecewise_const,\n  integral_piecewise (\ud835\udca2.le _ _ hs) (hint _).integrableOn (hint _).integrableOn, \u2190\n  integral_add_compl (\ud835\udca2.le _ _ hs) (hint j), add_le_add_iff_right] at hf ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.tendsto_condCdf_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 \u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1) ** have h_exists : \u2200 x : \u211d, \u2203 q : \u211a, x - 1 < q \u2227 \u2191q < x := fun x => exists_rat_btwn (sub_one_lt x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x \u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1) ** let qs : \u211d \u2192 \u211a := fun x => (h_exists x).choose ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) \u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1) ** have hqs_tendsto : Tendsto qs atTop atTop := by\n  rw [tendsto_atTop_atTop]\n  refine' fun q => \u27e8q + 1, fun y hy => _\u27e9\n  have h_le : y - 1 \u2264 qs y := (h_exists y).choose_spec.1.le\n  rw [sub_le_iff_le_add] at h_le\n  exact_mod_cast le_of_add_le_add_right (hy.trans h_le) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) hqs_tendsto : Tendsto qs atTop atTop \u22a2 Tendsto (\u2191(condCdf \u03c1 a)) atTop (\ud835\udcdd 1) ** refine'\n  tendsto_of_tendsto_of_tendsto_of_le_of_le ((tendsto_condCdfRat_atTop \u03c1 a).comp hqs_tendsto)\n    tendsto_const_nhds _ (condCdf_le_one \u03c1 a) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) hqs_tendsto : Tendsto qs atTop atTop \u22a2 condCdfRat \u03c1 a \u2218 qs \u2264 \u2191(condCdf \u03c1 a) ** intro x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) hqs_tendsto : Tendsto qs atTop atTop x : \u211d \u22a2 (condCdfRat \u03c1 a \u2218 qs) x \u2264 \u2191(condCdf \u03c1 a) x ** rw [Function.comp_apply, \u2190 condCdf_eq_condCdfRat] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) hqs_tendsto : Tendsto qs atTop atTop x : \u211d \u22a2 \u2191(condCdf \u03c1 a) \u2191(qs x) \u2264 \u2191(condCdf \u03c1 a) x ** exact (condCdf \u03c1 a).mono (le_of_lt (h_exists x).choose_spec.2) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) \u22a2 Tendsto qs atTop atTop ** rw [tendsto_atTop_atTop] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) \u22a2 \u2200 (b : \u211a), \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 b \u2264 qs a ** refine' fun q => \u27e8q + 1, fun y hy => _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) q : \u211a y : \u211d hy : \u2191q + 1 \u2264 y \u22a2 q \u2264 qs y ** have h_le : y - 1 \u2264 qs y := (h_exists y).choose_spec.1.le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) q : \u211a y : \u211d hy : \u2191q + 1 \u2264 y h_le : y - 1 \u2264 \u2191(qs y) \u22a2 q \u2264 qs y ** rw [sub_le_iff_le_add] at h_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 h_exists : \u2200 (x : \u211d), \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x qs : \u211d \u2192 \u211a := fun x => Exists.choose (_ : \u2203 q, x - 1 < \u2191q \u2227 \u2191q < x) q : \u211a y : \u211d hy : \u2191q + 1 \u2264 y h_le : y \u2264 \u2191(qs y) + 1 \u22a2 q \u2264 qs y ** exact_mod_cast le_of_add_le_add_right (hy.trans h_le) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.preimage_mul_left_one' ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : Group \u03b1 s t : Finset \u03b1 a b : \u03b1 \u22a2 preimage 1 ((fun x x_1 => x * x_1) a\u207b\u00b9)       (_ : Set.InjOn ((fun x x_1 => x * x_1) a\u207b\u00b9) ((fun x x_1 => x * x_1) a\u207b\u00b9 \u207b\u00b9' \u21911)) =     {a} ** rw [preimage_mul_left_one, inv_inv] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.IsSFiniteKernel.withDensity ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 \u22a2 IsSFiniteKernel (kernel.withDensity \u03ba f) ** have h_eq_sum : withDensity \u03ba f = kernel.sum fun i => withDensity (seq \u03ba i) f := by\n  rw [\u2190 withDensity_kernel_sum _ _]\n  congr\n  exact (kernel_sum_seq \u03ba).symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 h_eq_sum : kernel.withDensity \u03ba f = kernel.sum fun i => kernel.withDensity (seq \u03ba i) f \u22a2 IsSFiniteKernel (kernel.withDensity \u03ba f) ** rw [h_eq_sum] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 h_eq_sum : kernel.withDensity \u03ba f = kernel.sum fun i => kernel.withDensity (seq \u03ba i) f \u22a2 IsSFiniteKernel (kernel.sum fun i => kernel.withDensity (seq \u03ba i) f) ** exact isSFiniteKernel_sum fun n =>\n  isSFiniteKernel_withDensity_of_isFiniteKernel (seq \u03ba n) hf_ne_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 \u22a2 kernel.withDensity \u03ba f = kernel.sum fun i => kernel.withDensity (seq \u03ba i) f ** rw [\u2190 withDensity_kernel_sum _ _] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 \u22a2 kernel.withDensity \u03ba f = kernel.withDensity (kernel.sum fun i => seq \u03ba i) f ** congr ** case e_\u03ba \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba hf_ne_top : \u2200 (a : \u03b1) (b : \u03b2), f a b \u2260 \u22a4 \u22a2 \u03ba = kernel.sum fun i => seq \u03ba i ** exact (kernel_sum_seq \u03ba).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "ae_restrict_iff_subtype ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s p : \u03b1 \u2192 Prop \u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, p x) \u2194 \u2200\u1d50 (x : \u2191s) \u2202Measure.comap Subtype.val \u03bc, p \u2191x ** rw [\u2190 map_comap_subtype_coe hs, (MeasurableEmbedding.subtype_coe hs).ae_map_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.bit1_ne_zero ** m : \u2124 \u22a2 bit1 m \u2260 0 ** simpa only [bit0_zero] using bit1_ne_bit0 m 0 ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM1.stmts_trans ** \u0393 : Type u_1 inst\u271d : Inhabited \u0393 \u039b : Type u_2 \u03c3 : Type u_3 M : \u039b \u2192 Stmt\u2081 S : Finset \u039b q\u2081 q\u2082 : Stmt\u2081 h\u2081 : q\u2081 \u2208 stmts\u2081 q\u2082 \u22a2 some q\u2082 \u2208 stmts M S \u2192 some q\u2081 \u2208 stmts M S ** simp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq,\n  forall_eq', exists_imp, and_imp] ** \u0393 : Type u_1 inst\u271d : Inhabited \u0393 \u039b : Type u_2 \u03c3 : Type u_3 M : \u039b \u2192 Stmt\u2081 S : Finset \u039b q\u2081 q\u2082 : Stmt\u2081 h\u2081 : q\u2081 \u2208 stmts\u2081 q\u2082 \u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q\u2082 \u2208 stmts\u2081 (M x) \u2192 \u2203 a, a \u2208 S \u2227 q\u2081 \u2208 stmts\u2081 (M a) ** exact fun l ls h\u2082 \u21a6 \u27e8_, ls, stmts\u2081_trans h\u2082 h\u2081\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_injective ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 \u22a2 Injective (image f) \u2194 Injective f ** refine' \u27e8fun h x x' hx => _, Injective.image_injective\u27e9 ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Injective (image f) x x' : \u03b1 hx : f x = f x' \u22a2 x = x' ** rw [\u2190 singleton_eq_singleton_iff] ** \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Injective (image f) x x' : \u03b1 hx : f x = f x' \u22a2 {x} = {x'} ** apply h ** case a \u03b1 : Type u \u03b2 : Type v f : \u03b1 \u2192 \u03b2 h : Injective (image f) x x' : \u03b1 hx : f x = f x' \u22a2 f '' {x} = f '' {x'} ** rw [image_singleton, image_singleton, hx] ** Qed",
        "informal": ""
    },
    {
        "formal": "StieltjesFunction.iInf_rat_gt_eq ** f\u271d f : StieltjesFunction x : \u211d \u22a2 \u2a05 r, \u2191f \u2191\u2191r = \u2191f x ** rw [\u2190 iInf_Ioi_eq f x] ** f\u271d f : StieltjesFunction x : \u211d \u22a2 \u2a05 r, \u2191f \u2191\u2191r = \u2a05 r, \u2191f \u2191r ** refine' (Real.iInf_Ioi_eq_iInf_rat_gt _ _ f.mono).symm ** f\u271d f : StieltjesFunction x : \u211d \u22a2 BddBelow (\u2191f '' Ioi x) ** refine' \u27e8f x, fun y => _\u27e9 ** f\u271d f : StieltjesFunction x y : \u211d \u22a2 y \u2208 \u2191f '' Ioi x \u2192 \u2191f x \u2264 y ** rintro \u27e8y, hy_mem, rfl\u27e9 ** case intro.intro f\u271d f : StieltjesFunction x y : \u211d hy_mem : y \u2208 Ioi x \u22a2 \u2191f x \u2264 \u2191f y ** exact f.mono (le_of_lt hy_mem) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** let u n := disjointed (spanningSets \u03bc) n ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** have u_meas : \u2200 n, MeasurableSet (u n) := by\n  intro n\n  apply MeasurableSet.disjointed fun i => ?_\n  exact measurable_spanningSets \u03bc i ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** have A : s = \u22c3 n, s \u2229 u n := by\n  rw [\u2190 inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n \u22a2 \u2200 (n : \u2115), MeasurableSet (u n) ** intro n ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n n : \u2115 \u22a2 MeasurableSet (u n) ** apply MeasurableSet.disjointed fun i => ?_ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n n i : \u2115 \u22a2 MeasurableSet (spanningSets \u03bc i) ** exact measurable_spanningSets \u03bc i ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) \u22a2 s = \u22c3 n, s \u2229 u n ** rw [\u2190 inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n)) ** conv_lhs => rw [A, image_iUnion] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n \u22a2 \u2191\u2191\u03bc (\u22c3 i, f '' (s \u2229 u i)) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n)) ** exact measure_iUnion_le _ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n \u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n)) \u2264     \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** apply ENNReal.tsum_le_tsum fun n => ?_ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n n : \u2115 \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 u n)) \u2264 \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** apply\n  addHaar_image_le_lintegral_abs_det_fderiv_aux2 \u03bc (hs.inter (u_meas n)) _ fun x hx =>\n    (hf' x hx.1).mono (inter_subset_left _ _) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n n : \u2115 \u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4 ** have : \u03bc (u n) < \u221e :=\n  lt_of_le_of_lt (measure_mono (disjointed_subset _ _)) (measure_spanningSets_lt_top \u03bc n) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n n : \u2115 this : \u2191\u2191\u03bc (u n) < \u22a4 \u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4 ** exact ne_of_lt (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) this) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n \u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =     \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** conv_rhs => rw [A] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n \u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =     \u222b\u207b (x : E) in \u22c3 n, s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** rw [lintegral_iUnion] ** case hm E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n \u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 u i) ** intro n ** case hm E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n n : \u2115 \u22a2 MeasurableSet (s \u2229 u n) ** exact hs.inter (u_meas n) ** case hd E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x u : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n u_meas : \u2200 (n : \u2115), MeasurableSet (u n) A : s = \u22c3 n, s \u2229 u n \u22a2 Pairwise (Disjoint on fun n => s \u2229 u n) ** exact pairwise_disjoint_mono (disjoint_disjointed _) fun n => inter_subset_right _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.integral_eq_lintegral ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : SMulCommClass \u211d \ud835\udd5c E f : \u03b1 \u2192\u209b \u211d hf : Integrable \u2191f h_pos : 0 \u2264\u1d50[\u03bc] \u2191f \u22a2 integral \u03bc f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191f a) \u2202\u03bc) ** have : f =\u1d50[\u03bc] f.map (ENNReal.toReal \u2218 ENNReal.ofReal) :=\n  h_pos.mono fun a h => (ENNReal.toReal_ofReal h).symm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : SMulCommClass \u211d \ud835\udd5c E f : \u03b1 \u2192\u209b \u211d hf : Integrable \u2191f h_pos : 0 \u2264\u1d50[\u03bc] \u2191f this : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f) \u22a2 integral \u03bc f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191f a) \u2202\u03bc) ** rw [\u2190 integral_eq_lintegral' hf] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : SMulCommClass \u211d \ud835\udd5c E f : \u03b1 \u2192\u209b \u211d hf : Integrable \u2191f h_pos : 0 \u2264\u1d50[\u03bc] \u2191f this : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f) \u22a2 integral \u03bc f = integral \u03bc (map (ENNReal.toReal \u2218 ENNReal.ofReal) f)  case hg0 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : SMulCommClass \u211d \ud835\udd5c E f : \u03b1 \u2192\u209b \u211d hf : Integrable \u2191f h_pos : 0 \u2264\u1d50[\u03bc] \u2191f this : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f) \u22a2 ENNReal.ofReal 0 = 0  case ht \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : SMulCommClass \u211d \ud835\udd5c E f : \u03b1 \u2192\u209b \u211d hf : Integrable \u2191f h_pos : 0 \u2264\u1d50[\u03bc] \u2191f this : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f) \u22a2 \u2200 (b : \u211d), ENNReal.ofReal b \u2260 \u22a4 ** exacts [integral_congr hf this, ENNReal.ofReal_zero, fun b => ENNReal.ofReal_ne_top] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.IsPrefix.filter ** \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l\u2081 l\u2082 : List \u03b1 h : l\u2081 <+: l\u2082 \u22a2 List.filter p l\u2081 <+: List.filter p l\u2082 ** obtain \u27e8xs, rfl\u27e9 := h ** case intro \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l\u2081 xs : List \u03b1 \u22a2 List.filter p l\u2081 <+: List.filter p (l\u2081 ++ xs) ** rw [filter_append] ** case intro \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l\u2081 xs : List \u03b1 \u22a2 List.filter p l\u2081 <+: List.filter p l\u2081 ++ List.filter p xs ** apply prefix_append ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.isCompl_range_inl_range_inr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f : \u03b9 \u2192 \u03b1 s t : Set \u03b1 \u22a2 range Sum.inl \u2293 range Sum.inr \u2264 \u22a5 ** rintro y \u27e8\u27e8x\u2081, rfl\u27e9, \u27e8x\u2082, h\u27e9\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f : \u03b9 \u2192 \u03b1 s t : Set \u03b1 x\u2081 : \u03b1 x\u2082 : \u03b2 h : Sum.inr x\u2082 = Sum.inl x\u2081 \u22a2 Sum.inl x\u2081 \u2208 \u22a5 ** exact Sum.noConfusion h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f : \u03b9 \u2192 \u03b1 s t : Set \u03b1 \u22a2 \u22a4 \u2264 range Sum.inl \u2294 range Sum.inr ** rintro (x | y) - <;> [left; right] <;> exact mem_range_self _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Mem\u2112p.snorm_indicator_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** have h\u2112p := hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : Mem\u2112p f p \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p \u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8\u27e8f', hf', heq\u27e9, _\u27e9 := hf ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := (h\u2112p.ae_eq heq).snorm_indicator_le_of_meas \u03bc hp_one hp_top hf' h\u03b5 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5 \u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** refine' \u27e8\u03b4, h\u03b4pos, fun s hs h\u03bcs => _\u27e9 ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5 ** convert h\u03b4 s hs h\u03bcs using 1 ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 snorm (Set.indicator s f) p \u03bc = snorm (Set.indicator s f') p \u03bc ** rw [snorm_indicator_eq_snorm_restrict hs, snorm_indicator_eq_snorm_restrict hs] ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b1 \u2192 \u03b2 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 h\u2112p : Mem\u2112p f p right\u271d : snorm f p \u03bc < \u22a4 f' : \u03b1 \u2192 \u03b2 hf' : StronglyMeasurable f' heq : f =\u1d50[\u03bc] f' \u03b4 : \u211d h\u03b4pos : 0 < \u03b4 h\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5 s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u22a2 snorm f p (Measure.restrict \u03bc s) = snorm f' p (Measure.restrict \u03bc s) ** refine' snorm_congr_ae heq.restrict ** Qed",
        "informal": ""
    },
    {
        "formal": "UFModel.Models.push ** case g \u03b1 : Type u_1 arr : Array (UFNode \u03b1) n : \u2115 m : UFModel n H : Models arr m k : \u2115 hk : k = n + 1 x : \u03b1 \u22a2 (Agrees arr (fun x => x.rank) fun i => rank m \u2191i) \u2192     Agrees (Array.push arr { parent := n, value := x, rank := 0 }) (fun x => x.rank) fun i =>       rank (UFModel.push m k (_ : n \u2264 k)) \u2191i ** intro H ** case g \u03b1 : Type u_1 arr : Array (UFNode \u03b1) n : \u2115 m : UFModel n H\u271d : Models arr m k : \u2115 hk : k = n + 1 x : \u03b1 H : Agrees arr (fun x => x.rank) fun i => rank m \u2191i \u22a2 Agrees (Array.push arr { parent := n, value := x, rank := 0 }) (fun x => x.rank) fun i =>     rank (UFModel.push m k (_ : n \u2264 k)) \u2191i ** refine H.push _ hk _ _ (fun i h \u21a6 ?_) (fun h \u21a6 ?_) <;>\nsimp [UFModel.push, h, lt_irrefl] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0) ** rw [ENNReal.tendsto_atTop_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** intro \u03b5 h\u03b5 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** by_cases \u03b5 < \u221e ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u00ac\u03b5 < \u22a4 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** by_cases h\u03bc : \u03bc = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** have h\u03b5' : 0 < \u03b5.toReal / 3 :=\n  div_pos (ENNReal.toReal_pos (gt_iff_lt.1 h\u03b5).ne.symm h.ne) (by norm_num) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** have hdivp : 0 \u2264 1 / p.toReal := by\n  refine' one_div_nonneg.2 _\n  rw [\u2190 ENNReal.zero_toReal, ENNReal.toReal_le_toReal ENNReal.zero_ne_top hp']\n  exact le_trans (zero_le _) hp ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** have hpow : 0 < measureUnivNNReal \u03bc ^ (1 / p.toReal) :=\n  Real.rpow_pos_of_pos (measureUnivNNReal_pos h\u03bc) _ ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** obtain \u27e8\u03b4\u2081, h\u03b4\u2081, hsnorm\u2081\u27e9 := hui h\u03b5' ** case neg.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** obtain \u27e8\u03b4\u2082, h\u03b4\u2082, hsnorm\u2082\u27e9 := hg'.snorm_indicator_le \u03bc hp hp' h\u03b5' ** case neg.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** obtain \u27e8t, htm, ht\u2081, ht\u2082\u27e9 := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg (lt_min h\u03b4\u2081 h\u03b4\u2082) ** case neg.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) ht\u2082 : TendstoUniformlyOn (fun n => f n) g atTop t\u1d9c \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** rw [Metric.tendstoUniformlyOn_iff] at ht\u2082 ** case neg.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) ht\u2082 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < \u03b5 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** specialize ht\u2082 (\u03b5.toReal / (3 * measureUnivNNReal \u03bc ^ (1 / p.toReal)))\n  (div_pos (ENNReal.toReal_pos (gt_iff_lt.1 h\u03b5).ne.symm h.ne) (mul_pos (by norm_num) hpow)) ** case neg.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) ht\u2082 :   \u2200\u1da0 (n : \u2115) in atTop,     \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** obtain \u27e8N, hN\u27e9 := eventually_atTop.1 ht\u2082 ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) ht\u2082 :   \u2200\u1da0 (n : \u2115) in atTop,     \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** clear ht\u2082 ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** refine' \u27e8N, fun n hn => _\u27e9 ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N \u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5 ** rw [\u2190 t.indicator_self_add_compl (f n - g)] ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N \u22a2 snorm (indicator t (f n - g) + indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 ** refine' le_trans (snorm_add_le (((hf n).sub hg).indicator htm).aestronglyMeasurable\n  (((hf n).sub hg).indicator htm.compl).aestronglyMeasurable hp) _ ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N \u22a2 snorm (indicator t (f n - g)) p \u03bc + snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 ** rw [sub_eq_add_neg, Set.indicator_add' t, Set.indicator_neg'] ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N \u22a2 snorm (indicator t (f n) + -indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5 ** refine' le_trans (add_le_add_right (snorm_add_le ((hf n).indicator htm).aestronglyMeasurable\n  (hg.indicator htm).neg.aestronglyMeasurable hp) _) _ ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N \u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5 ** have hnf : snorm (t.indicator (f n)) p \u03bc \u2264 ENNReal.ofReal (\u03b5.toReal / 3) := by\n  refine' hsnorm\u2081 n t htm (le_trans ht\u2081 _)\n  rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2081.le]\n  exact min_le_left _ _ ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5 ** have hng : snorm (t.indicator g) p \u03bc \u2264 ENNReal.ofReal (\u03b5.toReal / 3) := by\n  refine' hsnorm\u2082 t htm (le_trans ht\u2081 _)\n  rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2082.le]\n  exact min_le_right _ _ ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5 ** have : ENNReal.ofReal (\u03b5.toReal / 3) = \u03b5 / 3 := by\n  rw [ENNReal.ofReal_div_of_pos (show (0 : \u211d) < 3 by norm_num), ENNReal.ofReal_toReal h.ne]\n  simp ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) this : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3 \u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5 ** rw [this] at hnf hng hlt ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 \u03b5 / 3 hng : snorm (indicator t g) p \u03bc \u2264 \u03b5 / 3 hlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 / 3 this : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3 \u22a2 snorm (indicator t (f n)) p \u03bc + snorm (-indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n + -g)) p \u03bc \u2264 \u03b5 ** rw [snorm_neg, \u2190 ENNReal.add_thirds \u03b5, \u2190 sub_eq_add_neg] ** case neg.intro.intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 \u03b5 / 3 hng : snorm (indicator t g) p \u03bc \u2264 \u03b5 / 3 hlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 / 3 this : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3 \u22a2 snorm (indicator t (f n)) p \u03bc + snorm (indicator t g) p \u03bc + snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 \u03b5 / 3 + \u03b5 / 3 + \u03b5 / 3 ** exact add_le_add_three hnf hng hlt ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u00ac\u03b5 < \u22a4 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** rw [not_lt, top_le_iff] at h ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 = \u22a4 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** exact \u27e80, fun n _ => by simp [h]\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 = \u22a4 n : \u2115 x\u271d : n \u2265 0 \u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5 ** simp [h] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u03bc = 0 \u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5 ** exact \u27e80, fun n _ => by simp [h\u03bc]\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u03bc = 0 n : \u2115 x\u271d : n \u2265 0 \u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5 ** simp [h\u03bc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 \u22a2 0 < 3 ** norm_num ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 \u22a2 0 \u2264 1 / ENNReal.toReal p ** refine' one_div_nonneg.2 _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 \u22a2 0 \u2264 ENNReal.toReal p ** rw [\u2190 ENNReal.zero_toReal, ENNReal.toReal_le_toReal ENNReal.zero_ne_top hp'] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 \u22a2 0 \u2264 p ** exact le_trans (zero_le _) hp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) ht\u2082 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < \u03b5 \u22a2 0 < 3 ** norm_num ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N \u22a2 snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) ** refine' hsnorm\u2081 n t htm (le_trans ht\u2081 _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N \u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2081 ** rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2081.le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N \u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2081 ** exact min_le_left _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) ** refine' hsnorm\u2082 t htm (le_trans ht\u2081 _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2082 ** rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2082.le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2082 ** exact min_le_right _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) ** specialize hN n hn ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) \u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) ** have : 0 \u2264 \u03b5.toReal / (3 * measureUnivNNReal \u03bc ^ (1 / p.toReal)) := by\n  rw [div_mul_eq_div_mul_one_div]\n  exact mul_nonneg h\u03b5'.le (one_div_nonneg.2 hpow.le) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) \u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) ** have := snorm_sub_le_of_dist_bdd \u03bc hp' htm.compl this fun x hx =>\n  (dist_comm (g x) (f n x) \u25b8 (hN x hx).le :\n    dist (f n x) (g x) \u2264 \u03b5.toReal / (3 * measureUnivNNReal \u03bc ^ (1 / p.toReal))) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this :   snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u22a2 snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) ** refine' le_trans this _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this :   snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u22a2 ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) ** rw [div_mul_eq_div_mul_one_div, \u2190 ENNReal.ofReal_toReal (measure_lt_top \u03bc t\u1d9c).ne,\n  ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg hdivp, \u2190 ENNReal.ofReal_mul, mul_assoc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) \u22a2 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) ** rw [div_mul_eq_div_mul_one_div] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) \u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) ** exact mul_nonneg h\u03b5'.le (one_div_nonneg.2 hpow.le) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this :   snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u22a2 ENNReal.ofReal       (ENNReal.toReal \u03b5 / 3 *         (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc t\u1d9c) ^ (1 / ENNReal.toReal p))) \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) ** refine' ENNReal.ofReal_le_ofReal (mul_le_of_le_one_right h\u03b5'.le _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this :   snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u22a2 1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc t\u1d9c) ^ (1 / ENNReal.toReal p) \u2264 1 ** rw [mul_comm, mul_one_div, div_le_one] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this :   snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u22a2 ENNReal.toReal (\u2191\u2191\u03bc t\u1d9c) ^ (1 / ENNReal.toReal p) \u2264 \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) ** refine' Real.rpow_le_rpow ENNReal.toReal_nonneg\n  (ENNReal.toReal_le_of_le_ofReal (measureUnivNNReal_pos h\u03bc).le _) hdivp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this :   snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u22a2 \u2191\u2191\u03bc t\u1d9c \u2264 ENNReal.ofReal \u2191(measureUnivNNReal \u03bc) ** rw [ENNReal.ofReal_coe_nnreal, coe_measureUnivNNReal] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this :   snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u22a2 \u2191\u2191\u03bc t\u1d9c \u2264 \u2191\u2191\u03bc univ ** exact measure_mono (Set.subset_univ _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this :   snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u22a2 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) ** exact Real.rpow_pos_of_pos (measureUnivNNReal_pos h\u03bc) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) this :   snorm (indicator t\u1d9c ((fun x => f n x) - fun x => g x)) p \u03bc \u2264     ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *       \u2191\u2191\u03bc t\u1d9c ^ (1 / ENNReal.toReal p) \u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) ** refine' mul_nonneg h\u03b5'.le (one_div_nonneg.2 hpow.le) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3 ** rw [ENNReal.ofReal_div_of_pos (show (0 : \u211d) < 3 by norm_num), ENNReal.ofReal_toReal h.ne] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 \u03b5 / ENNReal.ofReal 3 = \u03b5 / 3 ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e inst\u271d : IsFiniteMeasure \u03bc hp : 1 \u2264 p hp' : p \u2260 \u22a4 f : \u2115 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 hf : \u2200 (n : \u2115), StronglyMeasurable (f n) hg : StronglyMeasurable g hg' : Mem\u2112p g p hui : UnifIntegrable f p \u03bc hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 > 0 h : \u03b5 < \u22a4 h\u03bc : \u00ac\u03bc = 0 h\u03b5' : 0 < ENNReal.toReal \u03b5 / 3 hdivp : 0 \u2264 1 / ENNReal.toReal p hpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) \u03b4\u2081 : \u211d h\u03b4\u2081 : 0 < \u03b4\u2081 hsnorm\u2081 :   \u2200 (i : \u2115) (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u03b4\u2082 : \u211d h\u03b4\u2082 : 0 < \u03b4\u2082 hsnorm\u2082 :   \u2200 (s : Set \u03b1),     MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) t : Set \u03b1 htm : MeasurableSet t ht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) N : \u2115 hN :   \u2200 (b : \u2115),     b \u2265 N \u2192       \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)) n : \u2115 hn : n \u2265 N hnf : snorm (indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hng : snorm (indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) hlt : snorm (indicator t\u1d9c (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) \u22a2 0 < 3 ** norm_num ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s \u22a2 \u2191\u2191\u03bc s = 0 ** suffices H : \u2200 R, \u03bc (s \u2229 closedBall 0 R) = 0 ** case H E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s \u22a2 \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0 ** intro R ** case H E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s R : \u211d \u22a2 \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0 ** apply addHaar_eq_zero_of_disjoint_translates_aux \u03bc u\n  (isBounded_closedBall.subset (inter_subset_right _ _)) hu _ (h's.inter measurableSet_closedBall) ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s R : \u211d \u22a2 Pairwise (Disjoint on fun n => {u n} + s \u2229 closedBall 0 R) ** refine pairwise_disjoint_mono hs fun n => ?_ ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s R : \u211d n : \u2115 \u22a2 {u n} + s \u2229 closedBall 0 R \u2264 {u n} + s ** exact add_subset_add Subset.rfl (inter_subset_left _ _) ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s H : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0 \u22a2 \u2191\u2191\u03bc s = 0 ** apply le_antisymm _ (zero_le _) ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s H : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0 \u22a2 \u2191\u2191\u03bc s \u2264 0 ** calc\n  \u03bc s \u2264 \u2211' n : \u2115, \u03bc (s \u2229 closedBall 0 n) := by\n    conv_lhs => rw [\u2190 iUnion_inter_closedBall_nat s 0]\n    exact measure_iUnion_le _\n  _ = 0 := by simp only [H, tsum_zero] ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s H : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0 \u22a2 \u2191\u2191\u03bc s \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n) ** conv_lhs => rw [\u2190 iUnion_inter_closedBall_nat s 0] ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s H : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0 \u22a2 \u2191\u2191\u03bc (\u22c3 n, s \u2229 closedBall 0 \u2191n) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n) ** exact measure_iUnion_le _ ** E : Type u_1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : BorelSpace E inst\u271d\u00b9 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc s : Set E u : \u2115 \u2192 E hu : Bornology.IsBounded (range u) hs : Pairwise (Disjoint on fun n => {u n} + s) h's : MeasurableSet s H : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0 \u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n) = 0 ** simp only [H, tsum_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.norm_integral_condexpKernel ** \u03a9 : Type u_1 F : Type u_2 inst\u271d\u2077 : TopologicalSpace \u03a9 m m\u03a9 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsFiniteMeasure \u03bc inst\u271d\u00b2 : NormedAddCommGroup F f : \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hf_int : Integrable f \u22a2 Integrable fun \u03c9 => \u2016\u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9\u2016 ** rw [condexpKernel] ** \u03a9 : Type u_1 F : Type u_2 inst\u271d\u2077 : TopologicalSpace \u03a9 m m\u03a9 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsFiniteMeasure \u03bc inst\u271d\u00b2 : NormedAddCommGroup F f : \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hf_int : Integrable f \u22a2 Integrable fun \u03c9 => \u2016\u222b (y : \u03a9), f y \u2202\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9\u2016 ** exact Integrable.norm_integral_condDistrib\n  (aemeasurable_id'' \u03bc (inf_le_right : m \u2293 m\u03a9 \u2264 m\u03a9)) aemeasurable_id\n  (hf_int.comp_snd_map_prod_id (inf_le_right : m \u2293 m\u03a9 \u2264 m\u03a9)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.getD_eq_get? ** \u03b1 : Type u_1 a : Array \u03b1 n : Nat d : \u03b1 \u22a2 getD a n d = Option.getD (get? a n) d ** simp [get?, getD] ** \u03b1 : Type u_1 a : Array \u03b1 n : Nat d : \u03b1 \u22a2 (if h : n < size a then a[n] else d) = Option.getD (if h : n < size a then some a[n] else none) d ** split <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L2.snorm_inner_lt_top ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } \u22a2 snorm (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) 1 \u03bc < \u22a4 ** have h : \u2200 x, \u2016\u27eaf x, g x\u27eb\u2016 \u2264 \u2016\u2016f x\u2016 ^ (2 : \u211d) + \u2016g x\u2016 ^ (2 : \u211d)\u2016 := by\n  intro x\n  rw [\u2190 @Nat.cast_two \u211d, Real.rpow_nat_cast, Real.rpow_nat_cast]\n  calc\n    \u2016\u27eaf x, g x\u27eb\u2016 \u2264 \u2016f x\u2016 * \u2016g x\u2016 := norm_inner_le_norm _ _\n    _ \u2264 2 * \u2016f x\u2016 * \u2016g x\u2016 :=\n      (mul_le_mul_of_nonneg_right (le_mul_of_one_le_left (norm_nonneg _) one_le_two)\n        (norm_nonneg _))\n    _ \u2264 \u2016\u2016f x\u2016 ^ 2 + \u2016g x\u2016 ^ 2\u2016 := (two_mul_le_add_sq _ _).trans (le_abs_self _) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } h : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016 \u22a2 snorm (fun x => inner (\u2191\u2191f x) (\u2191\u2191g x)) 1 \u03bc < \u22a4 ** refine' (snorm_mono_ae (ae_of_all _ h)).trans_lt ((snorm_add_le _ _ le_rfl).trans_lt _) ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } h : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016 \u22a2 snorm (fun a => \u2016\u2191\u2191f a\u2016 ^ 2) 1 \u03bc + snorm (fun a => \u2016\u2191\u2191g a\u2016 ^ 2) 1 \u03bc < \u22a4 ** rw [ENNReal.add_lt_top] ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } h : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016 \u22a2 snorm (fun a => \u2016\u2191\u2191f a\u2016 ^ 2) 1 \u03bc < \u22a4 \u2227 snorm (fun a => \u2016\u2191\u2191g a\u2016 ^ 2) 1 \u03bc < \u22a4 ** exact \u27e8snorm_rpow_two_norm_lt_top f, snorm_rpow_two_norm_lt_top g\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } \u22a2 \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016 ** intro x ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } x : \u03b1 \u22a2 \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016 ** calc\n  \u2016\u27eaf x, g x\u27eb\u2016 \u2264 \u2016f x\u2016 * \u2016g x\u2016 := norm_inner_le_norm _ _\n  _ \u2264 2 * \u2016f x\u2016 * \u2016g x\u2016 :=\n    (mul_le_mul_of_nonneg_right (le_mul_of_one_le_left (norm_nonneg _) one_le_two)\n      (norm_nonneg _))\n  _ \u2264 \u2016\u2016f x\u2016 ^ 2 + \u2016g x\u2016 ^ 2\u2016 := (two_mul_le_add_sq _ _).trans (le_abs_self _) ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } h : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016 \u22a2 AEStronglyMeasurable (fun a => \u2016\u2191\u2191f a\u2016 ^ 2) \u03bc ** exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d : NormedAddCommGroup F f g : { x // x \u2208 Lp E 2 } h : \u2200 (x : \u03b1), \u2016inner (\u2191\u2191f x) (\u2191\u2191g x)\u2016 \u2264 \u2016\u2016\u2191\u2191f x\u2016 ^ 2 + \u2016\u2191\u2191g x\u2016 ^ 2\u2016 \u22a2 AEStronglyMeasurable (fun a => \u2016\u2191\u2191g a\u2016 ^ 2) \u03bc ** exact ((Lp.aestronglyMeasurable g).norm.aemeasurable.pow_const _).aestronglyMeasurable ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.filter_cons ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q s\u271d : Finset \u03b1 a : \u03b1 s : Finset \u03b1 ha : \u00aca \u2208 s \u22a2 _root_.Disjoint (if p a then {a} else \u2205) (filter p s) ** split_ifs ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q s\u271d : Finset \u03b1 a : \u03b1 s : Finset \u03b1 ha : \u00aca \u2208 s h\u271d : p a \u22a2 _root_.Disjoint {a} (filter p s) ** rw [disjoint_singleton_left] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q s\u271d : Finset \u03b1 a : \u03b1 s : Finset \u03b1 ha : \u00aca \u2208 s h\u271d : p a \u22a2 \u00aca \u2208 filter p s ** exact mem_filter.not.mpr <| mt And.left ha ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q s\u271d : Finset \u03b1 a : \u03b1 s : Finset \u03b1 ha : \u00aca \u2208 s h\u271d : \u00acp a \u22a2 _root_.Disjoint \u2205 (filter p s) ** exact disjoint_empty_left _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q s\u271d : Finset \u03b1 a : \u03b1 s : Finset \u03b1 ha : \u00aca \u2208 s \u22a2 filter p (cons a s ha) =     disjUnion (if p a then {a} else \u2205) (filter p s) (_ : _root_.Disjoint (if p a then {a} else \u2205) (filter p s)) ** split_ifs with h ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q s\u271d : Finset \u03b1 a : \u03b1 s : Finset \u03b1 ha : \u00aca \u2208 s h : p a \u22a2 filter p (cons a s ha) = disjUnion {a} (filter p s) (_ : _root_.Disjoint {a} (filter p s)) ** rw [filter_cons_of_pos _ _ _ ha h, singleton_disjUnion] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q s\u271d : Finset \u03b1 a : \u03b1 s : Finset \u03b1 ha : \u00aca \u2208 s h : \u00acp a \u22a2 filter p (cons a s ha) = disjUnion \u2205 (filter p s) (_ : _root_.Disjoint \u2205 (filter p s)) ** rw [filter_cons_of_neg _ _ _ ha h, empty_disjUnion] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc \u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc ** by_cases hab : a < b ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b \u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc ** simp_rw [upcrossings] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) \u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), \u2a06 N, \u2191(upcrossingsBefore a b f N \u03c9) \u2202\u03bc \u2264     \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc ** rw [lintegral_iSup'] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b \u22a2 \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) ** intro N ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b N : \u2115 \u22a2 \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) ** rw [ofReal_integral_eq_lintegral_ofReal] ** case hfi \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b N : \u2115 \u22a2 Integrable fun \u03c9 => (f N \u03c9 - a)\u207a ** exact (hf.sub_martingale (martingale_const _ _ _)).pos.integrable _ ** case f_nn \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b N : \u2115 \u22a2 0 \u2264\u1d50[\u03bc] fun \u03c9 => (f N \u03c9 - a)\u207a ** exact eventually_of_forall fun \u03c9 => LatticeOrderedGroup.pos_nonneg _ ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) \u22a2 ENNReal.ofReal (b - a) * \u2a06 n, \u222b\u207b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f n a_1) \u2202\u03bc \u2264     \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc ** simp_rw [this, ENNReal.mul_iSup, iSup_le_iff] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) \u22a2 \u2200 (i : \u2115),     ENNReal.ofReal (b - a) * \u222b\u207b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f i a_1) \u2202\u03bc \u2264       \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) ** intro N ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) N : \u2115 \u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc \u2264     \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) ** rw [(by simp :\n    \u222b\u207b \u03c9, upcrossingsBefore a b f N \u03c9 \u2202\u03bc = \u222b\u207b \u03c9, \u2191(upcrossingsBefore a b f N \u03c9 : \u211d\u22650) \u2202\u03bc),\n  lintegral_coe_eq_integral, \u2190 ENNReal.ofReal_mul (sub_pos.2 hab).le] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) N : \u2115 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(upcrossingsBefore a b f N \u03c9) \u2202\u03bc = \u222b\u207b (\u03c9 : \u03a9), \u2191\u2191(upcrossingsBefore a b f N \u03c9) \u2202\u03bc ** simp ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) N : \u2115 \u22a2 ENNReal.ofReal ((b - a) * \u222b (a_1 : \u03a9), \u2191\u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc) \u2264     \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) ** simp_rw [NNReal.coe_nat_cast] ** case pos \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) N : \u2115 \u22a2 ENNReal.ofReal ((b - a) * \u222b (a_1 : \u03a9), \u2191(upcrossingsBefore a b f N a_1) \u2202\u03bc) \u2264     \u2a06 N, ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) ** exact (ENNReal.ofReal_le_ofReal\n  (hf.mul_integral_upcrossingsBefore_le_integral_pos_part a b N)).trans\n    (le_iSup (\u03b1 := \u211d\u22650\u221e) _ N) ** case pos.hfi \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) N : \u2115 \u22a2 Integrable fun x => \u2191\u2191(upcrossingsBefore a b f N x) ** simp only [NNReal.coe_nat_cast, hf.adapted.integrable_upcrossingsBefore hab] ** case pos.hf \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) \u22a2 \u2200 (n : \u2115), AEMeasurable fun \u03c9 => \u2191(upcrossingsBefore a b f n \u03c9) ** exact fun n => measurable_from_top.comp_aemeasurable\n  (hf.adapted.measurable_upcrossingsBefore hab).aemeasurable ** case pos.h_mono \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) \u22a2 \u2200\u1d50 (x : \u03a9) \u2202\u03bc, Monotone fun n => \u2191(upcrossingsBefore a b f n x) ** refine' eventually_of_forall fun \u03c9 N M hNM => _ ** case pos.h_mono \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) \u03c9 : \u03a9 N M : \u2115 hNM : N \u2264 M \u22a2 (fun n => \u2191(upcrossingsBefore a b f n \u03c9)) N \u2264 (fun n => \u2191(upcrossingsBefore a b f n \u03c9)) M ** rw [Nat.cast_le] ** case pos.h_mono \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N\u271d n m : \u2115 \u03c9\u271d : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : a < b this : \u2200 (N : \u2115), \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc = ENNReal.ofReal (\u222b (\u03c9 : \u03a9), (f N \u03c9 - a)\u207a \u2202\u03bc) \u03c9 : \u03a9 N M : \u2115 hNM : N \u2264 M \u22a2 upcrossingsBefore a b f N \u03c9 \u2264 upcrossingsBefore a b f M \u03c9 ** exact upcrossingsBefore_mono hab hNM \u03c9 ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab : \u00aca < b \u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc ** rw [not_lt, \u2190 sub_nonpos] at hab ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab\u271d : b \u2264 a hab : b - a \u2264 0 \u22a2 ENNReal.ofReal (b - a) * \u222b\u207b (\u03c9 : \u03a9), upcrossings a b f \u03c9 \u2202\u03bc \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc ** rw [ENNReal.ofReal_of_nonpos hab, zero_mul] ** case neg \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a\u271d b\u271d : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc a b : \u211d hf : Submartingale f \u2131 \u03bc hab\u271d : b \u2264 a hab : b - a \u2264 0 \u22a2 0 \u2264 \u2a06 N, \u222b\u207b (\u03c9 : \u03a9), ENNReal.ofReal (f N \u03c9 - a)\u207a \u2202\u03bc ** exact zero_le _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.WF.mapVal ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m \u22a2 WF (Imp.mapVal f m) ** have \u27e8h\u2081, h\u2082\u27e9 := H.out ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets \u22a2 WF (Imp.mapVal f m) ** simp [Imp.mapVal, Buckets.mapVal, WF_iff, h\u2081] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets \u22a2 Buckets.size m.buckets =       Buckets.size         { val := Array.map (AssocList.mapVal f) m.buckets.val,           property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) } \u2227     Buckets.WF       { val := Array.map (AssocList.mapVal f) m.buckets.val,         property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) } ** refine \u27e8?_, ?_, fun i h => ?_\u27e9 ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets \u22a2 Buckets.size m.buckets =     Buckets.size       { val := Array.map (AssocList.mapVal f) m.buckets.val,         property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) } ** simp [Buckets.size] ** case refine_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) =     Nat.sum (List.map ((fun x => List.length (AssocList.toList x)) \u2218 AssocList.mapVal f) m.buckets.val.data) ** congr ** case refine_1.e_l.e_f \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets \u22a2 (fun x => List.length (AssocList.toList x)) = (fun x => List.length (AssocList.toList x)) \u2218 AssocList.mapVal f ** funext l ** case refine_1.e_l.e_f.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets l : AssocList \u03b1 \u03b2 \u22a2 List.length (AssocList.toList l) = ((fun x => List.length (AssocList.toList x)) \u2218 AssocList.mapVal f) l ** simp ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets \u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (bucket : AssocList \u03b1 \u03b3),     bucket \u2208         { val := Array.map (AssocList.mapVal f) m.buckets.val,               property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val.data \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList bucket) ** simp only [Array.map_data, List.forall_mem_map_iff] ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets \u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (j : AssocList \u03b1 \u03b2),     j \u2208 m.buckets.val.data \u2192       List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList (AssocList.mapVal f j)) ** simp [List.pairwise_map] ** case refine_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets \u22a2 \u2200 [inst : LawfulHashable \u03b1] [inst : PartialEquivBEq \u03b1] (j : AssocList \u03b1 \u03b2),     j \u2208 m.buckets.val.data \u2192 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList j) ** exact fun _ => h\u2082.1 _ ** case refine_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets i : Nat h :   i <     Array.size       { val := Array.map (AssocList.mapVal f) m.buckets.val,           property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val \u22a2 AssocList.All     (fun k x =>       USize.toNat           (UInt64.toUSize (hash k) %             Array.size               { val := Array.map (AssocList.mapVal f) m.buckets.val,                   property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val) =         i)     { val := Array.map (AssocList.mapVal f) m.buckets.val,           property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val[i] ** simp [AssocList.All] at h \u22a2 ** case refine_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 H : WF m h\u2081 : m.size = Buckets.size m.buckets h\u2082 : Buckets.WF m.buckets i : Nat h\u271d :   i <     Array.size       { val := Array.map (AssocList.mapVal f) m.buckets.val,           property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m.buckets.val)) }.val h : i < Array.size m.buckets.val \u22a2 \u2200 (a : \u03b1 \u00d7 \u03b2),     a \u2208 AssocList.toList m.buckets.val[i] \u2192 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m.buckets.val) = i ** intro a m ** case refine_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m\u271d : Imp \u03b1 \u03b2 H : WF m\u271d h\u2081 : m\u271d.size = Buckets.size m\u271d.buckets h\u2082 : Buckets.WF m\u271d.buckets i : Nat h\u271d :   i <     Array.size       { val := Array.map (AssocList.mapVal f) m\u271d.buckets.val,           property := (_ : 0 < Array.size (Array.map (AssocList.mapVal f) m\u271d.buckets.val)) }.val h : i < Array.size m\u271d.buckets.val a : \u03b1 \u00d7 \u03b2 m : a \u2208 AssocList.toList m\u271d.buckets.val[i] \u22a2 USize.toNat (UInt64.toUSize (hash a.fst) % Array.size m\u271d.buckets.val) = i ** apply h\u2082.2 _ _ _ m ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.withDensity_indicator ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 withDensity \u03bc (indicator s f) = withDensity (restrict \u03bc s) f ** ext1 t ht ** case h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d\u22650\u221e t : Set \u03b1 ht : MeasurableSet t \u22a2 \u2191\u2191(withDensity \u03bc (indicator s f)) t = \u2191\u2191(withDensity (restrict \u03bc s) f) t ** rw [withDensity_apply _ ht, lintegral_indicator _ hs, restrict_comm hs, \u2190\n  withDensity_apply _ ht] ** Qed",
        "informal": ""
    },
    {
        "formal": "borel_eq_generateFrom_of_subbasis ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 hu : TopologicalSpace.IsOpen u \u22a2 MeasurableSet u ** induction hu ** case basic \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d\u00b9 t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u s\u271d : Set \u03b1 a\u271d : s\u271d \u2208 s \u22a2 MeasurableSet s\u271d  case univ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 \u22a2 MeasurableSet univ  case inter \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateOpen s s\u271d a\u271d : GenerateOpen s t\u271d a_ih\u271d\u00b9 : MeasurableSet s\u271d a_ih\u271d : MeasurableSet t\u271d \u22a2 MeasurableSet (s\u271d \u2229 t\u271d)  case sUnion \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 S\u271d : Set (Set \u03b1) a\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1 a_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1 \u22a2 MeasurableSet (\u22c3\u2080 S\u271d) ** case basic u hu => exact GenerateMeasurable.basic u hu ** case univ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 \u22a2 MeasurableSet univ  case inter \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateOpen s s\u271d a\u271d : GenerateOpen s t\u271d a_ih\u271d\u00b9 : MeasurableSet s\u271d a_ih\u271d : MeasurableSet t\u271d \u22a2 MeasurableSet (s\u271d \u2229 t\u271d)  case sUnion \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 S\u271d : Set (Set \u03b1) a\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1 a_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1 \u22a2 MeasurableSet (\u22c3\u2080 S\u271d) ** case univ => exact @MeasurableSet.univ \u03b1 (generateFrom s) ** case inter \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateOpen s s\u271d a\u271d : GenerateOpen s t\u271d a_ih\u271d\u00b9 : MeasurableSet s\u271d a_ih\u271d : MeasurableSet t\u271d \u22a2 MeasurableSet (s\u271d \u2229 t\u271d)  case sUnion \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 S\u271d : Set (Set \u03b1) a\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1 a_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1 \u22a2 MeasurableSet (\u22c3\u2080 S\u271d) ** case inter s\u2081 s\u2082 _ _ hs\u2081 hs\u2082 => exact @MeasurableSet.inter \u03b1 (generateFrom s) _ _ hs\u2081 hs\u2082 ** case sUnion \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 S\u271d : Set (Set \u03b1) a\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1 a_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1 \u22a2 MeasurableSet (\u22c3\u2080 S\u271d) ** case\n  sUnion f hf ih =>\n  rcases isOpen_sUnion_countable f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9\n  rw [\u2190 vu]\n  exact @MeasurableSet.sUnion \u03b1 (generateFrom s) _ hv fun x xv => ih _ (vf xv) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d\u00b9 : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u\u271d u : Set \u03b1 hu : u \u2208 s \u22a2 MeasurableSet u ** exact GenerateMeasurable.basic u hu ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 \u22a2 MeasurableSet univ ** exact @MeasurableSet.univ \u03b1 (generateFrom s) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u s\u2081 s\u2082 : Set \u03b1 a\u271d\u00b9 : GenerateOpen s s\u2081 a\u271d : GenerateOpen s s\u2082 hs\u2081 : MeasurableSet s\u2081 hs\u2082 : MeasurableSet s\u2082 \u22a2 MeasurableSet (s\u2081 \u2229 s\u2082) ** exact @MeasurableSet.inter \u03b1 (generateFrom s) _ _ hs\u2081 hs\u2082 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 f : Set (Set \u03b1) hf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1 ih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1 \u22a2 MeasurableSet (\u22c3\u2080 f) ** rcases isOpen_sUnion_countable f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 f : Set (Set \u03b1) hf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1 ih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1 v : Set (Set \u03b1) hv : Set.Countable v vf : v \u2286 f vu : \u22c3\u2080 v = \u22c3\u2080 f \u22a2 MeasurableSet (\u22c3\u2080 f) ** rw [\u2190 vu] ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 f : Set (Set \u03b1) hf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1 ih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1 v : Set (Set \u03b1) hv : Set.Countable v vf : v \u2286 f vu : \u22c3\u2080 v = \u22c3\u2080 f \u22a2 MeasurableSet (\u22c3\u2080 v) ** exact @MeasurableSet.sUnion \u03b1 (generateFrom s) _ hv fun x xv => ih _ (vf xv) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 f : Set (Set \u03b1) hf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1 ih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1 \u22a2 \u2200 (s : Set \u03b1), s \u2208 f \u2192 IsOpen s ** rwa [hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u\u271d : Set \u03b1 s : Set (Set \u03b1) t : TopologicalSpace \u03b1 inst\u271d : SecondCountableTopology \u03b1 hs : t = TopologicalSpace.generateFrom s u : Set \u03b1 hu : u \u2208 s \u22a2 TopologicalSpace.IsOpen u ** exact GenerateOpen.basic _ hu ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.insert_union_distrib ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1 a\u271d b a : \u03b1 s t : Finset \u03b1 \u22a2 insert a (s \u222a t) = insert a s \u222a insert a t ** simp only [insert_union, union_insert, insert_idem] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.tendsto_preCdf_atTop_one ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** have h_mono := monotone_preCdf \u03c1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** have h_le_one := preCdf_le_one \u03c1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** classical\nlet F : \u03b1 \u2192 \u211d\u22650\u221e := fun a =>\n  if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then h.choose else 0\nhave h_tendsto_\u211a : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_exists] with a ha\n  simp_rw [dif_pos ha]\n  exact ha.choose_spec\nhave h_tendsto_\u2115 : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun n : \u2115 => preCdf \u03c1 n a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_tendsto_\u211a] with a ha using ha.comp tendsto_nat_cast_atTop_atTop\nhave hF_ae_meas : AEMeasurable F \u03c1.fst := by\n  refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto_\u211a\n  exact measurable_preCdf.aemeasurable\nhave hF_le_one : \u2200\u1d50 a \u2202\u03c1.fst, F a \u2264 1 := by\n  filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using le_of_tendsto' ha ha_le\nsuffices \u2200\u1d50 a \u2202\u03c1.fst, F a = 1 by\n  filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq\n  rwa [ha_eq] at ha_tendsto\nhave h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := by\n  have h_lintegral :\n    Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) := by\n    refine'\n      lintegral_tendsto_of_tendsto_of_monotone\n        (fun _ => measurable_preCdf.aemeasurable)\n        _ h_tendsto_\u2115\n    filter_upwards [h_mono] with a ha\n    refine' fun n m hnm => ha _\n    exact_mod_cast hnm\n  have h_lintegral' :\n    Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) := by\n    rw [lintegral_one, Measure.fst_univ]\n    exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop\n  exact tendsto_nhds_unique h_lintegral h_lintegral'\nhave : \u222b\u207b a, 1 - F a \u2202\u03c1.fst = 0 := by\n  rw [lintegral_sub' hF_ae_meas _ hF_le_one, h_lintegral_eq, tsub_self]\n  calc\n    \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n    _ = \u03c1.fst univ := lintegral_one\n    _ = \u03c1 univ := Measure.fst_univ\n    _ \u2260 \u221e := measure_ne_top \u03c1 _\nrw [lintegral_eq_zero_iff' (aemeasurable_const.sub hF_ae_meas)] at this\nfilter_upwards [this, hF_le_one] with ha h_one_sub_eq_zero h_le_one\nrw [Pi.zero_apply, tsub_eq_zero_iff_le] at h_one_sub_eq_zero\nexact le_antisymm h_le_one h_one_sub_eq_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) ** filter_upwards [h_mono, h_le_one] with a ha_mono ha_le_one ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 a : \u03b1 ha_mono : Monotone fun r => preCdf \u03c1 r a ha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) ** have h_tendsto :\n  Tendsto (fun r => preCdf \u03c1 r a) atTop atTop \u2228\n    \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) :=\n  tendsto_of_monotone ha_mono ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 a : \u03b1 ha_mono : Monotone fun r => preCdf \u03c1 r a ha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_tendsto : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop \u2228 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) ** cases' h_tendsto with h_absurd h_tendsto ** case h.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 a : \u03b1 ha_mono : Monotone fun r => preCdf \u03c1 r a ha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_absurd : Tendsto (fun r => preCdf \u03c1 r a) atTop atTop \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) ** rw [Monotone.tendsto_atTop_atTop_iff ha_mono] at h_absurd ** case h.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 a : \u03b1 ha_mono : Monotone fun r => preCdf \u03c1 r a ha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_absurd : \u2200 (b : \u211d\u22650\u221e), \u2203 a_1, b \u2264 preCdf \u03c1 a_1 a \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) ** obtain \u27e8r, hr\u27e9 := h_absurd 2 ** case h.inl.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 a : \u03b1 ha_mono : Monotone fun r => preCdf \u03c1 r a ha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_absurd : \u2200 (b : \u211d\u22650\u221e), \u2203 a_1, b \u2264 preCdf \u03c1 a_1 a r : \u211a hr : 2 \u2264 preCdf \u03c1 r a \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) ** exact absurd (hr.trans (ha_le_one r)) ENNReal.one_lt_two.not_le ** case h.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 a : \u03b1 ha_mono : Monotone fun r => preCdf \u03c1 r a ha_le_one : \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_tendsto : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) \u22a2 \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) ** exact h_tendsto ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** let F : \u03b1 \u2192 \u211d\u22650\u221e := fun a =>\n  if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then h.choose else 0 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** have h_tendsto_\u211a : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_exists] with a ha\n  simp_rw [dif_pos ha]\n  exact ha.choose_spec ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** have h_tendsto_\u2115 : \u2200\u1d50 a \u2202\u03c1.fst, Tendsto (fun n : \u2115 => preCdf \u03c1 n a) atTop (\ud835\udcdd (F a)) := by\n  filter_upwards [h_tendsto_\u211a] with a ha using ha.comp tendsto_nat_cast_atTop_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** have hF_ae_meas : AEMeasurable F \u03c1.fst := by\n  refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto_\u211a\n  exact measurable_preCdf.aemeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** have hF_le_one : \u2200\u1d50 a \u2202\u03c1.fst, F a \u2264 1 := by\n  filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using le_of_tendsto' ha ha_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** suffices \u2200\u1d50 a \u2202\u03c1.fst, F a = 1 by\n  filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq\n  rwa [ha_eq] at ha_tendsto ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1 ** have h_lintegral_eq : \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := by\n  have h_lintegral :\n    Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) := by\n    refine'\n      lintegral_tendsto_of_tendsto_of_monotone\n        (fun _ => measurable_preCdf.aemeasurable)\n        _ h_tendsto_\u2115\n    filter_upwards [h_mono] with a ha\n    refine' fun n m hnm => ha _\n    exact_mod_cast hnm\n  have h_lintegral' :\n    Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) := by\n    rw [lintegral_one, Measure.fst_univ]\n    exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop\n  exact tendsto_nhds_unique h_lintegral h_lintegral' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1 ** have : \u222b\u207b a, 1 - F a \u2202\u03c1.fst = 0 := by\n  rw [lintegral_sub' hF_ae_meas _ hF_le_one, h_lintegral_eq, tsub_self]\n  calc\n    \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n    _ = \u03c1.fst univ := lintegral_one\n    _ = \u03c1 univ := Measure.fst_univ\n    _ \u2260 \u221e := measure_ne_top \u03c1 _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 this : \u222b\u207b (a : \u03b1), 1 - F a \u2202Measure.fst \u03c1 = 0 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1 ** rw [lintegral_eq_zero_iff' (aemeasurable_const.sub hF_ae_meas)] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 this : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1 ** filter_upwards [this, hF_le_one] with ha h_one_sub_eq_zero h_le_one ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one\u271d : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 this : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0 ha : \u03b1 h_one_sub_eq_zero : 1 - F ha = OfNat.ofNat 0 ha h_le_one : F ha \u2264 1 \u22a2 F ha = 1 ** rw [Pi.zero_apply, tsub_eq_zero_iff_le] at h_one_sub_eq_zero ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one\u271d : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 this : (fun a => 1 - F a) =\u1d50[Measure.fst \u03c1] 0 ha : \u03b1 h_one_sub_eq_zero : 1 \u2264 F ha h_le_one : F ha \u2264 1 \u22a2 F ha = 1 ** exact le_antisymm h_le_one h_one_sub_eq_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) ** filter_upwards [h_exists] with a ha ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 a : \u03b1 ha : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) \u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) ** simp_rw [dif_pos ha] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 a : \u03b1 ha : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) \u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (Exists.choose ha)) ** exact ha.choose_spec ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) ** filter_upwards [h_tendsto_\u211a] with a ha using ha.comp tendsto_nat_cast_atTop_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) \u22a2 AEMeasurable F ** refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto_\u211a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) n : \u211a \u22a2 AEMeasurable fun x => preCdf \u03c1 n x ** exact measurable_preCdf.aemeasurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 ** filter_upwards [h_tendsto_\u211a, h_le_one] with a ha ha_le using le_of_tendsto' ha ha_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 this : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** filter_upwards [h_tendsto_\u211a, this] with a ha_tendsto ha_eq ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 this : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a = 1 a : \u03b1 ha_tendsto : Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) ha_eq : F a = 1 \u22a2 Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd 1) ** rwa [ha_eq] at ha_tendsto ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 \u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 ** have h_lintegral :\n  Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b a, F a \u2202\u03c1.fst)) := by\n  refine'\n    lintegral_tendsto_of_tendsto_of_monotone\n      (fun _ => measurable_preCdf.aemeasurable)\n      _ h_tendsto_\u2115\n  filter_upwards [h_mono] with a ha\n  refine' fun n m hnm => ha _\n  exact_mod_cast hnm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) \u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 ** have h_lintegral' :\n  Tendsto (fun r : \u2115 => \u222b\u207b a, preCdf \u03c1 r a \u2202\u03c1.fst) atTop (\ud835\udcdd (\u222b\u207b _, 1 \u2202\u03c1.fst)) := by\n  rw [lintegral_one, Measure.fst_univ]\n  exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) h_lintegral' : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1)) \u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 ** exact tendsto_nhds_unique h_lintegral h_lintegral' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 \u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) ** refine'\n  lintegral_tendsto_of_tendsto_of_monotone\n    (fun _ => measurable_preCdf.aemeasurable)\n    _ h_tendsto_\u2115 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, Monotone fun n => preCdf \u03c1 (\u2191n) x ** filter_upwards [h_mono] with a ha ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 a : \u03b1 ha : Monotone fun r => preCdf \u03c1 r a \u22a2 Monotone fun n => preCdf \u03c1 (\u2191n) a ** refine' fun n m hnm => ha _ ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 a : \u03b1 ha : Monotone fun r => preCdf \u03c1 r a n m : \u2115 hnm : n \u2264 m \u22a2 \u2191n \u2264 \u2191m ** exact_mod_cast hnm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) \u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1)) ** rw [lintegral_one, Measure.fst_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral : Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1)) \u22a2 Tendsto (fun r => \u222b\u207b (a : \u03b1), preCdf \u03c1 (\u2191r) a \u2202Measure.fst \u03c1) atTop (\ud835\udcdd (\u2191\u2191\u03c1 univ)) ** exact (tendsto_lintegral_preCdf_atTop \u03c1).comp tendsto_nat_cast_atTop_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 \u22a2 \u222b\u207b (a : \u03b1), 1 - F a \u2202Measure.fst \u03c1 = 0 ** rw [lintegral_sub' hF_ae_meas _ hF_le_one, h_lintegral_eq, tsub_self] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 h_mono : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Monotone fun r => preCdf \u03c1 r a h_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2200 (r : \u211a), preCdf \u03c1 r a \u2264 1 h_exists : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => if h : \u2203 l, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd l) then Exists.choose h else 0 h_tendsto_\u211a : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun r => preCdf \u03c1 r a) atTop (\ud835\udcdd (F a)) h_tendsto_\u2115 : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, Tendsto (fun n => preCdf \u03c1 (\u2191n) a) atTop (\ud835\udcdd (F a)) hF_ae_meas : AEMeasurable F hF_le_one : \u2200\u1d50 (a : \u03b1) \u2202Measure.fst \u03c1, F a \u2264 1 h_lintegral_eq : \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 = \u222b\u207b (x : \u03b1), 1 \u2202Measure.fst \u03c1 \u22a2 \u222b\u207b (a : \u03b1), F a \u2202Measure.fst \u03c1 \u2260 \u22a4 ** calc\n  \u222b\u207b a, F a \u2202\u03c1.fst = \u222b\u207b _, 1 \u2202\u03c1.fst := h_lintegral_eq\n  _ = \u03c1.fst univ := lintegral_one\n  _ = \u03c1 univ := Measure.fst_univ\n  _ \u2260 \u221e := measure_ne_top \u03c1 _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.ToPartrec.Code.zero_eval ** v : List \u2115 \u22a2 eval zero v = pure [0] ** simp [zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.all_ae_ofReal_f_le_bound ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) ** have F_le_bound := all_ae_ofReal_F_le_bound h_bound ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) F_le_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) ** rw [\u2190 ae_all_iff] at F_le_bound ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) F_le_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) ** apply F_le_bound.mp ((all_ae_tendsto_ofReal_norm h_lim).mono _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) F_le_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 \u2200 (x : \u03b1),     Tendsto (fun n => ENNReal.ofReal \u2016F n x\u2016) atTop (\ud835\udcdd (ENNReal.ofReal \u2016f x\u2016)) \u2192       (\u2200 (i : \u2115), ENNReal.ofReal \u2016F i x\u2016 \u2264 ENNReal.ofReal (bound x)) \u2192 ENNReal.ofReal \u2016f x\u2016 \u2264 ENNReal.ofReal (bound x) ** intro a tendsto_norm F_le_bound ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) F_le_bound\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a) a : \u03b1 tendsto_norm : Tendsto (fun n => ENNReal.ofReal \u2016F n a\u2016) atTop (\ud835\udcdd (ENNReal.ofReal \u2016f a\u2016)) F_le_bound : \u2200 (i : \u2115), ENNReal.ofReal \u2016F i a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) ** exact le_of_tendsto' tendsto_norm F_le_bound ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.NullMeasurableSet.const_smul ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : NullMeasurableSet s r : \u211d \u22a2 NullMeasurableSet (r \u2022 s) ** obtain rfl | hs' := s.eq_empty_or_nonempty ** case inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : NullMeasurableSet s r : \u211d hs' : Set.Nonempty s \u22a2 NullMeasurableSet (r \u2022 s) ** obtain rfl | hr := eq_or_ne r 0 ** case inr.inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : NullMeasurableSet s r : \u211d hs' : Set.Nonempty s hr : r \u2260 0 \u22a2 NullMeasurableSet (r \u2022 s) ** obtain \u27e8t, ht, hst\u27e9 := hs ** case inr.inr.intro.intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E r : \u211d hs' : Set.Nonempty s hr : r \u2260 0 t : Set E ht : MeasurableSet t hst : s =\u1da0[ae \u03bc] t \u22a2 NullMeasurableSet (r \u2022 s) ** refine' \u27e8_, ht.const_smul_of_ne_zero hr, _\u27e9 ** case inr.inr.intro.intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E r : \u211d hs' : Set.Nonempty s hr : r \u2260 0 t : Set E ht : MeasurableSet t hst : s =\u1da0[ae \u03bc] t \u22a2 r \u2022 s =\u1da0[ae \u03bc] r \u2022 t ** rw [\u2190 measure_symmDiff_eq_zero_iff] at hst \u22a2 ** case inr.inr.intro.intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E r : \u211d hs' : Set.Nonempty s hr : r \u2260 0 t : Set E ht : MeasurableSet t hst : \u2191\u2191\u03bc (s \u2206 t) = 0 \u22a2 \u2191\u2191\u03bc ((r \u2022 s) \u2206 (r \u2022 t)) = 0 ** rw [\u2190 smul_set_symmDiff\u2080 hr, addHaar_smul \u03bc, hst, mul_zero] ** case inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F r : \u211d hs : NullMeasurableSet \u2205 \u22a2 NullMeasurableSet (r \u2022 \u2205) ** simp ** case inr.inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : NullMeasurableSet s hs' : Set.Nonempty s \u22a2 NullMeasurableSet (0 \u2022 s) ** simpa [zero_smul_set hs'] using nullMeasurableSet_singleton _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.ssubset_iff_exists_cons_subset ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s t : Finset \u03b1 a b : \u03b1 \u22a2 s \u2282 t \u2194 \u2203 a h, cons a s h \u2286 t ** refine' \u27e8fun h => _, fun \u27e8a, ha, h\u27e9 => ssubset_of_ssubset_of_subset (ssubset_cons _) h\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s t : Finset \u03b1 a b : \u03b1 h : s \u2282 t \u22a2 \u2203 a h, cons a s h \u2286 t ** obtain \u27e8a, hs, ht\u27e9 := not_subset.1 h.2 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s t : Finset \u03b1 a\u271d b : \u03b1 h : s \u2282 t a : \u03b1 hs : a \u2208 t ht : \u00aca \u2208 s \u22a2 \u2203 a h, cons a s h \u2286 t ** exact \u27e8a, ht, cons_subset.2 \u27e8hs, h.subset\u27e9\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_univ_of_isMulLeftInvariant ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc \u22a2 \u2191\u2191\u03bc univ = \u22a4 ** obtain \u27e8K, hK, Kclosed, K1\u27e9 : \u2203 K : Set G, IsCompact K \u2227 IsClosed K \u2227 K \u2208 \ud835\udcdd 1 :=\n  exists_isCompact_isClosed_nhds_one G ** case intro.intro.intro \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 \u22a2 \u2191\u2191\u03bc univ = \u22a4 ** have K_pos : 0 < \u03bc K := measure_pos_of_nonempty_interior _ \u27e8_, mem_interior_iff_mem_nhds.2 K1\u27e9 ** case intro.intro.intro \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K \u22a2 \u2191\u2191\u03bc univ = \u22a4 ** have A : \u2200 L : Set G, IsCompact L \u2192 \u2203 g : G, Disjoint L (g \u2022 K) := fun L hL =>\n  exists_disjoint_smul_of_isCompact hL hK ** case intro.intro.intro \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K A : \u2200 (L : Set G), IsCompact L \u2192 \u2203 g, Disjoint L (g \u2022 K) \u22a2 \u2191\u2191\u03bc univ = \u22a4 ** choose! g hg using A ** case intro.intro.intro \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) \u22a2 \u2191\u2191\u03bc univ = \u22a4 ** set L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K ** case intro.intro.intro \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) M : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K \u22a2 \u2191\u2191\u03bc univ = \u22a4 ** have N : Tendsto (fun n => \u03bc (L n)) atTop (\ud835\udcdd (\u221e * \u03bc K)) := by\n  simp_rw [M]\n  apply ENNReal.Tendsto.mul_const _ (Or.inl ENNReal.top_ne_zero)\n  exact ENNReal.tendsto_nat_nhds_top.comp (tendsto_add_atTop_nat _) ** case intro.intro.intro \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) M : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K N : Tendsto (fun n => \u2191\u2191\u03bc (L n)) atTop (\ud835\udcdd (\u22a4 * \u2191\u2191\u03bc K)) \u22a2 \u2191\u2191\u03bc univ = \u22a4 ** simp only [ENNReal.top_mul', K_pos.ne', if_false] at N ** case intro.intro.intro \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) M : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K N : Tendsto (fun n => \u2191\u2191\u03bc ((fun T => T \u222a g T \u2022 K)^[n] K)) atTop (\ud835\udcdd \u22a4) \u22a2 \u2191\u2191\u03bc univ = \u22a4 ** apply top_le_iff.1 ** case intro.intro.intro \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) M : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K N : Tendsto (fun n => \u2191\u2191\u03bc ((fun T => T \u222a g T \u2022 K)^[n] K)) atTop (\ud835\udcdd \u22a4) \u22a2 \u22a4 \u2264 \u2191\u2191\u03bc univ ** exact le_of_tendsto' N fun n => measure_mono (subset_univ _) ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K \u22a2 \u2200 (n : \u2115), IsCompact (L n) ** intro n ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K n : \u2115 \u22a2 IsCompact (L n) ** induction' n with n IH ** case zero \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K \u22a2 IsCompact (L Nat.zero) ** exact hK ** case succ \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K n : \u2115 IH : IsCompact (L n) \u22a2 IsCompact (L (Nat.succ n)) ** simp_rw [iterate_succ'] ** case succ \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K n : \u2115 IH : IsCompact (L n) \u22a2 IsCompact (((fun T => T \u222a g T \u2022 K) \u2218 (fun T => T \u222a g T \u2022 K)^[n]) K) ** apply IsCompact.union IH (hK.smul (g (L n))) ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) \u22a2 \u2200 (n : \u2115), IsClosed (L n) ** intro n ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) n : \u2115 \u22a2 IsClosed (L n) ** induction' n with n IH ** case zero \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) \u22a2 IsClosed (L Nat.zero) ** exact Kclosed ** case succ \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) n : \u2115 IH : IsClosed (L n) \u22a2 IsClosed (L (Nat.succ n)) ** simp_rw [iterate_succ'] ** case succ \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) n : \u2115 IH : IsClosed (L n) \u22a2 IsClosed (((fun T => T \u222a g T \u2022 K) \u2218 (fun T => T \u222a g T \u2022 K)^[n]) K) ** apply IsClosed.union IH (Kclosed.smul (g (L n))) ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) \u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K ** intro n ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) n : \u2115 \u22a2 \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K ** induction' n with n IH ** case zero \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) \u22a2 \u2191\u2191\u03bc (L Nat.zero) = \u2191(Nat.zero + 1) * \u2191\u2191\u03bc K ** simp only [one_mul, Nat.cast_one, iterate_zero, id.def, Nat.zero_eq, Nat.zero_add] ** case succ \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) n : \u2115 IH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K \u22a2 \u2191\u2191\u03bc (L (Nat.succ n)) = \u2191(Nat.succ n + 1) * \u2191\u2191\u03bc K ** calc\n  \u03bc (L (n + 1)) = \u03bc (L n) + \u03bc (g (L n) \u2022 K) := by\n    simp_rw [iterate_succ']\n    exact measure_union' (hg _ (Lcompact _)) (Lclosed _).measurableSet\n  _ = (n + 1 + 1 : \u2115) * \u03bc K := by\n    simp only [IH, measure_smul, add_mul, Nat.cast_add, Nat.cast_one, one_mul] ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) n : \u2115 IH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K \u22a2 \u2191\u2191\u03bc (L (n + 1)) = \u2191\u2191\u03bc (L n) + \u2191\u2191\u03bc (g (L n) \u2022 K) ** simp_rw [iterate_succ'] ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) n : \u2115 IH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K \u22a2 \u2191\u2191\u03bc (((fun T => T \u222a g T \u2022 K) \u2218 (fun T => T \u222a g T \u2022 K)^[n]) K) =     \u2191\u2191\u03bc ((fun T => T \u222a g T \u2022 K)^[n] K) + \u2191\u2191\u03bc (g ((fun T => T \u222a g T \u2022 K)^[n] K) \u2022 K) ** exact measure_union' (hg _ (Lcompact _)) (Lclosed _).measurableSet ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) n : \u2115 IH : \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K \u22a2 \u2191\u2191\u03bc (L n) + \u2191\u2191\u03bc (g (L n) \u2022 K) = \u2191(n + 1 + 1) * \u2191\u2191\u03bc K ** simp only [IH, measure_smul, add_mul, Nat.cast_add, Nat.cast_one, one_mul] ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) M : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K \u22a2 Tendsto (fun n => \u2191\u2191\u03bc (L n)) atTop (\ud835\udcdd (\u22a4 * \u2191\u2191\u03bc K)) ** simp_rw [M] ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) M : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K \u22a2 Tendsto (fun n => \u2191(n + 1) * \u2191\u2191\u03bc K) atTop (\ud835\udcdd (\u22a4 * \u2191\u2191\u03bc K)) ** apply ENNReal.Tendsto.mul_const _ (Or.inl ENNReal.top_ne_zero) ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u00b9\u2070 : MeasurableSpace G inst\u271d\u2079 : MeasurableSpace H inst\u271d\u2078 : TopologicalSpace G inst\u271d\u2077 : BorelSpace G \u03bc\u271d : Measure G inst\u271d\u2076 : Group G inst\u271d\u2075 : TopologicalGroup G inst\u271d\u2074 : IsMulLeftInvariant \u03bc\u271d inst\u271d\u00b3 : WeaklyLocallyCompactSpace G inst\u271d\u00b2 : NoncompactSpace G \u03bc : Measure G inst\u271d\u00b9 : IsOpenPosMeasure \u03bc inst\u271d : IsMulLeftInvariant \u03bc K : Set G hK : IsCompact K Kclosed : IsClosed K K1 : K \u2208 \ud835\udcdd 1 K_pos : 0 < \u2191\u2191\u03bc K g : Set G \u2192 G hg : \u2200 (L : Set G), IsCompact L \u2192 Disjoint L (g L \u2022 K) L : \u2115 \u2192 Set G := fun n => (fun T => T \u222a g T \u2022 K)^[n] K Lcompact : \u2200 (n : \u2115), IsCompact (L n) Lclosed : \u2200 (n : \u2115), IsClosed (L n) M : \u2200 (n : \u2115), \u2191\u2191\u03bc (L n) = \u2191(n + 1) * \u2191\u2191\u03bc K \u22a2 Tendsto (fun x => \u2191(x + 1)) atTop (\ud835\udcdd \u22a4) ** exact ENNReal.tendsto_nat_nhds_top.comp (tendsto_add_atTop_nat _) ** Qed",
        "informal": ""
    },
    {
        "formal": "torusIntegral_succAbove ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) \u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =     \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y) ** set e : \u211d \u00d7 \u211d\u207f \u2243\u1d50 \u211d\u207f\u207a\u00b9 := (MeasurableEquiv.piFinSuccAboveEquiv (fun _ => \u211d) i).symm ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) \u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =     \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y) ** have hem : MeasurePreserving e :=\n  (volume_preserving_piFinSuccAboveEquiv (fun _ : Fin (n + 1) => \u211d) i).symm _ ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) hem : MeasurePreserving \u2191e \u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =     \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y) ** have he\u03c0 : (e \u207b\u00b9' Icc 0 fun _ => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc (0 : \u211d\u207f) fun _ => 2 * \u03c0 :=\n  ((OrderIso.piFinSuccAboveIso (fun _ => \u211d) i).symm.preimage_Icc _ _).trans (Icc_prod_eq _ _) ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) hem : MeasurePreserving \u2191e he\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0 \u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =     \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y) ** rw [torusIntegral, \u2190 hem.map_eq, set_integral_map_equiv, he\u03c0, Measure.volume_eq_prod,\n  set_integral_prod, circleIntegral_def_Icc] ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) hem : MeasurePreserving \u2191e he\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0 \u22a2 \u222b (x : \u211d) in Icc 0 (2 * \u03c0),       \u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,         (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e (x, y) i) * I) * I) \u2022 f (torusMap c R (\u2191e (x, y))) =     \u222b (\u03b8 : \u211d) in Icc 0 (2 * \u03c0),       deriv (circleMap (c i) (R i)) \u03b8 \u2022         \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i),           f (Fin.insertNth i (circleMap (c i) (R i) \u03b8) y) ** refine' set_integral_congr measurableSet_Icc fun \u03b8 _ => _ ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) hem : MeasurePreserving \u2191e he\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0 \u03b8 : \u211d x\u271d : \u03b8 \u2208 Icc 0 (2 * \u03c0) \u22a2 \u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,       (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e (\u03b8, y) i) * I) * I) \u2022 f (torusMap c R (\u2191e (\u03b8, y))) =     deriv (circleMap (c i) (R i)) \u03b8 \u2022       \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i (circleMap (c i) (R i) \u03b8) y) ** simp only [torusIntegral, \u2190 integral_smul, deriv_circleMap, i.prod_univ_succAbove _, smul_smul,\n  torusMap, circleMap_zero] ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) hem : MeasurePreserving \u2191e he\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0 \u03b8 : \u211d x\u271d : \u03b8 \u2208 Icc 0 (2 * \u03c0) \u22a2 (\u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,       (\u2191(R i) * cexp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y) i) * I) *             I *           \u220f i_1 : Fin n,             \u2191(R (Fin.succAbove i i_1)) *                 cexp                   (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y)                         (Fin.succAbove i i_1)) *                     I) *               I) \u2022         f fun i_1 =>           c i_1 +             \u2191(R i_1) *               cexp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y) i_1) * I)) =     \u222b (a : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,       (\u2191(R i) * cexp (\u2191\u03b8 * I) * I * \u220f i_1 : Fin n, \u2191((R \u2218 Fin.succAbove i) i_1) * cexp (\u2191(a i_1) * I) * I) \u2022         f           (Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>             (c \u2218 Fin.succAbove i) i_1 + \u2191((R \u2218 Fin.succAbove i) i_1) * cexp (\u2191(a i_1) * I)) ** refine' set_integral_congr measurableSet_Icc fun \u0398 _ => _ ** n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) hem : MeasurePreserving \u2191e he\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0 \u03b8 : \u211d x\u271d\u00b9 : \u03b8 \u2208 Icc 0 (2 * \u03c0) \u0398 : Fin n \u2192 \u211d x\u271d : \u0398 \u2208 Icc 0 fun x => 2 * \u03c0 \u22a2 ((\u2191(R i) * cexp (\u2191\u03b8 * I) * I * \u220f x : Fin n, \u2191(R (Fin.succAbove i x)) * cexp (\u2191(\u0398 x) * I) * I) \u2022       f fun i_1 => c i_1 + \u2191(R i_1) * cexp (\u2191(Fin.insertNth i \u03b8 \u0398 i_1) * I)) =     (\u2191(R i) * cexp (\u2191\u03b8 * I) * I * \u220f x : Fin n, \u2191(R (Fin.succAbove i x)) * cexp (\u2191(\u0398 x) * I) * I) \u2022       f         (Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>           c (Fin.succAbove i i_1) + \u2191(R (Fin.succAbove i i_1)) * cexp (\u2191(\u0398 i_1) * I)) ** congr 2 ** case e_a.e_a n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) hem : MeasurePreserving \u2191e he\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0 \u03b8 : \u211d x\u271d\u00b9 : \u03b8 \u2208 Icc 0 (2 * \u03c0) \u0398 : Fin n \u2192 \u211d x\u271d : \u0398 \u2208 Icc 0 fun x => 2 * \u03c0 \u22a2 (fun i_1 => c i_1 + \u2191(R i_1) * cexp (\u2191(Fin.insertNth i \u03b8 \u0398 i_1) * I)) =     Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>       c (Fin.succAbove i i_1) + \u2191(R (Fin.succAbove i i_1)) * cexp (\u2191(\u0398 i_1) * I) ** simp only [funext_iff, i.forall_iff_succAbove, circleMap, Fin.insertNth_apply_same,\n  eq_self_iff_true, Fin.insertNth_apply_succAbove, imp_true_iff, and_self_iff] ** case hf n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) hem : MeasurePreserving \u2191e he\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0 \u22a2 IntegrableOn (fun x => (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e x i) * I) * I) \u2022 f (torusMap c R (\u2191e x)))     (Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0) ** have := hf.function_integrable ** case hf n : \u2115 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f\u271d g : (Fin n \u2192 \u2102) \u2192 E c\u271d : Fin n \u2192 \u2102 R\u271d : Fin n \u2192 \u211d f : (Fin (n + 1) \u2192 \u2102) \u2192 E c : Fin (n + 1) \u2192 \u2102 R : Fin (n + 1) \u2192 \u211d hf : TorusIntegrable f c R i : Fin (n + 1) e : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i) hem : MeasurePreserving \u2191e he\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0 this :   IntegrableOn (fun \u03b8 => (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u03b8 i) * I) * I) \u2022 f (torusMap c R \u03b8))     (Icc 0 fun x => 2 * \u03c0) \u22a2 IntegrableOn (fun x => (\u220f i : Fin (n + 1), \u2191(R i) * cexp (\u2191(\u2191e x i) * I) * I) \u2022 f (torusMap c R (\u2191e x)))     (Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0) ** rwa [\u2190 hem.integrableOn_comp_preimage e.measurableEmbedding, he\u03c0] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.set_lintegral_condKernelReal_Iic ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d s : Set \u03b1 hs : MeasurableSet s \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2191(\u2191(condKernelReal \u03c1) a) (Iic x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** simp_rw [condKernelReal_Iic] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 x : \u211d s : Set \u03b1 hs : MeasurableSet s \u22a2 \u222b\u207b (a : \u03b1) in s, ENNReal.ofReal (\u2191(condCdf \u03c1 a) x) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic x) ** exact set_lintegral_condCdf \u03c1 x hs ** Qed",
        "informal": ""
    },
    {
        "formal": "List.pairwise_iff_get ** \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d \u22a2 Pairwise R l \u2194 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j) ** rw [pairwise_iff_forall_sublist] ** \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d \u22a2 (\u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b) \u2194 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j) ** constructor <;> intro h ** case mp \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b \u22a2 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j) ** intros i j h' ** case mp \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b i j : Fin (length l) h' : i < j \u22a2 R (get l i) (get l j) ** apply h ** case mp.a \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b i j : Fin (length l) h' : i < j \u22a2 [get l i, get l j] <+ l ** apply map_get_sublist (is := [i, j]) ** case mp.a \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b i j : Fin (length l) h' : i < j \u22a2 Pairwise (fun x x_1 => \u2191x < \u2191x_1) [i, j] ** rw [Fin.lt_def] at h' ** case mp.a \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b i j : Fin (length l) h' : \u2191i < \u2191j \u22a2 Pairwise (fun x x_1 => \u2191x < \u2191x_1) [i, j] ** simp [h'] ** case mpr \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j) \u22a2 \u2200 {a b : \u03b1\u271d}, [a, b] <+ l \u2192 R a b ** intros a b h' ** case mpr \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j) a b : \u03b1\u271d h' : [a, b] <+ l \u22a2 R a b ** have \u27e8is, h', hij\u27e9 := sublist_eq_map_get h' ** case mpr \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j) a b : \u03b1\u271d h'\u271d : [a, b] <+ l is : List (Fin (length l)) h' : [a, b] = map (get l) is hij : Pairwise (fun x x_1 => x < x_1) is \u22a2 R a b ** rcases is with \u27e8\u27e9 | \u27e8a', \u27e8\u27e9 | \u27e8b', \u27e8\u27e9\u27e9\u27e9 <;> simp at h' ** case mpr.cons.cons.nil \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j) a b : \u03b1\u271d h'\u271d : [a, b] <+ l a' b' : Fin (length l) hij : Pairwise (fun x x_1 => x < x_1) [a', b'] h' : a = get l a' \u2227 b = get l b' \u22a2 R a b ** rcases h' with \u27e8rfl, rfl\u27e9 ** case mpr.cons.cons.nil.intro \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j) a' b' : Fin (length l) hij : Pairwise (fun x x_1 => x < x_1) [a', b'] h' : [get l a', get l b'] <+ l \u22a2 R (get l a') (get l b') ** apply h ** case mpr.cons.cons.nil.intro._hij \u03b1\u271d : Type u_1 R : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop l : List \u03b1\u271d h : \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j) a' b' : Fin (length l) hij : Pairwise (fun x x_1 => x < x_1) [a', b'] h' : [get l a', get l b'] <+ l \u22a2 a' < b' ** simpa using hij ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0 : 0 \u2264 p f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0 ** rw [\u2190 @lintegral_zero_fun \u03b1 _ \u03bc] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0 : 0 \u2264 p f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = lintegral \u03bc 0 ** refine' lintegral_congr_ae _ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0 : 0 \u2264 p f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \u22a2 (fun a => (f * g) a) =\u1d50[\u03bc] fun a => OfNat.ofNat 0 a ** suffices h_mul_zero : f * g =\u1d50[\u03bc] 0 * g ** case h_mul_zero \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0 : 0 \u2264 p f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 \u22a2 f * g =\u1d50[\u03bc] 0 * g ** have hf_eq_zero : f =\u1d50[\u03bc] 0 := ae_eq_zero_of_lintegral_rpow_eq_zero hp0 hf hf_zero ** case h_mul_zero \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0 : 0 \u2264 p f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 hf_eq_zero : f =\u1d50[\u03bc] 0 \u22a2 f * g =\u1d50[\u03bc] 0 * g ** exact hf_eq_zero.mul (ae_eq_refl g) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0 : 0 \u2264 p f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0 h_mul_zero : f * g =\u1d50[\u03bc] 0 * g \u22a2 (fun a => (f * g) a) =\u1d50[\u03bc] fun a => OfNat.ofNat 0 a ** rwa [zero_mul] at h_mul_zero ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_mul_const_uIcc ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a : \u03b1 ha : a \u2260 0 b c : \u03b1 h : a < 0 \u22a2 (fun x => x * a) \u207b\u00b9' [[b, c]] = [[b / a, c / a]] ** simp [\u2190 Icc_min_max, h, h.le, min_div_div_right_of_nonpos, max_div_div_right_of_nonpos] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a : \u03b1 ha\u271d : a \u2260 0 b c : \u03b1 ha : 0 < a \u22a2 (fun x => x * a) \u207b\u00b9' [[b, c]] = [[b / a, c / a]] ** simp [\u2190 Icc_min_max, ha, ha.le, min_div_div_right, max_div_div_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "Measurable.iSup_Prop ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : p \u22a2 Measurable fun b => \u2a06 (_ : p), f b ** convert hf ** case h.e'_5.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : p x\u271d : \u03b4 \u22a2 \u2a06 (_ : p), f x\u271d = f x\u271d ** exact ciSup_pos h ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp \u22a2 Measurable fun b => \u2a06 (_ : p), f b ** convert measurable_const using 1 ** case h.e'_5 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp \u22a2 (fun b => \u2a06 (_ : p), f b) = fun x => ?convert_5  case convert_5 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp \u22a2 \u03b1 ** funext ** case h.e'_5.h \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp x\u271d : \u03b4 \u22a2 \u2a06 (_ : p), f x\u271d = ?convert_5  case convert_5 \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1\u271d inst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d inst\u271d\u00b9\u2070 : MeasurableSpace \u03b1\u271d inst\u271d\u2079 : BorelSpace \u03b1\u271d inst\u271d\u2078 : TopologicalSpace \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : BorelSpace \u03b2 inst\u271d\u2075 : TopologicalSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b3 inst\u271d\u00b3 : BorelSpace \u03b3 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03b1 : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : ConditionallyCompleteLattice \u03b1 p : Prop f : \u03b4 \u2192 \u03b1 hf : Measurable f h : \u00acp \u22a2 \u03b1 ** exact ciSup_neg h ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.addCases_right ** m n : Nat motive : Fin (m + n) \u2192 Sort u_1 left : (i : Fin m) \u2192 motive (castAdd n i) right : (i : Fin n) \u2192 motive (natAdd m i) i : Fin n \u22a2 addCases left right (natAdd m i) = right i ** have : \u00ac(natAdd m i : Nat) < m := Nat.not_lt.2 (le_coe_natAdd ..) ** m n : Nat motive : Fin (m + n) \u2192 Sort u_1 left : (i : Fin m) \u2192 motive (castAdd n i) right : (i : Fin n) \u2192 motive (natAdd m i) i : Fin n this : \u00ac\u2191(natAdd m i) < m \u22a2 addCases left right (natAdd m i) = right i ** rw [addCases, dif_neg this] ** m n : Nat motive : Fin (m + n) \u2192 Sort u_1 left : (i : Fin m) \u2192 motive (castAdd n i) right : (i : Fin n) \u2192 motive (natAdd m i) i : Fin n this : \u00ac\u2191(natAdd m i) < m \u22a2 (_ : natAdd m (subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i))) = natAdd m i) \u25b8       right (subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i))) =     right i ** exact eq_of_heq <| (eqRec_heq _ _).trans (by congr 1; simp) ** m n : Nat motive : Fin (m + n) \u2192 Sort u_1 left : (i : Fin m) \u2192 motive (castAdd n i) right : (i : Fin n) \u2192 motive (natAdd m i) i : Fin n this : \u00ac\u2191(natAdd m i) < m \u22a2 HEq (right (subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i)))) (right i) ** congr 1 ** case e_1 m n : Nat motive : Fin (m + n) \u2192 Sort u_1 left : (i : Fin m) \u2192 motive (castAdd n i) right : (i : Fin n) \u2192 motive (natAdd m i) i : Fin n this : \u00ac\u2191(natAdd m i) < m \u22a2 subNat m (cast (_ : m + n = n + m) (natAdd m i)) (_ : m \u2264 \u2191(natAdd m i)) = i ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Pos.zero_addChar_byteIdx ** c : Char \u22a2 (0 + c).byteIdx = csize c ** simp only [addChar_byteIdx, byteIdx_zero, Nat.zero_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.Path.ins_eq_fill ** \u03b1 : Type u_1 c\u2080 : RBColor n\u2080 : Nat c : RBColor n\u271d : Nat path : Path \u03b1 t : RBNode \u03b1 n : Nat x\u271d : RBNode \u03b1 v\u271d : \u03b1 parent\u271d : Path \u03b1 ha : Balanced x\u271d black n H : Path.Balanced c\u2080 n\u2080 parent\u271d red n hb : Balanced t black n \u22a2 ins (right red x\u271d v\u271d parent\u271d) t = setBlack (fill (right red x\u271d v\u271d parent\u271d) t) ** unfold ins ** \u03b1 : Type u_1 c\u2080 : RBColor n\u2080 : Nat c : RBColor n\u271d : Nat path : Path \u03b1 t : RBNode \u03b1 n : Nat x\u271d : RBNode \u03b1 v\u271d : \u03b1 parent\u271d : Path \u03b1 ha : Balanced x\u271d black n H : Path.Balanced c\u2080 n\u2080 parent\u271d red n hb : Balanced t black n \u22a2 ins parent\u271d (node red x\u271d v\u271d t) = setBlack (fill (right red x\u271d v\u271d parent\u271d) t) ** exact ins_eq_fill H (.red ha hb) ** \u03b1 : Type u_1 c\u2080 : RBColor n\u2080 : Nat c\u271d : RBColor n\u271d : Nat path : Path \u03b1 t : RBNode \u03b1 n : Nat c : RBColor y\u271d : RBNode \u03b1 c\u2082\u271d : RBColor parent\u271d : Path \u03b1 v\u271d : \u03b1 hb : Balanced y\u271d c\u2082\u271d n H : Path.Balanced c\u2080 n\u2080 parent\u271d black (n + 1) ha : Balanced t c n \u22a2 ins (left black parent\u271d v\u271d y\u271d) t = setBlack (fill (left black parent\u271d v\u271d y\u271d) t) ** rw [ins, fill, \u2190 ins_eq_fill H (.black ha hb), balance1_eq ha] ** \u03b1 : Type u_1 c\u2080 : RBColor n\u2080 : Nat c\u271d : RBColor n\u271d : Nat path : Path \u03b1 t : RBNode \u03b1 n : Nat c : RBColor x\u271d : RBNode \u03b1 c\u2081\u271d : RBColor v\u271d : \u03b1 parent\u271d : Path \u03b1 ha : Balanced x\u271d c\u2081\u271d n H : Path.Balanced c\u2080 n\u2080 parent\u271d black (n + 1) hb : Balanced t c n \u22a2 ins (right black x\u271d v\u271d parent\u271d) t = setBlack (fill (right black x\u271d v\u271d parent\u271d) t) ** rw [ins, fill, \u2190 ins_eq_fill H (.black ha hb), balance2_eq hb] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.pow_subset_pow_of_one_mem ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s t : Finset \u03b1 a : \u03b1 m n : \u2115 hs : 1 \u2208 s \u22a2 m \u2264 n \u2192 s ^ m \u2286 s ^ n ** apply Nat.le_induction ** case base F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s t : Finset \u03b1 a : \u03b1 m n : \u2115 hs : 1 \u2208 s \u22a2 s ^ m \u2286 s ^ m ** exact fun _ hn => hn ** case succ F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s t : Finset \u03b1 a : \u03b1 m n : \u2115 hs : 1 \u2208 s \u22a2 \u2200 (n : \u2115), m \u2264 n \u2192 s ^ m \u2286 s ^ n \u2192 s ^ m \u2286 s ^ (n + 1) ** intro n _ hmn ** case succ F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s t : Finset \u03b1 a : \u03b1 m n\u271d : \u2115 hs : 1 \u2208 s n : \u2115 hn\u271d : m \u2264 n hmn : s ^ m \u2286 s ^ n \u22a2 s ^ m \u2286 s ^ (n + 1) ** rw [pow_succ] ** case succ F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : Monoid \u03b1 s t : Finset \u03b1 a : \u03b1 m n\u271d : \u2115 hs : 1 \u2208 s n : \u2115 hn\u271d : m \u2264 n hmn : s ^ m \u2286 s ^ n \u22a2 s ^ m \u2286 s * s ^ n ** exact hmn.trans (subset_mul_right (s ^ n) hs) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_le_ncard_of_injOn ** \u03b1 : Type u_2 s t\u271d : Set \u03b1 \u03b2 : Type u_1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t f_inj : InjOn f s ht : autoParam (Set.Finite t) _auto\u271d \u22a2 ncard s \u2264 ncard t ** have hle := encard_le_encard_of_injOn hf f_inj ** \u03b1 : Type u_2 s t\u271d : Set \u03b1 \u03b2 : Type u_1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t f_inj : InjOn f s ht : autoParam (Set.Finite t) _auto\u271d hle : encard s \u2264 encard t \u22a2 ncard s \u2264 ncard t ** to_encard_tac ** \u03b1 : Type u_2 s t\u271d : Set \u03b1 \u03b2 : Type u_1 t : Set \u03b2 f : \u03b1 \u2192 \u03b2 hf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t f_inj : InjOn f s ht : autoParam (Set.Finite t) _auto\u271d hle : encard s \u2264 encard t \u22a2 \u2191(ncard s) \u2264 \u2191(ncard t) ** rwa [ht.cast_ncard_eq, (ht.finite_of_encard_le hle).cast_ncard_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_le_one_iff_eq ** \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard s \u2264 1 \u2194 s = \u2205 \u2228 \u2203 a, s = {a} ** obtain rfl | \u27e8x, hx\u27e9 := s.eq_empty_or_nonempty ** case inr.intro \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d x : \u03b1 hx : x \u2208 s \u22a2 ncard s \u2264 1 \u2194 s = \u2205 \u2228 \u2203 a, s = {a} ** rw [ncard_le_one_iff hs] ** case inr.intro \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d x : \u03b1 hx : x \u2208 s \u22a2 (\u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b) \u2194 s = \u2205 \u2228 \u2203 a, s = {a} ** refine' \u27e8fun h \u21a6 Or.inr \u27e8x, (singleton_subset_iff.mpr hx).antisymm' fun y hy \u21a6 h hy hx\u27e9, _\u27e9 ** case inr.intro \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d x : \u03b1 hx : x \u2208 s \u22a2 (s = \u2205 \u2228 \u2203 a, s = {a}) \u2192 \u2200 {a b : \u03b1}, a \u2208 s \u2192 b \u2208 s \u2192 a = b ** rintro (rfl | \u27e8a, rfl\u27e9) ** case inr.intro.inr.intro \u03b1 : Type u_1 t : Set \u03b1 x a : \u03b1 hs : autoParam (Set.Finite {a}) _auto\u271d hx : x \u2208 {a} \u22a2 \u2200 {a_1 b : \u03b1}, a_1 \u2208 {a} \u2192 b \u2208 {a} \u2192 a_1 = b ** simp_rw [mem_singleton_iff] at hx \u22a2 ** case inr.intro.inr.intro \u03b1 : Type u_1 t : Set \u03b1 x a : \u03b1 hs : autoParam (Set.Finite {a}) _auto\u271d hx : x = a \u22a2 \u2200 {a_1 b : \u03b1}, a_1 = a \u2192 b = a \u2192 a_1 = b ** subst hx ** case inr.intro.inr.intro \u03b1 : Type u_1 t : Set \u03b1 x : \u03b1 hs : autoParam (Set.Finite {x}) _auto\u271d \u22a2 \u2200 {a b : \u03b1}, a = x \u2192 b = x \u2192 a = b ** simp only [forall_eq_apply_imp_iff, imp_self, implies_true] ** case inl \u03b1 : Type u_1 t : Set \u03b1 hs : autoParam (Set.Finite \u2205) _auto\u271d \u22a2 ncard \u2205 \u2264 1 \u2194 \u2205 = \u2205 \u2228 \u2203 a, \u2205 = {a} ** exact iff_of_true (by simp) (Or.inl rfl) ** \u03b1 : Type u_1 t : Set \u03b1 hs : autoParam (Set.Finite \u2205) _auto\u271d \u22a2 ncard \u2205 \u2264 1 ** simp ** case inr.intro.inl \u03b1 : Type u_1 t : Set \u03b1 x : \u03b1 hs : autoParam (Set.Finite \u2205) _auto\u271d hx : x \u2208 \u2205 \u22a2 \u2200 {a b : \u03b1}, a \u2208 \u2205 \u2192 b \u2208 \u2205 \u2192 a = b ** exact (not_mem_empty _ hx).elim ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Content.contentRegular_exists_compact ** G : Type w inst\u271d : TopologicalSpace G \u03bc : Content G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 K', K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 ** by_contra hc ** G : Type w inst\u271d : TopologicalSpace G \u03bc : Content G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 hc : \u00ac\u2203 K', K.carrier \u2286 interior K'.carrier \u2227 (fun s => \u2191(toFun \u03bc s)) K' \u2264 (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u22a2 False ** simp only [not_exists, not_and, not_le] at hc ** G : Type w inst\u271d : TopologicalSpace G \u03bc : Content G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 hc : \u2200 (x : Compacts G), K.carrier \u2286 interior x.carrier \u2192 \u2191(toFun \u03bc K) + \u2191\u03b5 < \u2191(toFun \u03bc x) \u22a2 False ** have lower_bound_iInf : \u03bc K + \u03b5 \u2264\n    \u2a05 (K' : TopologicalSpace.Compacts G) (_ : (K : Set G) \u2286 interior (K' : Set G)), \u03bc K' :=\n  le_iInf fun K' => le_iInf fun K'_hyp => le_of_lt (hc K' K'_hyp) ** G : Type w inst\u271d : TopologicalSpace G \u03bc : Content G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 hc : \u2200 (x : Compacts G), K.carrier \u2286 interior x.carrier \u2192 \u2191(toFun \u03bc K) + \u2191\u03b5 < \u2191(toFun \u03bc x) lower_bound_iInf : (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u2264 \u2a05 K', \u2a05 (_ : \u2191K \u2286 interior \u2191K'), (fun s => \u2191(toFun \u03bc s)) K' \u22a2 False ** rw [\u2190 H] at lower_bound_iInf ** G : Type w inst\u271d : TopologicalSpace G \u03bc : Content G H : ContentRegular \u03bc K : Compacts G \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 hc : \u2200 (x : Compacts G), K.carrier \u2286 interior x.carrier \u2192 \u2191(toFun \u03bc K) + \u2191\u03b5 < \u2191(toFun \u03bc x) lower_bound_iInf : (fun s => \u2191(toFun \u03bc s)) K + \u2191\u03b5 \u2264 (fun s => \u2191(toFun \u03bc s)) K \u22a2 False ** exact (lt_self_iff_false (\u03bc K)).mp (lt_of_le_of_lt' lower_bound_iInf\n  (ENNReal.lt_add_right (ne_top_of_lt (\u03bc.lt_top K)) (ENNReal.coe_ne_zero.mpr h\u03b5))) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.mem_smul_finset ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b9 : DecidableEq \u03b2 inst\u271d : SMul \u03b1 \u03b2 s s\u2081 s\u2082 t u : Finset \u03b2 a : \u03b1 b x : \u03b2 \u22a2 x \u2208 a \u2022 s \u2194 \u2203 y, y \u2208 s \u2227 a \u2022 y = x ** simp only [Finset.smul_finset_def, and_assoc, mem_image, exists_prop, Prod.exists, mem_product] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.DominatedFinMeasAdditive.add ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' \u22a2 DominatedFinMeasAdditive \u03bc (T + T') (C + C') ** refine' \u27e8hT.1.add hT'.1, fun s hs h\u03bcs => _\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2016(T + T') s\u2016 \u2264 (C + C') * ENNReal.toReal (\u2191\u2191\u03bc s) ** rw [Pi.add_apply, add_mul] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2016T s + T' s\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s) + C' * ENNReal.toReal (\u2191\u2191\u03bc s) ** exact (norm_add_le _ _).trans (add_le_add (hT.2 s hs h\u03bcs) (hT'.2 s hs h\u03bcs)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsFundamentalDomain.exists_ne_one_smul_eq ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s htm : NullMeasurableSet t ht : \u2191\u2191\u03bc s < \u2191\u2191\u03bc t \u22a2 \u2203 x, x \u2208 t \u2227 \u2203 y, y \u2208 t \u2227 \u2203 g, g \u2260 1 \u2227 g \u2022 x = y ** contrapose! ht ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s htm : NullMeasurableSet t ht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y \u22a2 \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s ** refine' hs.measure_le_of_pairwise_disjoint htm (Pairwise.aedisjoint fun g\u2081 g\u2082 hne => _) ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s htm : NullMeasurableSet t ht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y g\u2081 g\u2082 : G hne : g\u2081 \u2260 g\u2082 \u22a2 (Disjoint on fun g => g \u2022 t \u2229 s) g\u2081 g\u2082 ** dsimp [Function.onFun] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s htm : NullMeasurableSet t ht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y g\u2081 g\u2082 : G hne : g\u2081 \u2260 g\u2082 \u22a2 Disjoint (g\u2081 \u2022 t \u2229 s) (g\u2082 \u2022 t \u2229 s) ** refine' (Disjoint.inf_left _ _).inf_right _ ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s htm : NullMeasurableSet t ht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y g\u2081 g\u2082 : G hne : g\u2081 \u2260 g\u2082 \u22a2 Disjoint (g\u2081 \u2022 t) (g\u2082 \u2022 t) ** rw [Set.disjoint_left] ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s htm : NullMeasurableSet t ht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y g\u2081 g\u2082 : G hne : g\u2081 \u2260 g\u2082 \u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 g\u2081 \u2022 t \u2192 \u00aca \u2208 g\u2082 \u2022 t ** rintro _ \u27e8x, hx, rfl\u27e9 \u27e8y, hy, hxy : g\u2082 \u2022 y = g\u2081 \u2022 x\u27e9 ** case intro.intro.intro.intro G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s htm : NullMeasurableSet t ht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y g\u2081 g\u2082 : G hne : g\u2081 \u2260 g\u2082 x : \u03b1 hx : x \u2208 t y : \u03b1 hy : y \u2208 t hxy : g\u2082 \u2022 y = g\u2081 \u2022 x \u22a2 False ** refine' ht x hx y hy (g\u2082\u207b\u00b9 * g\u2081) (mt inv_mul_eq_one.1 hne.symm) _ ** case intro.intro.intro.intro G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u00b2 : Group G inst\u271d\u00b9\u00b9 : Group H inst\u271d\u00b9\u2070 : MulAction G \u03b1 inst\u271d\u2079 : MeasurableSpace \u03b1 inst\u271d\u2078 : MulAction H \u03b2 inst\u271d\u2077 : MeasurableSpace \u03b2 inst\u271d\u2076 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : MeasurableSpace G inst\u271d\u2074 : MeasurableSMul G \u03b1 inst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d\u00b2 : Countable G \u03bd : Measure \u03b1 inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E hs : IsFundamentalDomain G s htm : NullMeasurableSet t ht : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 \u2200 (g : G), g \u2260 1 \u2192 g \u2022 x \u2260 y g\u2081 g\u2082 : G hne : g\u2081 \u2260 g\u2082 x : \u03b1 hx : x \u2208 t y : \u03b1 hy : y \u2208 t hxy : g\u2082 \u2022 y = g\u2081 \u2022 x \u22a2 (g\u2082\u207b\u00b9 * g\u2081) \u2022 x = y ** rw [mul_smul, \u2190 hxy, inv_smul_smul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_mul_const_uIcc ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a b c : \u03b1 ha : a = 0 \u22a2 (fun x => x * a) '' [[b, c]] = [[b * a, c * a]] ** simp [ha] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a b c : \u03b1 ha : \u00aca = 0 \u22a2 (fun x => x * a\u207b\u00b9) \u207b\u00b9' [[b, c]] = (fun x => x / a) \u207b\u00b9' [[b, c]] ** simp only [div_eq_mul_inv] ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.apply_eq_one_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 \u22a2 \u2191p a = 1 \u2194 support p = {a} ** refine' \u27e8fun h => Set.Subset.antisymm (fun a' ha' => by_contra fun ha => _)\n  fun a' ha' => ha'.symm \u25b8 (p.mem_support_iff a).2 fun ha => zero_ne_one <| ha.symm.trans h,\n  fun h => _root_.trans (symm <| tsum_eq_single a\n    fun a' ha' => (p.apply_eq_zero_iff a').2 (h.symm \u25b8 ha')) p.tsum_coe\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 h : \u2191p a = 1 a' : \u03b1 ha' : a' \u2208 support p ha : \u00aca' \u2208 {a} \u22a2 False ** suffices : 1 < \u2211' a, p a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 h : \u2191p a = 1 a' : \u03b1 ha' : a' \u2208 support p ha : \u00aca' \u2208 {a} this : 1 < \u2211' (a : \u03b1), \u2191p a \u22a2 False  case this \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 h : \u2191p a = 1 a' : \u03b1 ha' : a' \u2208 support p ha : \u00aca' \u2208 {a} \u22a2 1 < \u2211' (a : \u03b1), \u2191p a ** exact ne_of_lt this p.tsum_coe.symm ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 h : \u2191p a = 1 a' : \u03b1 ha' : a' \u2208 support p ha : \u00aca' \u2208 {a} \u22a2 1 < \u2211' (a : \u03b1), \u2191p a ** have : 0 < \u2211' b, ite (b = a) 0 (p b) := lt_of_le_of_ne' zero_le'\n  ((tsum_ne_zero_iff ENNReal.summable).2\n    \u27e8a', ite_ne_left_iff.2 \u27e8ha, Ne.symm <| (p.mem_support_iff a').2 ha'\u27e9\u27e9) ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 h : \u2191p a = 1 a' : \u03b1 ha' : a' \u2208 support p ha : \u00aca' \u2208 {a} this : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b \u22a2 1 < \u2211' (a : \u03b1), \u2191p a ** calc\n  1 = 1 + 0 := (add_zero 1).symm\n  _ < p a + \u2211' b, ite (b = a) 0 (p b) :=\n    (ENNReal.add_lt_add_of_le_of_lt ENNReal.one_ne_top (le_of_eq h.symm) this)\n  _ = ite (a = a) (p a) 0 + \u2211' b, ite (b = a) 0 (p b) := by rw [eq_self_iff_true, if_true]\n  _ = (\u2211' b, ite (b = a) (p b) 0) + \u2211' b, ite (b = a) 0 (p b) := by\n    congr\n    exact symm (tsum_eq_single a fun b hb => if_neg hb)\n  _ = \u2211' b, (ite (b = a) (p b) 0 + ite (b = a) 0 (p b)) := ENNReal.tsum_add.symm\n  _ = \u2211' b, p b := tsum_congr fun b => by split_ifs <;> simp only [zero_add, add_zero, le_rfl] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 h : \u2191p a = 1 a' : \u03b1 ha' : a' \u2208 support p ha : \u00aca' \u2208 {a} this : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b \u22a2 (\u2191p a + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b) = (if a = a then \u2191p a else 0) + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b ** rw [eq_self_iff_true, if_true] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 h : \u2191p a = 1 a' : \u03b1 ha' : a' \u2208 support p ha : \u00aca' \u2208 {a} this : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b \u22a2 ((if a = a then \u2191p a else 0) + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b) =     (\u2211' (b : \u03b1), if b = a then \u2191p b else 0) + \u2211' (b : \u03b1), if b = a then 0 else \u2191p b ** congr ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 h : \u2191p a = 1 a' : \u03b1 ha' : a' \u2208 support p ha : \u00aca' \u2208 {a} this : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b \u22a2 (if a = a then \u2191p a else 0) = \u2211' (b : \u03b1), if b = a then \u2191p b else 0 ** exact symm (tsum_eq_single a fun b hb => if_neg hb) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : PMF \u03b1 a : \u03b1 h : \u2191p a = 1 a' : \u03b1 ha' : a' \u2208 support p ha : \u00aca' \u2208 {a} this : 0 < \u2211' (b : \u03b1), if b = a then 0 else \u2191p b b : \u03b1 \u22a2 ((if b = a then \u2191p b else 0) + if b = a then 0 else \u2191p b) = \u2191p b ** split_ifs <;> simp only [zero_add, add_zero, le_rfl] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_comp_mul_deriv_Ioi ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' g : \u211d \u2192 \u211d a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => (g \u2218 f) x * f' x) (Ici a) \u22a2 \u222b (x : \u211d) in Ioi a, (g \u2218 f) x * f' x = \u222b (u : \u211d) in Ioi (f a), g u ** have hg2' : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) := by simpa [mul_comm] using hg2 ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' g : \u211d \u2192 \u211d a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => (g \u2218 f) x * f' x) (Ici a) hg2' : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) \u22a2 \u222b (x : \u211d) in Ioi a, (g \u2218 f) x * f' x = \u222b (u : \u211d) in Ioi (f a), g u ** simpa [mul_comm] using integral_comp_smul_deriv_Ioi hf hft hff' hg_cont hg1 hg2' ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' g : \u211d \u2192 \u211d a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => (g \u2218 f) x * f' x) (Ici a) \u22a2 IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) ** simpa [mul_comm] using hg2 ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.foldl ** \u03b1 : Type u_1 l m r : List Char f : \u03b1 \u2192 Char \u2192 \u03b1 init : \u03b1 \u22a2 Substring.foldl f init       { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },         stopPos := { byteIdx := utf8Len l + utf8Len m } } =     List.foldl f init m ** simp [-List.append_assoc, Substring.foldl, foldlAux_of_valid] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.exists_compl_positive_negative ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 \u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c ** obtain \u27e8f, _, hf\u2082, hf\u2081\u27e9 :=\n  exists_seq_tendsto_sInf \u27e80, @zero_mem_measureOfNegatives _ _ s\u27e9 bddBelow_measureOfNegatives ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) hf\u2081 : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c ** choose B hB using hf\u2081 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n \u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c ** have hB\u2081 : \u2200 n, MeasurableSet (B n) := fun n => (hB n).1.1 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) \u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c ** have hB\u2082 : \u2200 n, s \u2264[B n] 0 := fun n => (hB n).1.2 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) \u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c ** set A := \u22c3 n, B n with hA ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n \u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c ** have hA\u2081 : MeasurableSet A := MeasurableSet.iUnion hB\u2081 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A \u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c ** have hA\u2082 : s \u2264[A] 0 := restrict_le_restrict_iUnion _ _ hB\u2081 hB\u2082 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) \u22a2 \u2203 i, MeasurableSet i \u2227 restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c ** refine' \u27e8A\u1d9c, hA\u2081.compl, _, (compl_compl A).symm \u25b8 hA\u2082\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) \u22a2 restrict 0 A\u1d9c \u2264 restrict s A\u1d9c ** rw [restrict_le_restrict_iff _ _ hA\u2081.compl] ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) \u22a2 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 A\u1d9c \u2192 \u21910 j \u2264 \u2191s j ** intro C _ hC\u2081 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) C : Set \u03b1 a\u271d : MeasurableSet C hC\u2081 : C \u2286 A\u1d9c \u22a2 \u21910 C \u2264 \u2191s C ** by_contra' hC\u2082 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) C : Set \u03b1 a\u271d : MeasurableSet C hC\u2081 : C \u2286 A\u1d9c hC\u2082 : \u2191s C < \u21910 C \u22a2 False ** rcases exists_subset_restrict_nonpos hC\u2082 with \u27e8D, hD\u2081, hD, hD\u2082, hD\u2083\u27e9 ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) C : Set \u03b1 a\u271d : MeasurableSet C hC\u2081 : C \u2286 A\u1d9c hC\u2082 : \u2191s C < \u21910 C D : Set \u03b1 hD\u2081 : MeasurableSet D hD : D \u2286 C hD\u2082 : restrict s D \u2264 restrict 0 D hD\u2083 : \u2191s D < 0 \u22a2 False ** have : s (A \u222a D) < sInf s.measureOfNegatives := by\n  rw [\u2190 hA\u2083,\n    of_union (Set.disjoint_of_subset_right (Set.Subset.trans hD hC\u2081) disjoint_compl_right) hA\u2081\n      hD\u2081]\n  linarith ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) C : Set \u03b1 a\u271d : MeasurableSet C hC\u2081 : C \u2286 A\u1d9c hC\u2082 : \u2191s C < \u21910 C D : Set \u03b1 hD\u2081 : MeasurableSet D hD : D \u2286 C hD\u2082 : restrict s D \u2264 restrict 0 D hD\u2083 : \u2191s D < 0 this : \u2191s (A \u222a D) < sInf (measureOfNegatives s) \u22a2 False ** refine' not_le.2 this _ ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) C : Set \u03b1 a\u271d : MeasurableSet C hC\u2081 : C \u2286 A\u1d9c hC\u2082 : \u2191s C < \u21910 C D : Set \u03b1 hD\u2081 : MeasurableSet D hD : D \u2286 C hD\u2082 : restrict s D \u2264 restrict 0 D hD\u2083 : \u2191s D < 0 this : \u2191s (A \u222a D) < sInf (measureOfNegatives s) \u22a2 sInf (measureOfNegatives s) \u2264 \u2191s (A \u222a D) ** refine' csInf_le bddBelow_measureOfNegatives \u27e8A \u222a D, \u27e8_, _\u27e9, rfl\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A \u22a2 \u2191s A = sInf (measureOfNegatives s) ** apply le_antisymm ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A \u22a2 \u2191s A \u2264 sInf (measureOfNegatives s) ** refine' le_of_tendsto_of_tendsto tendsto_const_nhds hf\u2082 (eventually_of_forall fun n => _) ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A n : \u2115 \u22a2 (fun x => \u2191s A) n \u2264 f n ** rw [\u2190 (hB n).2, hA, \u2190 Set.diff_union_of_subset (Set.subset_iUnion _ n),\n  of_union Set.disjoint_sdiff_left _ (hB\u2081 n)] ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A n : \u2115 \u22a2 (fun x => \u2191s ((\u22c3 i, B i) \\ B n) + \u2191s (B n)) n \u2264 \u2191s (B n) ** refine' add_le_of_nonpos_left _ ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A n : \u2115 \u22a2 \u2191s ((\u22c3 i, B i) \\ B n) \u2264 0 ** have : s \u2264[A] 0 :=\n  restrict_le_restrict_iUnion _ _ hB\u2081 fun m =>\n    let \u27e8_, h\u27e9 := (hB m).1\n    h ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A n : \u2115 this : restrict s A \u2264 restrict 0 A \u22a2 \u2191s ((\u22c3 i, B i) \\ B n) \u2264 0 ** refine'\n  nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this) ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A n : \u2115 this : restrict s A \u2264 restrict 0 A \u22a2 MeasurableSet (\u22c3 i, B i) ** exact MeasurableSet.iUnion hB\u2081 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A n : \u2115 \u22a2 MeasurableSet ((\u22c3 i, B i) \\ B n) ** exact (MeasurableSet.iUnion hB\u2081).diff (hB\u2081 n) ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A \u22a2 sInf (measureOfNegatives s) \u2264 \u2191s A ** exact csInf_le bddBelow_measureOfNegatives \u27e8A, \u27e8hA\u2081, hA\u2082\u27e9, rfl\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) C : Set \u03b1 a\u271d : MeasurableSet C hC\u2081 : C \u2286 A\u1d9c hC\u2082 : \u2191s C < \u21910 C D : Set \u03b1 hD\u2081 : MeasurableSet D hD : D \u2286 C hD\u2082 : restrict s D \u2264 restrict 0 D hD\u2083 : \u2191s D < 0 \u22a2 \u2191s (A \u222a D) < sInf (measureOfNegatives s) ** rw [\u2190 hA\u2083,\n  of_union (Set.disjoint_of_subset_right (Set.Subset.trans hD hC\u2081) disjoint_compl_right) hA\u2081\n    hD\u2081] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) C : Set \u03b1 a\u271d : MeasurableSet C hC\u2081 : C \u2286 A\u1d9c hC\u2082 : \u2191s C < \u21910 C D : Set \u03b1 hD\u2081 : MeasurableSet D hD : D \u2286 C hD\u2082 : restrict s D \u2264 restrict 0 D hD\u2083 : \u2191s D < 0 \u22a2 \u2191s A + \u2191s D < \u2191s A ** linarith ** case intro.intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) C : Set \u03b1 a\u271d : MeasurableSet C hC\u2081 : C \u2286 A\u1d9c hC\u2082 : \u2191s C < \u21910 C D : Set \u03b1 hD\u2081 : MeasurableSet D hD : D \u2286 C hD\u2082 : restrict s D \u2264 restrict 0 D hD\u2083 : \u2191s D < 0 this : \u2191s (A \u222a D) < sInf (measureOfNegatives s) \u22a2 MeasurableSet (A \u222a D) ** exact hA\u2081.union hD\u2081 ** case intro.intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i j : Set \u03b1 s : SignedMeasure \u03b1 f : \u2115 \u2192 \u211d left\u271d : Antitone f hf\u2082 : Tendsto f atTop (nhds (sInf (measureOfNegatives s))) B : \u2115 \u2192 Set \u03b1 hB : \u2200 (n : \u2115), B n \u2208 {B | MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B} \u2227 \u2191s (B n) = f n hB\u2081 : \u2200 (n : \u2115), MeasurableSet (B n) hB\u2082 : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hA\u2081 : MeasurableSet A hA\u2082 : restrict s A \u2264 restrict 0 A hA\u2083 : \u2191s A = sInf (measureOfNegatives s) C : Set \u03b1 a\u271d : MeasurableSet C hC\u2081 : C \u2286 A\u1d9c hC\u2082 : \u2191s C < \u21910 C D : Set \u03b1 hD\u2081 : MeasurableSet D hD : D \u2286 C hD\u2082 : restrict s D \u2264 restrict 0 D hD\u2083 : \u2191s D < 0 this : \u2191s (A \u222a D) < sInf (measureOfNegatives s) \u22a2 restrict s (A \u222a D) \u2264 restrict 0 (A \u222a D) ** exact restrict_le_restrict_union _ _ hA\u2081 hA\u2082 hD\u2081 hD\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.id_traverse ** \u03b1 \u03b2 \u03b3 : Type u F G : Type u \u2192 Type u inst\u271d\u2074 : Applicative F inst\u271d\u00b3 : Applicative G inst\u271d\u00b2 : CommApplicative F inst\u271d\u00b9 : CommApplicative G inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 \u22a2 traverse pure s = s ** rw [traverse, Multiset.id_traverse] ** \u03b1 \u03b2 \u03b3 : Type u F G : Type u \u2192 Type u inst\u271d\u2074 : Applicative F inst\u271d\u00b3 : Applicative G inst\u271d\u00b2 : CommApplicative F inst\u271d\u00b9 : CommApplicative G inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 \u22a2 Multiset.toFinset <$> s.val = s ** exact s.val_toFinset ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.set_lintegral_iInf_gt_preCdf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s \u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 = \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s ** refine' le_antisymm _ _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s h : \u2200 (q : \u2191(Ioi t)), \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s \u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s ** calc\n  \u222b\u207b x in s, \u2a05 r : Ioi t, preCdf \u03c1 r x \u2202\u03c1.fst \u2264 \u2a05 q : Ioi t, \u03c1.IicSnd q s := le_iInf h\n  _ = \u03c1.IicSnd t s := Measure.iInf_IicSnd_gt \u03c1 t hs ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2200 (q : \u2191(Ioi t)), \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s ** intro q ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s q : \u2191(Ioi t) \u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u2191\u2191(Measure.IicSnd \u03c1 \u2191\u2191q) s ** rw [\u2190 set_lintegral_preCdf_fst \u03c1 _ hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s q : \u2191(Ioi t) \u22a2 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 (\u2191q) x \u2202Measure.fst \u03c1 ** refine' set_lintegral_mono_ae _ measurable_preCdf _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s q : \u2191(Ioi t) \u22a2 Measurable fun x => \u2a05 r, preCdf \u03c1 (\u2191r) x ** exact measurable_iInf fun _ => measurable_preCdf ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s q : \u2191(Ioi t) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, x \u2208 s \u2192 \u2a05 r, preCdf \u03c1 (\u2191r) x \u2264 preCdf \u03c1 (\u2191q) x ** filter_upwards [monotone_preCdf _] with a _ ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s q : \u2191(Ioi t) a : \u03b1 a\u271d : Monotone fun r => preCdf \u03c1 r a \u22a2 a \u2208 s \u2192 \u2a05 r, preCdf \u03c1 (\u2191r) a \u2264 preCdf \u03c1 (\u2191q) a ** exact fun _ => iInf_le _ q ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(Measure.IicSnd \u03c1 \u2191t) s \u2264 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 ** rw [(set_lintegral_preCdf_fst \u03c1 t hs).symm] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s \u22a2 \u222b\u207b (x : \u03b1) in s, preCdf \u03c1 t x \u2202Measure.fst \u03c1 \u2264 \u222b\u207b (x : \u03b1) in s, \u2a05 r, preCdf \u03c1 (\u2191r) x \u2202Measure.fst \u03c1 ** refine' set_lintegral_mono_ae measurable_preCdf _ _ ** case refine'_2.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s \u22a2 Measurable fun x => \u2a05 r, preCdf \u03c1 (\u2191r) x ** exact measurable_iInf fun _ => measurable_preCdf ** case refine'_2.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.fst \u03c1, x \u2208 s \u2192 preCdf \u03c1 t x \u2264 \u2a05 r, preCdf \u03c1 (\u2191r) x ** filter_upwards [monotone_preCdf _] with a ha_mono ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 t : \u211a s : Set \u03b1 hs : MeasurableSet s a : \u03b1 ha_mono : Monotone fun r => preCdf \u03c1 r a \u22a2 a \u2208 s \u2192 preCdf \u03c1 t a \u2264 \u2a05 r, preCdf \u03c1 (\u2191r) a ** exact fun _ => le_iInf fun r => ha_mono (le_of_lt r.prop) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.extend_mono ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),     Pairwise (Disjoint on f) \u2192       m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) s\u2081 s\u2082 : Set \u03b1 h\u2081 : MeasurableSet s\u2081 hs : s\u2081 \u2286 s\u2082 \u22a2 extend m s\u2081 \u2264 extend m s\u2082 ** refine' le_iInf _ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),     Pairwise (Disjoint on f) \u2192       m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) s\u2081 s\u2082 : Set \u03b1 h\u2081 : MeasurableSet s\u2081 hs : s\u2081 \u2286 s\u2082 \u22a2 \u2200 (i : (fun s => MeasurableSet s) s\u2082), extend m s\u2081 \u2264 m s\u2082 i ** intro h\u2082 ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),     Pairwise (Disjoint on f) \u2192       m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) s\u2081 s\u2082 : Set \u03b1 h\u2081 : MeasurableSet s\u2081 hs : s\u2081 \u2286 s\u2082 h\u2082 : (fun s => MeasurableSet s) s\u2082 \u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082 ** have :=\n  extend_union MeasurableSet.empty m0 MeasurableSet.iUnion mU disjoint_sdiff_self_right h\u2081\n    (h\u2082.diff h\u2081) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),     Pairwise (Disjoint on f) \u2192       m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) s\u2081 s\u2082 : Set \u03b1 h\u2081 : MeasurableSet s\u2081 hs : s\u2081 \u2286 s\u2082 h\u2082 : (fun s => MeasurableSet s) s\u2082 this : extend m (s\u2081 \u222a s\u2082 \\ s\u2081) = extend m s\u2081 + extend m (s\u2082 \\ s\u2081) \u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082 ** rw [union_diff_cancel hs] at this ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),     Pairwise (Disjoint on f) \u2192       m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) s\u2081 s\u2082 : Set \u03b1 h\u2081 : MeasurableSet s\u2081 hs : s\u2081 \u2286 s\u2082 h\u2082 : (fun s => MeasurableSet s) s\u2082 this : extend m s\u2082 = extend m s\u2081 + extend m (s\u2082 \\ s\u2081) \u22a2 extend m s\u2081 \u2264 m s\u2082 h\u2082 ** rw [\u2190 extend_eq m] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU :   \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)),     Pairwise (Disjoint on f) \u2192       m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) s\u2081 s\u2082 : Set \u03b1 h\u2081 : MeasurableSet s\u2081 hs : s\u2081 \u2286 s\u2082 h\u2082 : (fun s => MeasurableSet s) s\u2082 this : extend m s\u2082 = extend m s\u2081 + extend m (s\u2082 \\ s\u2081) \u22a2 extend m s\u2081 \u2264 extend m s\u2082 ** exact le_iff_exists_add.2 \u27e8_, this\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "PMF.bernoulli_expectation ** p : \u211d\u22650\u221e h : p \u2264 1 \u22a2 \u222b (b : Bool), bif b then 1 else 0 \u2202toMeasure (bernoulli p h) = ENNReal.toReal p ** simp [integral_eq_sum] ** Qed",
        "informal": ""
    },
    {
        "formal": "generateFrom_eq_pi ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b9 : Finite \u03b9 inst\u271d : Finite \u03b9' h : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) C : (i : \u03b9) \u2192 Set (Set (\u03b1 i)) hC : \u2200 (i : \u03b9), generateFrom (C i) = h i h2C : \u2200 (i : \u03b9), IsCountablySpanning (C i) \u22a2 generateFrom (Set.pi univ '' Set.pi univ C) = MeasurableSpace.pi ** rw [\u2190 funext hC, generateFrom_pi_eq h2C] ** Qed",
        "informal": ""
    },
    {
        "formal": "Fin.pred_mk_succ ** n i : Nat h : i < n + 1 \u22a2 pred { val := i + 1, isLt := (_ : i + 1 < n + 1 + 1) } (_ : \u00ac{ val := i + 1, isLt := (_ : i + 1 < n + 1 + 1) } = 0) =     { val := i, isLt := h } ** simp only [ext_iff, coe_pred, Nat.add_sub_cancel] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.smul_ofFunction ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 \u22a2 (c \u2022 m) \u2205 = 0 ** simp [m_empty] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 \u22a2 c \u2022 OuterMeasure.ofFunction m m_empty = OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0) ** ext1 s ** case h \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 s : Set \u03b1 \u22a2 \u2191(c \u2022 OuterMeasure.ofFunction m m_empty) s = \u2191(OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)) s ** haveI : Nonempty { t : \u2115 \u2192 Set \u03b1 // s \u2286 \u22c3 i, t i } := \u27e8\u27e8fun _ => s, subset_iUnion (fun _ => s) 0\u27e9\u27e9 ** case h \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 s : Set \u03b1 this : Nonempty { t // s \u2286 \u22c3 i, t i } \u22a2 \u2191(c \u2022 OuterMeasure.ofFunction m m_empty) s = \u2191(OuterMeasure.ofFunction (c \u2022 m) (_ : c \u2022 m \u2205 = 0)) s ** simp only [smul_apply, ofFunction_apply, ENNReal.tsum_mul_left, Pi.smul_apply, smul_eq_mul,\niInf_subtype'] ** case h \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 c : \u211d\u22650\u221e hc : c \u2260 \u22a4 s : Set \u03b1 this : Nonempty { t // s \u2286 \u22c3 i, t i } \u22a2 c * \u2a05 x, \u2211' (n : \u2115), m (\u2191x n) = \u2a05 x, c * \u2211' (i : \u2115), m (\u2191x i) ** rw [ENNReal.iInf_mul_left fun h => (hc h).elim] ** Qed",
        "informal": ""
    },
    {
        "formal": "ComputablePred.rice ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) h : ComputablePred fun c => eval c \u2208 C f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C \u22a2 g \u2208 C ** cases' h with _ h ** case intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C w\u271d : DecidablePred fun c => eval c \u2208 C h : Computable fun a => decide ((fun c => eval c \u2208 C) a) \u22a2 g \u2208 C ** obtain \u27e8c, e\u27e9 :=\n  fixed_point\u2082\n    (Partrec.cond (h.comp fst) ((Partrec.nat_iff.2 hg).comp snd).to\u2082\n        ((Partrec.nat_iff.2 hf).comp snd).to\u2082).to\u2082 ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C w\u271d : DecidablePred fun c => eval c \u2208 C h : Computable fun a => decide ((fun c => eval c \u2208 C) a) c : Code e :   eval c = fun b =>     bif decide ((fun c => eval c \u2208 C) (c, b).1) then (fun a b => g (a, b).2) (c, b).1 (c, b).2     else (fun a b => f (a, b).2) (c, b).1 (c, b).2 \u22a2 g \u2208 C ** simp at e ** case intro.intro \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C w\u271d : DecidablePred fun c => eval c \u2208 C h : Computable fun a => decide ((fun c => eval c \u2208 C) a) c : Code e : eval c = fun b => if eval c \u2208 C then g b else f b \u22a2 g \u2208 C ** by_cases H : eval c \u2208 C ** case pos \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C w\u271d : DecidablePred fun c => eval c \u2208 C h : Computable fun a => decide ((fun c => eval c \u2208 C) a) c : Code e : eval c = fun b => if eval c \u2208 C then g b else f b H : eval c \u2208 C \u22a2 g \u2208 C ** simp only [H, if_true] at e ** case pos \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C w\u271d : DecidablePred fun c => eval c \u2208 C h : Computable fun a => decide ((fun c => eval c \u2208 C) a) c : Code H : eval c \u2208 C e : eval c = fun b => g b \u22a2 g \u2208 C ** change (fun b => g b) \u2208 C ** case pos \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C w\u271d : DecidablePred fun c => eval c \u2208 C h : Computable fun a => decide ((fun c => eval c \u2208 C) a) c : Code H : eval c \u2208 C e : eval c = fun b => g b \u22a2 (fun b => g b) \u2208 C ** rwa [\u2190 e] ** case neg \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C w\u271d : DecidablePred fun c => eval c \u2208 C h : Computable fun a => decide ((fun c => eval c \u2208 C) a) c : Code e : eval c = fun b => if eval c \u2208 C then g b else f b H : \u00aceval c \u2208 C \u22a2 g \u2208 C ** simp only [H, if_false] at e ** case neg \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C w\u271d : DecidablePred fun c => eval c \u2208 C h : Computable fun a => decide ((fun c => eval c \u2208 C) a) c : Code H : \u00aceval c \u2208 C e : eval c = fun b => f b \u22a2 g \u2208 C ** rw [e] at H ** case neg \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 C : Set (\u2115 \u2192. \u2115) f g : \u2115 \u2192. \u2115 hf : Nat.Partrec f hg : Nat.Partrec g fC : f \u2208 C w\u271d : DecidablePred fun c => eval c \u2208 C h : Computable fun a => decide ((fun c => eval c \u2208 C) a) c : Code H : \u00ac(fun b => f b) \u2208 C e : eval c = fun b => f b \u22a2 g \u2208 C ** contradiction ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.prehaar_empty ** G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : TopologicalSpace G K\u2080 : PositiveCompacts G U : Set G \u22a2 prehaar (\u2191K\u2080) U \u22a5 = 0 ** rw [prehaar, Compacts.coe_bot, index_empty, Nat.cast_zero, zero_div] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_condexp_unique ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** by_cases hm : m \u2264 m0 ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm) ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m]  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** refine' (condexp_ae_eq_condexpL1 hm f).trans ((condexp_ae_eq_condexpL1 hm g).trans _).symm ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u2191\u2191(condexpL1 hm \u03bc g) =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc f) ** rw [\u2190 Lp.ext_iff] ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f ** have hn_eq : \u2200 n, condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n) := by\n  intro n\n  ext1\n  refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _)\n  exact condexp_ae_eq_condexpL1 hm (fs n) ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) hn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n) \u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f ** have hcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f)) :=\n  tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hfs_int n).1) h_int_bound_fs\n    hfs_bound hfs ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) hn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n) hcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f)) \u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f ** have hcond_gs : Tendsto (fun n => condexpL1 hm \u03bc (gs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc g)) :=\n  tendsto_condexpL1_of_dominated_convergence hm _ (fun n => (hgs_int n).1) h_int_bound_gs\n    hgs_bound hgs ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) hn_eq : \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n) hcond_fs : Tendsto (fun n => condexpL1 hm \u03bc (fs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc f)) hcond_gs : Tendsto (fun n => condexpL1 hm \u03bc (gs n)) atTop (\ud835\udcdd (condexpL1 hm \u03bc g)) \u22a2 condexpL1 hm \u03bc g = condexpL1 hm \u03bc f ** exact tendsto_nhds_unique_of_eventuallyEq hcond_gs hcond_fs (eventually_of_forall hn_eq) ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** simp_rw [condexp_of_not_le hm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : \u00acm \u2264 m0 \u22a2 0 =\u1d50[\u03bc] 0 ** rfl ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] =\u1d50[\u03bc] \u03bc[g|m] ** simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 0 =\u1d50[\u03bc] 0 ** rfl ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u2200 (n : \u2115), condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n) ** intro n ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) n : \u2115 \u22a2 condexpL1 hm \u03bc (gs n) = condexpL1 hm \u03bc (fs n) ** ext1 ** case h \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) n : \u2115 \u22a2 \u2191\u2191(condexpL1 hm \u03bc (gs n)) =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n)) ** refine' (condexp_ae_eq_condexpL1 hm (gs n)).symm.trans ((hfg n).symm.trans _) ** case h \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \ud835\udd5c F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 fs gs : \u2115 \u2192 \u03b1 \u2192 F' f g : \u03b1 \u2192 F' hfs_int : \u2200 (n : \u2115), Integrable (fs n) hgs_int : \u2200 (n : \u2115), Integrable (gs n) hfs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) hgs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => gs n x) atTop (\ud835\udcdd (g x)) bound_fs : \u03b1 \u2192 \u211d h_int_bound_fs : Integrable bound_fs bound_gs : \u03b1 \u2192 \u211d h_int_bound_gs : Integrable bound_gs hfs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016fs n x\u2016 \u2264 bound_fs x hgs_bound : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016gs n x\u2016 \u2264 bound_gs x hfg : \u2200 (n : \u2115), \u03bc[fs n|m] =\u1d50[\u03bc] \u03bc[gs n|m] hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) n : \u2115 \u22a2 \u03bc[fs n|m] =\u1d50[\u03bc] \u2191\u2191(condexpL1 hm \u03bc (fs n)) ** exact condexp_ae_eq_condexpL1 hm (fs n) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_ne_zero_iff_nonempty_of_isMulLeftInvariant ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : MeasurableSpace H inst\u271d\u2075 : TopologicalSpace G inst\u271d\u2074 : BorelSpace G \u03bc : Measure G inst\u271d\u00b3 : Group G inst\u271d\u00b2 : TopologicalGroup G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : Regular \u03bc h\u03bc : \u03bc \u2260 0 s : Set G hs : IsOpen s \u22a2 \u2191\u2191\u03bc s \u2260 0 \u2194 Set.Nonempty s ** simpa [null_iff_of_isMulLeftInvariant (\u03bc := \u03bc) hs, h\u03bc] using nonempty_iff_ne_empty.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEStronglyMeasurable.integral_condexpKernel ** \u03a9 : Type u_1 F : Type u_2 inst\u271d\u2077 : TopologicalSpace \u03a9 m m\u03a9 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsFiniteMeasure \u03bc inst\u271d\u00b2 : NormedAddCommGroup F f : \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hf : AEStronglyMeasurable f \u03bc \u22a2 AEStronglyMeasurable (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condexpKernel \u03bc m) \u03c9) \u03bc ** simp_rw [condexpKernel_apply_eq_condDistrib] ** \u03a9 : Type u_1 F : Type u_2 inst\u271d\u2077 : TopologicalSpace \u03a9 m m\u03a9 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : IsFiniteMeasure \u03bc inst\u271d\u00b2 : NormedAddCommGroup F f : \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hf : AEStronglyMeasurable f \u03bc \u22a2 AEStronglyMeasurable (fun \u03c9 => \u222b (y : \u03a9), f y \u2202\u2191(condDistrib id id \u03bc) (id \u03c9)) \u03bc ** exact AEStronglyMeasurable.integral_condDistrib\n  (aemeasurable_id'' \u03bc (inf_le_right : m \u2293 m\u03a9 \u2264 m\u03a9)) aemeasurable_id\n  (hf.comp_snd_map_prod_id inf_le_right) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.ofReal_norm_eq_lintegral ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : { x // x \u2208 Lp \u03b2 1 } \u22a2 ENNReal.ofReal \u2016f\u2016 = \u222b\u207b (x : \u03b1), \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc ** rw [norm_def, ENNReal.ofReal_toReal] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : { x // x \u2208 Lp \u03b2 1 } \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191f a\u2016\u208a \u2202\u03bc \u2260 \u22a4 ** exact ne_of_lt (hasFiniteIntegral_coeFn f) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.chaar_nonneg ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G \u22a2 0 \u2264 chaar K\u2080 K ** have := chaar_mem_haarProduct K\u2080 K (mem_univ _) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G this : chaar K\u2080 K \u2208 (fun K => Icc 0 \u2191(index \u2191K \u2191K\u2080)) K \u22a2 0 \u2264 chaar K\u2080 K ** rw [mem_Icc] at this ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K : Compacts G this : 0 \u2264 chaar K\u2080 K \u2227 chaar K\u2080 K \u2264 \u2191(index \u2191K \u2191K\u2080) \u22a2 0 \u2264 chaar K\u2080 K ** exact this.1 ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_non_aestronglyMeasurable_of_le ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d h : a \u2264 b hf : \u00acAEStronglyMeasurable f (Measure.restrict \u03bc (Ioc a b)) \u22a2 \u00acAEStronglyMeasurable (fun x => f x) (Measure.restrict \u03bc (\u0399 a b)) ** rwa [uIoc_of_le h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) ** have f_measurable : AEStronglyMeasurable f \u03bc :=\n  aestronglyMeasurable_of_tendsto_ae _ F_measurable h_lim ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) ** let b a := 2 * ENNReal.ofReal (bound a) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) ** have h : \u2200\u1d50 a \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) := by\n  rw [\u2190 ENNReal.ofReal_zero]\n  refine' h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp _\n  rwa [\u2190 tendsto_iff_norm_sub_tendsto_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) ** suffices h : Tendsto (fun n => \u222b\u207b a, ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b _ : \u03b1, 0 \u2202\u03bc)) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 0 \u2202\u03bc)) ** refine' tendsto_lintegral_of_dominated_convergence' _ _ hb _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) \u22a2 \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) n : \u2115 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a ** filter_upwards [all_ae_ofReal_F_le_bound h_bound n,\n  all_ae_ofReal_f_le_bound h_bound h_lim] with a h\u2081 h\u2082 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) n : \u2115 a : \u03b1 h\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a) h\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 ENNReal.ofReal \u2016F n a\u2016 + ENNReal.ofReal \u2016f a\u2016 ** rw [\u2190 ENNReal.ofReal_add] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) n : \u2115 a : \u03b1 h\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a) h\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 ENNReal.ofReal (\u2016F n a\u2016 + \u2016f a\u2016)  case hp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) n : \u2115 a : \u03b1 h\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a) h\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 0 \u2264 \u2016F n a\u2016  case hq \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) n : \u2115 a : \u03b1 h\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a) h\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 0 \u2264 \u2016f a\u2016 ** apply ofReal_le_ofReal ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) n : \u2115 a : \u03b1 h\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a) h\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 \u2016F n a - f a\u2016 \u2264 \u2016F n a\u2016 + \u2016f a\u2016 ** apply norm_sub_le ** case hp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) n : \u2115 a : \u03b1 h\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a) h\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 0 \u2264 \u2016F n a\u2016 ** exact norm_nonneg _ ** case hq \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) n : \u2115 a : \u03b1 h\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a) h\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 0 \u2264 \u2016f a\u2016 ** exact norm_nonneg _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) n : \u2115 a : \u03b1 h\u2081 : ENNReal.ofReal \u2016F n a\u2016 \u2264 ENNReal.ofReal (bound a) h\u2082 : ENNReal.ofReal \u2016f a\u2016 \u2264 ENNReal.ofReal (bound a) \u22a2 ENNReal.ofReal (bound a) + ENNReal.ofReal (bound a) = b a ** rw [\u2190 two_mul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) ** rw [\u2190 ENNReal.ofReal_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd (ENNReal.ofReal 0)) ** refine' h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a a : \u03b1 h : Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) \u22a2 Tendsto (fun n => \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) ** rwa [\u2190 tendsto_iff_norm_sub_tendsto_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h\u271d : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) h : Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (x : \u03b1), 0 \u2202\u03bc)) \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016F n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) ** rwa [lintegral_zero] at h ** case h.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) \u22a2 \u2200 (n : \u2115), AEMeasurable fun a => ENNReal.ofReal \u2016F n a - f a\u2016 ** exact fun n =>\n  measurable_ofReal.comp_aemeasurable ((F_measurable n).sub f_measurable).norm.aemeasurable ** case h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) \u22a2 \u222b\u207b (a : \u03b1), b a \u2202\u03bc \u2260 \u22a4 ** rw [hasFiniteIntegral_iff_ofReal] at bound_hasFiniteIntegral ** case h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) \u22a2 0 \u2264\u1d50[\u03bc] bound ** filter_upwards [h_bound 0] with _ h using le_trans (norm_nonneg _) h ** case h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc < \u22a4 h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) \u22a2 \u222b\u207b (a : \u03b1), b a \u2202\u03bc \u2260 \u22a4 ** calc\n  \u222b\u207b a, b a \u2202\u03bc = 2 * \u222b\u207b a, ENNReal.ofReal (bound a) \u2202\u03bc := by\n    rw [lintegral_const_mul']\n    exact coe_ne_top\n  _ \u2260 \u221e := mul_ne_top coe_ne_top bound_hasFiniteIntegral.ne ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc < \u22a4 h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) \u22a2 \u222b\u207b (a : \u03b1), b a \u2202\u03bc = 2 * \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc ** rw [lintegral_const_mul'] ** case hr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : \u222b\u207b (a : \u03b1), ENNReal.ofReal (bound a) \u2202\u03bc < \u22a4 h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) \u22a2 2 \u2260 \u22a4 ** exact coe_ne_top ** case h.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 F\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2 f\u271d : \u03b1 \u2192 \u03b2 bound\u271d : \u03b1 \u2192 \u211d F : \u2115 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 bound : \u03b1 \u2192 \u211d F_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (F n) \u03bc bound_hasFiniteIntegral : HasFiniteIntegral bound h_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F n a\u2016 \u2264 bound a h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) f_measurable : AEStronglyMeasurable f \u03bc b : \u03b1 \u2192 \u211d\u22650\u221e := fun a => 2 * ENNReal.ofReal (bound a) hb : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ENNReal.ofReal \u2016F n a - f a\u2016 \u2264 b a h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a - f a\u2016) atTop (\ud835\udcdd 0) ** exact h ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.eq_of_inclusion_surjective ** \u03b1 : Type u_1 s\u271d t\u271d u s t : Set \u03b1 h : s \u2286 t h_surj : Surjective (inclusion h) \u22a2 s = t ** refine' Set.Subset.antisymm h (fun x hx => _) ** \u03b1 : Type u_1 s\u271d t\u271d u s t : Set \u03b1 h : s \u2286 t h_surj : Surjective (inclusion h) x : \u03b1 hx : x \u2208 t \u22a2 x \u2208 s ** obtain \u27e8y, hy\u27e9 := h_surj \u27e8x, hx\u27e9 ** case intro \u03b1 : Type u_1 s\u271d t\u271d u s t : Set \u03b1 h : s \u2286 t h_surj : Surjective (inclusion h) x : \u03b1 hx : x \u2208 t y : \u2191s hy : inclusion h y = { val := x, property := hx } \u22a2 x \u2208 s ** exact mem_of_eq_of_mem (congr_arg Subtype.val hy).symm y.prop ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.Invariant.comp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 } \u03bc : Measure \u03b1 inst\u271d : IsSFiniteKernel \u03ba h\u03ba : Invariant \u03ba \u03bc h\u03b7 : Invariant \u03b7 \u03bc \u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc ** cases' isEmpty_or_nonempty \u03b1 with _ h\u03b1 ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 } \u03bc : Measure \u03b1 inst\u271d : IsSFiniteKernel \u03ba h\u03ba : Invariant \u03ba \u03bc h\u03b7 : Invariant \u03b7 \u03bc h\u271d : IsEmpty \u03b1 \u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc ** exact Subsingleton.elim _ _ ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 \u03ba \u03b7 : { x // x \u2208 kernel \u03b1 \u03b1 } \u03bc : Measure \u03b1 inst\u271d : IsSFiniteKernel \u03ba h\u03ba : Invariant \u03ba \u03bc h\u03b7 : Invariant \u03b7 \u03bc h\u03b1 : Nonempty \u03b1 \u22a2 Invariant (\u03ba \u2218\u2096 \u03b7) \u03bc ** simp_rw [Invariant, \u2190 comp_const_apply_eq_bind (\u03ba \u2218\u2096 \u03b7) \u03bc h\u03b1.some, comp_assoc, h\u03b7.comp_const,\n  h\u03ba.comp_const, const_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.mem_inter_iff ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 x : \u03b1 l\u2081 l\u2082 : List \u03b1 \u22a2 x \u2208 l\u2081 \u2229 l\u2082 \u2194 x \u2208 l\u2081 \u2227 x \u2208 l\u2082 ** cases l\u2081 <;> simp [List.inter_def, mem_filter] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.modifyNth_eq_modifyNthTR ** \u22a2 @modifyNth = @modifyNthTR ** funext \u03b1 f n l ** case h.h.h.h \u03b1 : Type u_1 f : \u03b1 \u2192 \u03b1 n : Nat l : List \u03b1 \u22a2 modifyNth f n l = modifyNthTR f n l ** simp [modifyNthTR, modifyNthTR_go_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.nonempty_iff_eq_singleton_default ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 s\u271d : Finset \u03b1 a b : \u03b1 inst\u271d : Unique \u03b1 s : Finset \u03b1 \u22a2 Finset.Nonempty s \u2194 s = {default} ** simp [eq_singleton_iff_nonempty_unique_mem] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.natAbs_dvd ** a b : Int e : a = \u2191(natAbs a) \u22a2 \u2191(natAbs a) \u2223 b \u2194 a \u2223 b ** rw [\u2190 e] ** a b : Int e : a = -\u2191(natAbs a) \u22a2 \u2191(natAbs a) \u2223 b \u2194 a \u2223 b ** rw [\u2190 Int.neg_dvd, \u2190 e] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.continuousWithinAt_condCdf'_Ici ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d \u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ici x) x ** rw [\u2190 continuousWithinAt_Ioi_iff_Ici] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d \u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x ** convert Monotone.tendsto_nhdsWithin_Ioi (monotone_condCdf' \u03c1 a) x ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d \u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (sInf (condCdf' \u03c1 a '' Ioi x))) ** rw [sInf_image'] ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d \u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 a_1, condCdf' \u03c1 a \u2191a_1)) ** have h' : \u2a05 r : Ioi x, condCdf' \u03c1 a r = \u2a05 r : { r' : \u211a // x < r' }, condCdf' \u03c1 a r := by\n  refine' Real.iInf_Ioi_eq_iInf_rat_gt x _ (monotone_condCdf' \u03c1 a)\n  refine' \u27e80, fun z => _\u27e9\n  rintro \u27e8u, -, rfl\u27e9\n  exact condCdf'_nonneg \u03c1 a u ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d h' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r \u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 a_1, condCdf' \u03c1 a \u2191a_1)) ** have h'' :\n  \u2a05 r : { r' : \u211a // x < r' }, condCdf' \u03c1 a r =\n    \u2a05 r : { r' : \u211a // x < r' }, condCdfRat \u03c1 a r := by\n  congr with r\n  exact condCdf'_eq_condCdfRat \u03c1 a r ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d h' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r h'' : \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r \u22a2 ContinuousWithinAt (condCdf' \u03c1 a) (Ioi x) x \u2194 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 a_1, condCdf' \u03c1 a \u2191a_1)) ** rw [h', h'', ContinuousWithinAt] ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d h' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r h'' : \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r \u22a2 Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (condCdf' \u03c1 a x)) \u2194     Tendsto (condCdf' \u03c1 a) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (\u2a05 r, condCdfRat \u03c1 a \u2191r)) ** congr! ** case a.a.h.e'_5.h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d h' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r h'' : \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r \u22a2 condCdf' \u03c1 a x = \u2a05 r, condCdfRat \u03c1 a \u2191r ** exact condCdf'_def' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d \u22a2 \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r ** refine' Real.iInf_Ioi_eq_iInf_rat_gt x _ (monotone_condCdf' \u03c1 a) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d \u22a2 BddBelow (condCdf' \u03c1 a '' Ioi x) ** refine' \u27e80, fun z => _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x z : \u211d \u22a2 z \u2208 condCdf' \u03c1 a '' Ioi x \u2192 0 \u2264 z ** rintro \u27e8u, -, rfl\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x u : \u211d \u22a2 0 \u2264 condCdf' \u03c1 a u ** exact condCdf'_nonneg \u03c1 a u ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d h' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r \u22a2 \u2a05 r, condCdf' \u03c1 a \u2191\u2191r = \u2a05 r, condCdfRat \u03c1 a \u2191r ** congr with r ** case e_s.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 x : \u211d h' : \u2a05 r, condCdf' \u03c1 a \u2191r = \u2a05 r, condCdf' \u03c1 a \u2191\u2191r r : { r' // x < \u2191r' } \u22a2 condCdf' \u03c1 a \u2191\u2191r = condCdfRat \u03c1 a \u2191r ** exact condCdf'_eq_condCdfRat \u03c1 a r ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.cgf_undef ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d hX : \u00acIntegrable fun \u03c9 => rexp (t * X \u03c9) \u22a2 cgf X \u03bc t = 0 ** simp only [cgf, mgf_undef hX, log_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.generateFrom_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u s : Set \u03b1 \u22a2 generateFrom {s} = MeasurableSpace.comap (fun x => x \u2208 s) \u22a4 ** classical\nletI : MeasurableSpace \u03b1 := generateFrom {s}\nrefine' le_antisymm (generateFrom_le fun t ht => \u27e8{True}, trivial, by simp [ht.symm]\u27e9) _\nrintro _ \u27e8u, -, rfl\u27e9\nexact (show MeasurableSet s from GenerateMeasurable.basic _ <| mem_singleton s).mem trivial ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u s : Set \u03b1 \u22a2 generateFrom {s} = MeasurableSpace.comap (fun x => x \u2208 s) \u22a4 ** letI : MeasurableSpace \u03b1 := generateFrom {s} ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u s : Set \u03b1 this : MeasurableSpace \u03b1 := generateFrom {s} \u22a2 generateFrom {s} = MeasurableSpace.comap (fun x => x \u2208 s) \u22a4 ** refine' le_antisymm (generateFrom_le fun t ht => \u27e8{True}, trivial, by simp [ht.symm]\u27e9) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u s : Set \u03b1 this : MeasurableSpace \u03b1 := generateFrom {s} \u22a2 MeasurableSpace.comap (fun x => x \u2208 s) \u22a4 \u2264 generateFrom {s} ** rintro _ \u27e8u, -, rfl\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u\u271d s : Set \u03b1 this : MeasurableSpace \u03b1 := generateFrom {s} u : Set Prop \u22a2 MeasurableSet ((fun x => x \u2208 s) \u207b\u00b9' u) ** exact (show MeasurableSet s from GenerateMeasurable.basic _ <| mem_singleton s).mem trivial ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t\u271d u s : Set \u03b1 this : MeasurableSpace \u03b1 := generateFrom {s} t : Set \u03b1 ht : t \u2208 {s} \u22a2 (fun x => x \u2208 s) \u207b\u00b9' {True} = t ** simp [ht.symm] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.const_unit_eq_compProd ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : MeasurableSpace \u03a9 inst\u271d\u00b2 : BorelSpace \u03a9 inst\u271d\u00b9 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d : IsFiniteMeasure \u03c1 \u22a2 const Unit \u03c1 = const Unit (Measure.fst \u03c1) \u2297\u2096 prodMkLeft Unit (Measure.condKernel \u03c1) ** simp_rw [condKernel_def] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03a9 : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : MeasurableSpace \u03a9 inst\u271d\u00b2 : BorelSpace \u03a9 inst\u271d\u00b9 : Nonempty \u03a9 \u03c1 : Measure (\u03b1 \u00d7 \u03a9) inst\u271d : IsFiniteMeasure \u03c1 \u22a2 const Unit \u03c1 =     const Unit (Measure.fst \u03c1) \u2297\u2096       prodMkLeft Unit (Exists.choose (_ : \u2203 \u03b7 _h, const Unit \u03c1 = const Unit (Measure.fst \u03c1) \u2297\u2096 prodMkLeft Unit \u03b7)) ** exact (exists_cond_kernel \u03c1 Unit).choose_spec.choose_spec ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsStoppingTime.measurableSet_inter_le ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : MeasurableSet s \u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) ** simp_rw [IsStoppingTime.measurableSet] at hs \u22a2 ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) \u22a2 \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i}) ** intro i ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 this :   s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =     s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i} \u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i}) ** rw [this] ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 this :   s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =     s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i} \u22a2 MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i}) ** refine' ((hs i).inter ((h\u03c4.min h\u03c0) i)).inter _ ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 this :   s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =     s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i} \u22a2 MeasurableSet {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i} ** apply @measurableSet_le _ _ _ _ _ (Filtration.seq f i) _ _ _ _ _ ?_ ?_ ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u22a2 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =     s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i} ** ext1 \u03c9 ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 \u22a2 \u03c9 \u2208 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2194     \u03c9 \u2208 s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i} ** simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,\n  and_true_iff, true_or_iff, or_true_iff] ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 \u22a2 (\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 \u03c0 \u03c9) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i) \u2194 ((\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 i) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i)) \u2227 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9) ** by_cases h\u03c4i : \u03c4 \u03c9 \u2264 i ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i \u22a2 (\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 \u03c0 \u03c9) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i) \u2194 ((\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 i) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i)) \u2227 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9) ** simp only [h\u03c4i, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,\n  and_imp] ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i \u22a2 \u03c9 \u2208 s \u2192 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2192 i < \u03c0 \u03c9 ** refine' fun _ h\u03c4_le_\u03c0 => lt_of_lt_of_le _ h\u03c4_le_\u03c0 ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i x\u271d : \u03c9 \u2208 s h\u03c4_le_\u03c0 : \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u22a2 i < \u03c4 \u03c9 ** rw [\u2190 not_le] ** case neg \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u00ac\u03c4 \u03c9 \u2264 i x\u271d : \u03c9 \u2208 s h\u03c4_le_\u03c0 : \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u22a2 \u00ac\u03c4 \u03c9 \u2264 i ** exact h\u03c4i ** case pos \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u03c4 \u03c9 \u2264 i \u22a2 (\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 \u03c0 \u03c9) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i) \u2194 ((\u03c9 \u2208 s \u2227 \u03c4 \u03c9 \u2264 i) \u2227 (\u03c4 \u03c9 \u2264 i \u2228 \u03c0 \u03c9 \u2264 i)) \u2227 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9) ** simp only [h\u03c4i, true_or_iff, and_true_iff, and_congr_right_iff] ** case pos \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u03c4 \u03c9 \u2264 i \u22a2 \u03c9 \u2208 s \u2192 (\u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2194 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9) ** intro ** case pos \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u03c4 \u03c9 \u2264 i a\u271d : \u03c9 \u2208 s \u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2194 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9 ** constructor <;> intro h ** case pos.mp \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u03c4 \u03c9 \u2264 i a\u271d : \u03c9 \u2208 s h : \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9 ** exact Or.inl h ** case pos.mpr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u03c4 \u03c9 \u2264 i a\u271d : \u03c9 \u2208 s h : \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u2228 i \u2264 \u03c0 \u03c9 \u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 ** cases' h with h h ** case pos.mpr.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u03c4 \u03c9 \u2264 i a\u271d : \u03c9 \u2208 s h : \u03c4 \u03c9 \u2264 \u03c0 \u03c9 \u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 ** exact h ** case pos.mpr.inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 \u03c9 : \u03a9 h\u03c4i : \u03c4 \u03c9 \u2264 i a\u271d : \u03c9 \u2208 s h : i \u2264 \u03c0 \u03c9 \u22a2 \u03c4 \u03c9 \u2264 \u03c0 \u03c9 ** exact h\u03c4i.trans h ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 this :   s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =     s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i} \u22a2 Measurable fun a => min (\u03c4 a) i ** exact (h\u03c4.min_const i).measurable_of_le fun _ => min_le_right _ _ ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u2075 : LinearOrder \u03b9 f : Filtration \u03b9 m \u03c4 \u03c0 : \u03a9 \u2192 \u03b9 inst\u271d\u2074 : TopologicalSpace \u03b9 inst\u271d\u00b3 : SecondCountableTopology \u03b9 inst\u271d\u00b2 : OrderTopology \u03b9 inst\u271d\u00b9 : MeasurableSpace \u03b9 inst\u271d : BorelSpace \u03b9 h\u03c4 : IsStoppingTime f \u03c4 h\u03c0 : IsStoppingTime f \u03c0 s : Set \u03a9 hs : \u2200 (i : \u03b9), MeasurableSet (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) i : \u03b9 this :   s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} =     s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) (\u03c0 \u03c9) \u2264 i} \u2229 {\u03c9 | min (\u03c4 \u03c9) i \u2264 min (min (\u03c4 \u03c9) (\u03c0 \u03c9)) i} \u22a2 Measurable fun a => min (min (\u03c4 a) (\u03c0 a)) i ** exact ((h\u03c4.min h\u03c0).min_const i).measurable_of_le fun _ => min_le_right _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.integral_add_measure ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : SMulCommClass \u211d \ud835\udd5c E \u03bd : Measure \u03b1 f : \u03b1 \u2192\u209b E hf : Integrable \u2191f \u22a2 integral (\u03bc + \u03bd) f = integral \u03bc f + integral \u03bd f ** simp_rw [integral_def] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : SMulCommClass \u211d \ud835\udd5c E \u03bd : Measure \u03b1 f : \u03b1 \u2192\u209b E hf : Integrable \u2191f \u22a2 setToSimpleFunc (weightedSMul (\u03bc + \u03bd)) f = setToSimpleFunc (weightedSMul \u03bc) f + setToSimpleFunc (weightedSMul \u03bd) f ** refine' setToSimpleFunc_add_left'\n  (weightedSMul \u03bc) (weightedSMul \u03bd) (weightedSMul (\u03bc + \u03bd)) (fun s _ h\u03bc\u03bds => _) hf ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : SMulCommClass \u211d \ud835\udd5c E \u03bd : Measure \u03b1 f : \u03b1 \u2192\u209b E hf : Integrable \u2191f s : Set \u03b1 x\u271d : MeasurableSet s h\u03bc\u03bds : \u2191\u2191(\u03bc + \u03bd) s < \u22a4 \u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s ** rw [lt_top_iff_ne_top, Measure.coe_add, Pi.add_apply, ENNReal.add_ne_top] at h\u03bc\u03bds ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F p : \u211d\u22650\u221e G : Type u_5 F' : Type u_6 inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedAddCommGroup F' inst\u271d\u2074 : NormedSpace \u211d F' m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : NormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : SMulCommClass \u211d \ud835\udd5c E \u03bd : Measure \u03b1 f : \u03b1 \u2192\u209b E hf : Integrable \u2191f s : Set \u03b1 x\u271d : MeasurableSet s h\u03bc\u03bds : \u2191\u2191\u03bc s \u2260 \u22a4 \u2227 \u2191\u2191\u03bd s \u2260 \u22a4 \u22a2 weightedSMul (\u03bc + \u03bd) s = weightedSMul \u03bc s + weightedSMul \u03bd s ** rw [weightedSMul_add_measure _ _ h\u03bc\u03bds.1 h\u03bc\u03bds.2] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Submartingale.exists_ae_trim_tendsto_of_bdd ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc (_ : sSup (Set.range fun n => \u2191\u2131 n) \u2264 m0), \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** letI := (\u2a06 n, \u2131 n) ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R this : MeasurableSpace \u03a9 := \u2a06 n, \u2191\u2131 n \u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc (_ : sSup (Set.range fun n => \u2191\u2131 n) \u2264 m0), \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) ** rw [ae_iff, trim_measurableSet_eq] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R this : MeasurableSpace \u03a9 := \u2a06 n, \u2191\u2131 n \u22a2 \u2191\u2191\u03bc {a | \u00ac\u2203 c, Tendsto (fun n => f n a) atTop (\ud835\udcdd c)} = 0 ** exact hf.exists_ae_tendsto_of_bdd hbdd ** case hs \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 inst\u271d : IsFiniteMeasure \u03bc hf : Submartingale f \u2131 \u03bc hbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R this : MeasurableSpace \u03a9 := \u2a06 n, \u2191\u2131 n \u22a2 MeasurableSet {a | \u00ac\u2203 c, Tendsto (fun n => f n a) atTop (\ud835\udcdd c)} ** exact MeasurableSet.compl $ measurableSet_exists_tendsto\n  fun n => (hf.stronglyMeasurable n).measurable.mono (le_sSup \u27e8n, rfl\u27e9) le_rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) C : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) = \u2211' (n : \u2124), \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) \u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2211' (n : \u2124), \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) ** rw [A, B, C, add_assoc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t \u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) ** rw [\u2190 measure_union] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t \u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0) ** congr 1 ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t \u22a2 s = s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0 ** ext x ** case e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t x : \u03b1 \u22a2 x \u2208 s \u2194 x \u2208 s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0 ** have : 0 = f x \u2228 0 < f x := eq_or_lt_of_le bot_le ** case e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t x : \u03b1 this : 0 = f x \u2228 0 < f x \u22a2 x \u2208 s \u2194 x \u2208 s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0 ** rw [eq_comm] at this ** case e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t x : \u03b1 this : f x = 0 \u2228 0 < f x \u22a2 x \u2208 s \u2194 x \u2208 s \u2229 f \u207b\u00b9' {0} \u222a s \u2229 f \u207b\u00b9' Ioi 0 ** simp only [\u2190 and_or_left, this, mem_singleton_iff, mem_inter_iff, and_true_iff, mem_union,\n  mem_Ioi, mem_preimage] ** case hd \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t \u22a2 Disjoint (s \u2229 f \u207b\u00b9' {0}) (s \u2229 f \u207b\u00b9' Ioi 0) ** refine disjoint_left.2 fun x hx h'x => ?_ ** case hd \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' {0} h'x : x \u2208 s \u2229 f \u207b\u00b9' Ioi 0 \u22a2 False ** have : 0 < f x := h'x.2 ** case hd \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' {0} h'x : x \u2208 s \u2229 f \u207b\u00b9' Ioi 0 this : 0 < f x \u22a2 False ** exact lt_irrefl 0 (this.trans_le hx.2.le) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t \u22a2 MeasurableSet (s \u2229 f \u207b\u00b9' Ioi 0) ** exact hs.inter (hf measurableSet_Ioi) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) \u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) ** rw [\u2190 measure_union] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) \u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4} \u222a s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) ** rw [\u2190 inter_union_distrib_left] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) \u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 (f \u207b\u00b9' {\u22a4} \u222a f \u207b\u00b9' Ioo 0 \u22a4)) ** congr ** case e_a.e_a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) \u22a2 f \u207b\u00b9' Ioi 0 = f \u207b\u00b9' {\u22a4} \u222a f \u207b\u00b9' Ioo 0 \u22a4 ** ext x ** case e_a.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) x : \u03b1 \u22a2 x \u2208 f \u207b\u00b9' Ioi 0 \u2194 x \u2208 f \u207b\u00b9' {\u22a4} \u222a f \u207b\u00b9' Ioo 0 \u22a4 ** simp only [mem_singleton_iff, mem_union, mem_Ioo, mem_Ioi, mem_preimage] ** case e_a.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) x : \u03b1 \u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4 ** have H : f x = \u221e \u2228 f x < \u221e := eq_or_lt_of_le le_top ** case e_a.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) x : \u03b1 H : f x = \u22a4 \u2228 f x < \u22a4 \u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4 ** cases' H with H H ** case e_a.e_a.h.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) x : \u03b1 H : f x = \u22a4 \u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4 ** simp only [H, eq_self_iff_true, or_false_iff, WithTop.zero_lt_top, not_top_lt,\n  and_false_iff] ** case e_a.e_a.h.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) x : \u03b1 H : f x < \u22a4 \u22a2 0 < f x \u2194 f x = \u22a4 \u2228 0 < f x \u2227 f x < \u22a4 ** simp only [H, H.ne, and_true_iff, false_or_iff] ** case hd \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) \u22a2 Disjoint (s \u2229 f \u207b\u00b9' {\u22a4}) (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) ** refine disjoint_left.2 fun x hx h'x => ?_ ** case hd \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' {\u22a4} h'x : x \u2208 s \u2229 f \u207b\u00b9' Ioo 0 \u22a4 \u22a2 False ** have : f x < \u221e := h'x.2.2 ** case hd \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' {\u22a4} h'x : x \u2208 s \u2229 f \u207b\u00b9' Ioo 0 \u22a4 this : f x < \u22a4 \u22a2 False ** exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm)) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) \u22a2 MeasurableSet (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) ** exact hs.inter (hf measurableSet_Ioo) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) \u22a2 \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) = \u2211' (n : \u2124), \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) ** rw [\u2190 measure_iUnion,\n  ENNReal.Ioo_zero_top_eq_iUnion_Ico_zpow (ENNReal.one_lt_coe_iff.2 ht) ENNReal.coe_ne_top,\n  preimage_iUnion, inter_iUnion] ** case hn \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) \u22a2 Pairwise (Disjoint on fun n => s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) ** intro i j ** case hn \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) i j : \u2124 \u22a2 i \u2260 j \u2192 (Disjoint on fun n => s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) i j ** simp only [Function.onFun] ** case hn \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) i j : \u2124 \u22a2 i \u2260 j \u2192 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1))) ** intro hij ** case hn \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) i j : \u2124 hij : i \u2260 j \u22a2 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1))) ** wlog h : i < j generalizing i j ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) i j : \u2124 hij : i \u2260 j h : i < j \u22a2 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1))) ** refine disjoint_left.2 fun x hx h'x => lt_irrefl (f x) ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) i j : \u2124 hij : i \u2260 j h : i < j x : \u03b1 hx : x \u2208 s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1)) h'x : x \u2208 s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1)) \u22a2 f x < f x ** calc\n  f x < (t : \u211d\u22650\u221e) ^ (i + 1) := hx.2.2\n  _ \u2264 (t : \u211d\u22650\u221e) ^ j := (ENNReal.zpow_le_of_le (ENNReal.one_le_coe_iff.2 ht.le) h)\n  _ \u2264 f x := h'x.2.1 ** case hn.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) i j : \u2124 hij : i \u2260 j this :   \u2200 \u2983i j : \u2124\u2984, i \u2260 j \u2192 i < j \u2192 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1))) h : \u00aci < j \u22a2 Disjoint (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ j) (\u2191t ^ (j + 1))) ** exact (this hij.symm (hij.lt_or_lt.resolve_left h)).symm ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) \u22a2 \u2200 (i : \u2124), MeasurableSet (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ i) (\u2191t ^ (i + 1))) ** intro n ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t\u271d u : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f s : Set \u03b1 hs : MeasurableSet s t : \u211d\u22650 ht : 1 < t A : \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {0}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) B : \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' {\u22a4}) + \u2191\u2191\u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u22a4) n : \u2124 \u22a2 MeasurableSet (s \u2229 f \u207b\u00b9' Ico (\u2191t ^ n) (\u2191t ^ (n + 1))) ** exact hs.inter (hf measurableSet_Ico) ** Qed",
        "informal": ""
    },
    {
        "formal": "Tree.numLeaves_pos ** \u03b1 : Type u x : Tree \u03b1 \u22a2 0 < numLeaves x ** rw [numLeaves_eq_numNodes_succ] ** \u03b1 : Type u x : Tree \u03b1 \u22a2 0 < numNodes x + 1 ** exact x.numNodes.zero_lt_succ ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.sub_ediv_of_dvd ** a b c : Int hcb : c \u2223 b \u22a2 (a - b) / c = a / c - b / c ** rw [Int.sub_eq_add_neg, Int.sub_eq_add_neg, Int.add_ediv_of_dvd_right (Int.dvd_neg.2 hcb)] ** a b c : Int hcb : c \u2223 b \u22a2 a / c + -b / c = a / c + -(b / c) ** congr ** case e_a a b c : Int hcb : c \u2223 b \u22a2 -b / c = -(b / c) ** exact Int.neg_ediv_of_dvd hcb ** Qed",
        "informal": ""
    },
    {
        "formal": "Partrec.bind_decode\u2082_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192. \u03c3 h : Nat.Partrec fun n => Part.bind \u2191(decode\u2082 \u03b1 n) fun a => Part.map encode (f a) \u22a2 Partrec fun a => Part.map encode (f a) ** simpa [encodek\u2082] using (nat_iff.2 h).comp (@Computable.encode \u03b1 _) ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hab : a \u2264 b hcont : ContinuousOn f (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x f'int : IntervalIntegrable f' volume a b \u22a2 \u222b (y : \u211d) in a..b, f' y = f b - f a ** refine' (NormedSpace.eq_iff_forall_dual_eq \u211d).2 fun g => _ ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hab : a \u2264 b hcont : ContinuousOn f (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x f'int : IntervalIntegrable f' volume a b g : NormedSpace.Dual \u211d E \u22a2 \u2191g (\u222b (y : \u211d) in a..b, f' y) = \u2191g (f b - f a) ** rw [\u2190 g.intervalIntegral_comp_comm f'int, g.map_sub] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d : \u211d \u2192 E g' g\u271d \u03c6 : \u211d \u2192 \u211d f f' : \u211d \u2192 E a b : \u211d hab : a \u2264 b hcont : ContinuousOn f (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x f'int : IntervalIntegrable f' volume a b g : NormedSpace.Dual \u211d E \u22a2 \u222b (x : \u211d) in a..b, \u2191g (f' x) = \u2191g (f b) - \u2191g (f a) ** exact integral_eq_sub_of_hasDeriv_right_of_le_real hab (g.continuous.comp_continuousOn hcont)\n  (fun x hx => g.hasFDerivAt.comp_hasDerivWithinAt x (hderiv x hx))\n  (g.integrable_comp ((intervalIntegrable_iff_integrable_Icc_of_le hab).1 f'int)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Holor.cprankMax_nil ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : AddMonoid \u03b1 x : Holor \u03b1 [] \u22a2 CPRankMax 1 x ** have h := CPRankMax.succ 0 x 0 (CPRankMax1.nil x) CPRankMax.zero ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : AddMonoid \u03b1 x : Holor \u03b1 [] h : CPRankMax (0 + 1) (x + 0) \u22a2 CPRankMax 1 x ** rwa [add_zero x, zero_add] at h ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.one_lt_encard_iff ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 1 < encard s \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b ** rw [\u2190not_iff_not, not_exists, not_lt, encard_le_one_iff] ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 (\u2200 (a b : \u03b1), a \u2208 s \u2192 b \u2208 s \u2192 a = b) \u2194 \u2200 (x : \u03b1), \u00ac\u2203 b, x \u2208 s \u2227 b \u2208 s \u2227 x \u2260 b ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.hasFiniteIntegral_prod_mk_left ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 s : Set (\u03b2 \u00d7 \u03b3) h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4 \u22a2 HasFiniteIntegral fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) ** let t := toMeasurable ((\u03ba \u2297\u2096 \u03b7) a) s ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 s : Set (\u03b2 \u00d7 \u03b3) h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4 t : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u22a2 HasFiniteIntegral fun b => ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) ** simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 s : Set (\u03b2 \u00d7 \u03b3) h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4 t : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u22a2 \u222b\u207b (a_1 : \u03b2), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s))) \u2202\u2191\u03ba a < \u22a4 ** calc\n  \u222b\u207b b, ENNReal.ofReal (\u03b7 (a, b) (Prod.mk b \u207b\u00b9' s)).toReal \u2202\u03ba a\n  _ \u2264 \u222b\u207b b, \u03b7 (a, b) (Prod.mk b \u207b\u00b9' t) \u2202\u03ba a := by\n    refine' lintegral_mono_ae _\n    filter_upwards [ae_kernel_lt_top a h2s] with b hb\n    rw [ofReal_toReal hb.ne]\n    exact measure_mono (preimage_mono (subset_toMeasurable _ _))\n  _ \u2264 (\u03ba \u2297\u2096 \u03b7) a t := (le_compProd_apply _ _ _ _)\n  _ = (\u03ba \u2297\u2096 \u03b7) a s := (measure_toMeasurable s)\n  _ < \u22a4 := h2s.lt_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 s : Set (\u03b2 \u00d7 \u03b3) h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4 t : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u22a2 \u222b\u207b (b : \u03b2), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))) \u2202\u2191\u03ba a \u2264     \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t) \u2202\u2191\u03ba a ** refine' lintegral_mono_ae _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 s : Set (\u03b2 \u00d7 \u03b3) h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4 t : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u22a2 \u2200\u1d50 (a_1 : \u03b2) \u2202\u2191\u03ba a,     ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' s))) \u2264 \u2191\u2191(\u2191\u03b7 (a, a_1)) (Prod.mk a_1 \u207b\u00b9' t) ** filter_upwards [ae_kernel_lt_top a h2s] with b hb ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 s : Set (\u03b2 \u00d7 \u03b3) h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4 t : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s b : \u03b2 hb : \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) < \u22a4 \u22a2 ENNReal.ofReal (ENNReal.toReal (\u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s))) \u2264 \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t) ** rw [ofReal_toReal hb.ne] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 E : Type u_4 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 inst\u271d\u00b2 : NormedAddCommGroup E \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 s : Set (\u03b2 \u00d7 \u03b3) h2s : \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s \u2260 \u22a4 t : Set (\u03b2 \u00d7 \u03b3) := toMeasurable (\u2191(\u03ba \u2297\u2096 \u03b7) a) s b : \u03b2 hb : \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) < \u22a4 \u22a2 \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s) \u2264 \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' t) ** exact measure_mono (preimage_mono (subset_toMeasurable _ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSet.analyticSet ** \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 t : TopologicalSpace \u03b1 inst\u271d\u00b2 : PolishSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : BorelSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s \u22a2 AnalyticSet s ** obtain \u27e8t', t't, t'_polish, s_closed, _\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b1, t' \u2264 t \u2227 @PolishSpace \u03b1 t' \u2227 IsClosed[t'] s \u2227 IsOpen[t'] s :=\n  hs.isClopenable ** case intro.intro.intro.intro \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 t : TopologicalSpace \u03b1 inst\u271d\u00b2 : PolishSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : BorelSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s t' : TopologicalSpace \u03b1 t't : t' \u2264 t t'_polish : PolishSpace \u03b1 s_closed : IsClosed s right\u271d : IsOpen s \u22a2 AnalyticSet s ** have A := @IsClosed.analyticSet \u03b1 t' t'_polish s s_closed ** case intro.intro.intro.intro \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 t : TopologicalSpace \u03b1 inst\u271d\u00b2 : PolishSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : BorelSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s t' : TopologicalSpace \u03b1 t't : t' \u2264 t t'_polish : PolishSpace \u03b1 s_closed : IsClosed s right\u271d : IsOpen s A : AnalyticSet s \u22a2 AnalyticSet s ** convert @AnalyticSet.image_of_continuous \u03b1 t' \u03b1 t s A id (continuous_id_of_le t't) ** case h.e'_3 \u03b1\u271d : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : TopologicalSpace \u03b1\u271d \u03b1 : Type u_3 t : TopologicalSpace \u03b1 inst\u271d\u00b2 : PolishSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : BorelSpace \u03b1 s : Set \u03b1 hs : MeasurableSet s t' : TopologicalSpace \u03b1 t't : t' \u2264 t t'_polish : PolishSpace \u03b1 s_closed : IsClosed s right\u271d : IsOpen s A : AnalyticSet s \u22a2 s = id '' s ** simp only [id.def, image_id'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.inv_mem_inv ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Inv \u03b1 a : Part \u03b1 ma : \u03b1 ha : ma \u2208 a \u22a2 ma\u207b\u00b9 \u2208 a\u207b\u00b9 ** simp [inv_def] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Inv \u03b1 a : Part \u03b1 ma : \u03b1 ha : ma \u2208 a \u22a2 \u2203 a_1, a_1 \u2208 a \u2227 a_1\u207b\u00b9 = ma\u207b\u00b9 ** aesop ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Nat.mem_antidiagonal ** n : \u2115 x : \u2115 \u00d7 \u2115 \u22a2 x \u2208 antidiagonal n \u2194 x.1 + x.2 = n ** rw [antidiagonal, mem_def, Multiset.Nat.mem_antidiagonal] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_le_ncard_insert ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 \u22a2 ncard s \u2264 ncard (insert a s) ** classical\nrefine'\n  s.finite_or_infinite.elim (fun h \u21a6 _) (fun h \u21a6 by (rw [h.ncard]; exact Nat.zero_le _))\nrw [ncard_insert_eq_ite h]; split_ifs <;> simp ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 \u22a2 ncard s \u2264 ncard (insert a s) ** refine'\n  s.finite_or_infinite.elim (fun h \u21a6 _) (fun h \u21a6 by (rw [h.ncard]; exact Nat.zero_le _)) ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 h : Set.Finite s \u22a2 ncard s \u2264 ncard (insert a s) ** rw [ncard_insert_eq_ite h] ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 h : Set.Finite s \u22a2 ncard s \u2264 if a \u2208 s then ncard s else ncard s + 1 ** split_ifs <;> simp ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 h : Set.Infinite s \u22a2 ncard s \u2264 ncard (insert a s) ** (rw [h.ncard]; exact Nat.zero_le _) ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 h : Set.Infinite s \u22a2 ncard s \u2264 ncard (insert a s) ** rw [h.ncard] ** \u03b1 : Type u_1 s\u271d t : Set \u03b1 a : \u03b1 s : Set \u03b1 h : Set.Infinite s \u22a2 0 \u2264 ncard (insert a s) ** exact Nat.zero_le _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.tendsto_IicSnd_atBot ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 Tendsto (fun r => \u2191\u2191(IicSnd \u03c1 \u2191r) s) atBot (\ud835\udcdd 0) ** simp_rw [\u03c1.IicSnd_apply _ hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd 0) ** have h_empty : \u03c1 (s \u00d7\u02e2 \u2205) = 0 := by simp only [prod_empty, measure_empty] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd 0) ** rw [\u2190 h_empty, \u2190 Real.iInter_Iic_rat, prod_iInter] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i))) ** suffices h_neg :\n  Tendsto (fun r : \u211a => \u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u03c1 (\u22c2 r : \u211a, s \u00d7\u02e2 Iic \u2191(-r)))) ** case h_neg \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r)))) ** refine' tendsto_measure_iInter (fun q => hs.prod measurableSet_Iic) _ \u27e80, measure_ne_top \u03c1 _\u27e9 ** case h_neg \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 \u22a2 Antitone fun r => s \u00d7\u02e2 Iic \u2191(-r) ** refine' fun q r hqr => prod_subset_prod_iff.mpr (Or.inl \u27e8subset_rfl, fun x hx => _\u27e9) ** case h_neg \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 q r : \u211a hqr : q \u2264 r x : \u211d hx : x \u2208 Iic \u2191(-r) \u22a2 x \u2208 Iic \u2191(-q) ** simp only [Rat.cast_neg, mem_Iic] at hx \u22a2 ** case h_neg \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 q r : \u211a hqr : q \u2264 r x : \u211d hx : x \u2264 -\u2191r \u22a2 x \u2264 -\u2191q ** refine' hx.trans (neg_le_neg _) ** case h_neg \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 q r : \u211a hqr : q \u2264 r x : \u211d hx : x \u2264 -\u2191r \u22a2 \u2191q \u2264 \u2191r ** exact_mod_cast hqr ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 ** simp only [prod_empty, measure_empty] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r)))) h_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i))) ** rw [h_inter_eq] at h_neg ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r))) h_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i))) ** have h_fun_eq : (fun r : \u211a => \u03c1 (s \u00d7\u02e2 Iic (r : \u211d))) = fun r : \u211a => \u03c1 (s \u00d7\u02e2 Iic \u2191(- -r)) := by\n  simp_rw [neg_neg] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r))) h_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r h_fun_eq : (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r)) \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i))) ** rw [h_fun_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r))) h_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r h_fun_eq : (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r)) \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r))) atBot (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 i, s \u00d7\u02e2 Iic \u2191i))) ** exact h_neg.comp tendsto_neg_atBot_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r)))) \u22a2 \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r ** ext1 x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r)))) x : \u03b1 \u00d7 \u211d \u22a2 x \u2208 \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) \u2194 x \u2208 \u22c2 r, s \u00d7\u02e2 Iic \u2191r ** simp only [Rat.cast_eq_id, id.def, mem_iInter, mem_prod, mem_Iic] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r)))) x : \u03b1 \u00d7 \u211d \u22a2 (\u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i)) \u2194 \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191i ** refine' \u27e8fun h i => \u27e8(h i).1, _\u27e9, fun h i => \u27e8(h i).1, _\u27e9\u27e9 <;> have h' := h (-i) ** case h.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r)))) x : \u03b1 \u00d7 \u211d h : \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i) i : \u211a h' : x.1 \u2208 s \u2227 x.2 \u2264 \u2191(- -i) \u22a2 x.2 \u2264 \u2191i ** rw [neg_neg] at h' ** case h.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r)))) x : \u03b1 \u00d7 \u211d h : \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i) i : \u211a h' : x.1 \u2208 s \u2227 x.2 \u2264 \u2191i \u22a2 x.2 \u2264 \u2191i ** exact h'.2 ** case h.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191(-r)))) x : \u03b1 \u00d7 \u211d h : \u2200 (i : \u211a), x.1 \u2208 s \u2227 x.2 \u2264 \u2191i i : \u211a h' : x.1 \u2208 s \u2227 x.2 \u2264 \u2191(-i) \u22a2 x.2 \u2264 \u2191(-i) ** exact h'.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set \u03b1 hs : MeasurableSet s h_empty : \u2191\u2191\u03c1 (s \u00d7\u02e2 \u2205) = 0 h_neg : Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(-r))) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c2 r, s \u00d7\u02e2 Iic \u2191r))) h_inter_eq : \u22c2 r, s \u00d7\u02e2 Iic \u2191(-r) = \u22c2 r, s \u00d7\u02e2 Iic \u2191r \u22a2 (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) = fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191(- -r)) ** simp_rw [neg_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "Computable.option_casesOn ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 o : \u03b1 \u2192 Option \u03b2 f : \u03b1 \u2192 \u03c3 g : \u03b1 \u2192 \u03b2 \u2192 \u03c3 ho : Computable o hf : Computable f hg : Computable\u2082 g a : \u03b1 \u22a2 (Nat.casesOn (encode (o a)) (Option.some (f a)) fun n => Option.map (g a) (decode n)) =     Option.some (Option.casesOn (o a) (f a) (g a)) ** cases o a <;> simp [encodek] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.exists_rename_eq_of_vars_subset_range ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c4 \u2192 \u03c3 hfi : Injective f hf : \u2191(vars p) \u2286 range f \u22a2 \u2191(rename f) (\u2191(aeval fun i => Option.elim' 0 X (partialInv f i)) p) = p ** show (rename f).toRingHom.comp _ p = RingHom.id _ p ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c4 \u2192 \u03c3 hfi : Injective f hf : \u2191(vars p) \u2286 range f \u22a2 \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) p = \u2191(RingHom.id (MvPolynomial \u03c3 R)) p ** refine' hom_congr_vars _ _ _ ** case refine'_1 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c4 \u2192 \u03c3 hfi : Injective f hf : \u2191(vars p) \u2286 range f \u22a2 RingHom.comp (RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) C =     RingHom.comp (RingHom.id (MvPolynomial \u03c3 R)) C ** ext1 ** case refine'_1.a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c4 \u2192 \u03c3 hfi : Injective f hf : \u2191(vars p) \u2286 range f x\u271d : R \u22a2 \u2191(RingHom.comp (RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) C) x\u271d =     \u2191(RingHom.comp (RingHom.id (MvPolynomial \u03c3 R)) C) x\u271d ** simp [algebraMap_eq] ** case refine'_2 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c4 \u2192 \u03c3 hfi : Injective f hf : \u2191(vars p) \u2286 range f \u22a2 \u2200 (i : \u03c3),     i \u2208 vars p \u2192       i \u2208 vars p \u2192         \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =           \u2191(RingHom.id (MvPolynomial \u03c3 R)) (X i) ** intro i hip _ ** case refine'_2 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c4 \u2192 \u03c3 hfi : Injective f hf : \u2191(vars p) \u2286 range f i : \u03c3 hip a\u271d : i \u2208 vars p \u22a2 \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) (X i) =     \u2191(RingHom.id (MvPolynomial \u03c3 R)) (X i) ** rcases hf hip with \u27e8i, rfl\u27e9 ** case refine'_2.intro R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c4 \u2192 \u03c3 hfi : Injective f hf : \u2191(vars p) \u2286 range f i : \u03c4 hip a\u271d : f i \u2208 vars p \u22a2 \u2191(RingHom.comp \u2191(rename f) \u2191(aeval fun i => Option.elim' 0 X (partialInv f i))) (X (f i)) =     \u2191(RingHom.id (MvPolynomial \u03c3 R)) (X (f i)) ** simp [partialInv_left hfi] ** case refine'_3 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q : MvPolynomial \u03c3 R inst\u271d : CommSemiring S p : MvPolynomial \u03c3 R f : \u03c4 \u2192 \u03c3 hfi : Injective f hf : \u2191(vars p) \u2286 range f \u22a2 p = p ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.mem_list_toFinmap ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 xs : List (Sigma \u03b2) \u22a2 a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs ** induction' xs with x xs ** case cons \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 x : Sigma \u03b2 xs : List (Sigma \u03b2) tail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs \u22a2 a \u2208 List.toFinmap (x :: xs) \u2194 \u2203 b, { fst := a, snd := b } \u2208 x :: xs ** cases' x with fst_i snd_i ** case cons.mk \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 xs : List (Sigma \u03b2) tail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs fst_i : \u03b1 snd_i : \u03b2 fst_i \u22a2 a \u2208 List.toFinmap ({ fst := fst_i, snd := snd_i } :: xs) \u2194     \u2203 b, { fst := a, snd := b } \u2208 { fst := fst_i, snd := snd_i } :: xs ** simp only [toFinmap_cons, *, exists_or, mem_cons, mem_insert, exists_and_left, Sigma.mk.inj_iff] ** case cons.mk \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 xs : List (Sigma \u03b2) tail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs fst_i : \u03b1 snd_i : \u03b2 fst_i \u22a2 (a = fst_i \u2228 \u2203 b, { fst := a, snd := b } \u2208 xs) \u2194 (a = fst_i \u2227 \u2203 x, HEq x snd_i) \u2228 \u2203 x, { fst := a, snd := x } \u2208 xs ** refine (or_congr_left <| and_iff_left_of_imp ?_).symm ** case cons.mk \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 xs : List (Sigma \u03b2) tail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs fst_i : \u03b1 snd_i : \u03b2 fst_i \u22a2 a = fst_i \u2192 \u2203 x, HEq x snd_i ** rintro rfl ** case cons.mk \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 xs : List (Sigma \u03b2) tail_ih\u271d : a \u2208 List.toFinmap xs \u2194 \u2203 b, { fst := a, snd := b } \u2208 xs snd_i : \u03b2 a \u22a2 \u2203 x, HEq x snd_i ** simp only [exists_eq, heq_iff_eq] ** case nil \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 \u22a2 a \u2208 List.toFinmap [] \u2194 \u2203 b, { fst := a, snd := b } \u2208 [] ** simp only [toFinmap_nil, not_mem_empty, find?, not_mem_nil, exists_false] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.nullMeasurableSet ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t : Set \u03b1 inst\u271d : MeasurableSingletonClass (NullMeasurableSpace \u03b1) s : Finset \u03b1 \u22a2 NullMeasurableSet \u2191s ** apply Finset.measurableSet ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.tendsto_IicSnd_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u03b1 hs : MeasurableSet s \u22a2 Tendsto (fun r => \u2191\u2191(IicSnd \u03c1 \u2191r) s) atTop (\ud835\udcdd (\u2191\u2191(fst \u03c1) s)) ** simp_rw [\u03c1.IicSnd_apply _ hs, fst_apply hs, \u2190 prod_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u03b1 hs : MeasurableSet s \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (s \u00d7\u02e2 univ))) ** rw [\u2190 Real.iUnion_Iic_rat, prod_iUnion] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u03b1 hs : MeasurableSet s \u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (s \u00d7\u02e2 Iic \u2191r)) atTop (\ud835\udcdd (\u2191\u2191\u03c1 (\u22c3 i, s \u00d7\u02e2 Iic \u2191i))) ** refine' tendsto_measure_iUnion fun r q hr_le_q x => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u03b1 hs : MeasurableSet s r q : \u211a hr_le_q : r \u2264 q x : \u03b1 \u00d7 \u211d \u22a2 x \u2208 s \u00d7\u02e2 Iic \u2191r \u2192 x \u2208 s \u00d7\u02e2 Iic \u2191q ** simp only [mem_prod, mem_Iic, and_imp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u03b1 hs : MeasurableSet s r q : \u211a hr_le_q : r \u2264 q x : \u03b1 \u00d7 \u211d \u22a2 x.1 \u2208 s \u2192 x.2 \u2264 \u2191r \u2192 x.1 \u2208 s \u2227 x.2 \u2264 \u2191q ** refine' fun hxs hxr => \u27e8hxs, hxr.trans _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) s : Set \u03b1 hs : MeasurableSet s r q : \u211a hr_le_q : r \u2264 q x : \u03b1 \u00d7 \u211d hxs : x.1 \u2208 s hxr : x.2 \u2264 \u2191r \u22a2 \u2191r \u2264 \u2191q ** exact_mod_cast hr_le_q ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.supported_le_supported_iff ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 inst\u271d : Nontrivial R \u22a2 supported R s \u2264 supported R t \u2194 s \u2286 t ** constructor ** case mp \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 inst\u271d : Nontrivial R \u22a2 supported R s \u2264 supported R t \u2192 s \u2286 t ** intro h i ** case mp \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 inst\u271d : Nontrivial R h : supported R s \u2264 supported R t i : \u03c3 \u22a2 i \u2208 s \u2192 i \u2208 t ** simpa using @h (X i) ** case mpr \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d\u00b9 : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 inst\u271d : Nontrivial R \u22a2 s \u2286 t \u2192 supported R s \u2264 supported R t ** exact supported_mono ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.range_list_getD ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : List \u03b1 d : \u03b1 \u22a2 (fun o => Option.getD o d) '' range (get? l) = insert d {x | x \u2208 l} ** simp only [range_list_get?, image_insert_eq, Option.getD, image_image, image_id'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pred_ncard_le_ncard_diff_singleton ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 a : \u03b1 \u22a2 ncard s - 1 \u2264 ncard (s \\ {a}) ** cases' s.finite_or_infinite with hs hs ** case inr \u03b1 : Type u_1 s\u271d t s : Set \u03b1 a : \u03b1 hs : Set.Infinite s \u22a2 ncard s - 1 \u2264 ncard (s \\ {a}) ** convert Nat.zero_le _ ** case h.e'_3 \u03b1 : Type u_1 s\u271d t s : Set \u03b1 a : \u03b1 hs : Set.Infinite s \u22a2 ncard s - 1 = 0 ** rw [hs.ncard] ** case inl \u03b1 : Type u_1 s\u271d t s : Set \u03b1 a : \u03b1 hs : Set.Finite s \u22a2 ncard s - 1 \u2264 ncard (s \\ {a}) ** by_cases h : a \u2208 s ** case neg \u03b1 : Type u_1 s\u271d t s : Set \u03b1 a : \u03b1 hs : Set.Finite s h : \u00aca \u2208 s \u22a2 ncard s - 1 \u2264 ncard (s \\ {a}) ** rw [diff_singleton_eq_self h] ** case neg \u03b1 : Type u_1 s\u271d t s : Set \u03b1 a : \u03b1 hs : Set.Finite s h : \u00aca \u2208 s \u22a2 ncard s - 1 \u2264 ncard s ** apply Nat.pred_le ** case pos \u03b1 : Type u_1 s\u271d t s : Set \u03b1 a : \u03b1 hs : Set.Finite s h : a \u2208 s \u22a2 ncard s - 1 \u2264 ncard (s \\ {a}) ** rw [ncard_diff_singleton_of_mem h hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.forall_mem_pwFilter ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a : \u03b1 l : List \u03b1 \u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2194 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b ** refine \u27e8?_, fun h b hb => h _ <| pwFilter_subset (R := R) _ hb\u27e9 ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a : \u03b1 l : List \u03b1 \u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b ** induction l with\n| nil => exact fun _ _ h => (not_mem_nil _ h).elim\n| cons x l IH =>\n  simp only [forall_mem_cons]\n  if h : \u2200 y \u2208 pwFilter R l, R x y then\n    simpa [pwFilter_cons_of_pos h] using fun r H => \u27e8r, IH H\u27e9\n  else\n    refine pwFilter_cons_of_neg h \u25b8 fun H => \u27e8?_, IH H\u27e9\n    match e : find? (fun y => \u00acR x y) (pwFilter R l) with\n    | none => exact h.elim fun y hy => by simpa using find?_eq_none.1 e y hy\n    | some k =>\n      have := find?_some e\n      apply (neg_trans (H k (mem_of_find?_eq_some e))).resolve_right\n      rw [decide_eq_true_iff] at this; exact this ** case nil \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a : \u03b1 \u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R [] \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 [] \u2192 R a b ** exact fun _ _ h => (not_mem_nil _ h).elim ** case cons \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b \u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 x :: l \u2192 R a b ** simp only [forall_mem_cons] ** case cons \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b \u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x ** if h : \u2200 y \u2208 pwFilter R l, R x y then\n  simpa [pwFilter_cons_of_pos h] using fun r H => \u27e8r, IH H\u27e9\nelse\n  refine pwFilter_cons_of_neg h \u25b8 fun H => \u27e8?_, IH H\u27e9\n  match e : find? (fun y => \u00acR x y) (pwFilter R l) with\n  | none => exact h.elim fun y hy => by simpa using find?_eq_none.1 e y hy\n  | some k =>\n    have := find?_some e\n    apply (neg_trans (H k (mem_of_find?_eq_some e))).resolve_right\n    rw [decide_eq_true_iff] at this; exact this ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b h : \u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y \u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x ** simpa [pwFilter_cons_of_pos h] using fun r H => \u27e8r, IH H\u27e9 ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b h : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y \u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x ** refine pwFilter_cons_of_neg h \u25b8 fun H => \u27e8?_, IH H\u27e9 ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b h : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y H : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b \u22a2 R a x ** match e : find? (fun y => \u00acR x y) (pwFilter R l) with\n| none => exact h.elim fun y hy => by simpa using find?_eq_none.1 e y hy\n| some k =>\n  have := find?_some e\n  apply (neg_trans (H k (mem_of_find?_eq_some e))).resolve_right\n  rw [decide_eq_true_iff] at this; exact this ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b h : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y H : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b e : find? (fun y => decide \u00acR x y) (pwFilter R l) = none \u22a2 R a x ** exact h.elim fun y hy => by simpa using find?_eq_none.1 e y hy ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b h : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y H : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b e : find? (fun y => decide \u00acR x y) (pwFilter R l) = none y : \u03b1 hy : y \u2208 pwFilter R l \u22a2 R x y ** simpa using find?_eq_none.1 e y hy ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b h : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y H : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b k : \u03b1 e : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k \u22a2 R a x ** have := find?_some e ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b h : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y H : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b k : \u03b1 e : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k this : (decide \u00acR x k) = true \u22a2 R a x ** apply (neg_trans (H k (mem_of_find?_eq_some e))).resolve_right ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b h : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y H : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b k : \u03b1 e : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k this : (decide \u00acR x k) = true \u22a2 \u00acR x k ** rw [decide_eq_true_iff] at this ** \u03b1 : Type u_1 R : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel R neg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z a x : \u03b1 l : List \u03b1 IH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b h : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y H : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b k : \u03b1 e : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k this : \u00acR x k \u22a2 \u00acR x k ** exact this ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.testAgainstNN_add ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2075 : SMul R \u211d\u22650 inst\u271d\u2074 : SMul R \u211d\u22650\u221e inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 f\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 testAgainstNN \u03bc (f\u2081 + f\u2082) = testAgainstNN \u03bc f\u2081 + testAgainstNN \u03bc f\u2082 ** simp only [\u2190 ENNReal.coe_eq_coe, BoundedContinuousFunction.coe_add, ENNReal.coe_add, Pi.add_apply,\n  testAgainstNN_coe_eq] ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2075 : SMul R \u211d\u22650 inst\u271d\u2074 : SMul R \u211d\u22650\u221e inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : FiniteMeasure \u03a9 f\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) + \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc = \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) \u2202\u2191\u03bc + \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc ** exact lintegral_add_left (BoundedContinuousFunction.measurable_coe_ennreal_comp _) _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.toGerm_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 inst\u271d : TopologicalSpace \u03b4 f : \u03b1 \u2192\u2098[\u03bc] \u03b2 \u22a2 toGerm f = \u2191\u2191f ** rw [\u2190 mk_toGerm, mk_coeFn] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.powerset_eq_singleton_empty ** \u03b1 : Type u_1 s t : Finset \u03b1 \u22a2 powerset s = {\u2205} \u2194 s = \u2205 ** rw [\u2190 powerset_empty, powerset_inj] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.offDiag_insert ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 x : \u03b1 \u00d7 \u03b1 a : \u03b1 has : \u00aca \u2208 s \u22a2 offDiag (insert a s) = offDiag s \u222a {a} \u00d7\u02e2 s \u222a s \u00d7\u02e2 {a} ** rw [insert_eq, union_comm, offDiag_union (disjoint_singleton_right.2 has), offDiag_singleton,\n  union_empty, union_right_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measurePreserving_piCongrLeft ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b2 : Fintype \u03b9 m\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) m : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d : Fintype \u03b9' f : \u03b9' \u2243 \u03b9 \u22a2 Measure.map (\u2191(MeasurableEquiv.piCongrLeft \u03b1 f)) (Measure.pi fun i' => \u03bc (\u2191f i')) = Measure.pi \u03bc ** refine' (pi_eq fun s _ => _).symm ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 inst\u271d\u00b2 : Fintype \u03b9 m\u271d : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i) m : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i) \u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i) inst\u271d\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i) inst\u271d : Fintype \u03b9' f : \u03b9' \u2243 \u03b9 s : (i : \u03b9) \u2192 Set (\u03b1 i) x\u271d : \u2200 (i : \u03b9), MeasurableSet (s i) \u22a2 \u2191\u2191(Measure.map (\u2191(MeasurableEquiv.piCongrLeft \u03b1 f)) (Measure.pi fun i' => \u03bc (\u2191f i'))) (Set.pi univ s) =     \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i) ** rw [MeasurableEquiv.map_apply, MeasurableEquiv.coe_piCongrLeft f,\n  Equiv.piCongrLeft_preimage_univ_pi, pi_pi _ _, f.prod_comp (fun i => \u03bc i (s i))] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.Heap.WF.singleton ** \u03b1\u271d : Type u_1 a : \u03b1\u271d le : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool \u22a2 0 \u2264 0 ** decide ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FinStronglyMeasurable.neg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : TopologicalAddGroup \u03b2 hf : FinStronglyMeasurable f \u03bc \u22a2 FinStronglyMeasurable (-f) \u03bc ** refine' \u27e8fun n => -hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).neg\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : TopologicalAddGroup \u03b2 hf : FinStronglyMeasurable f \u03bc n : \u2115 \u22a2 \u2191\u2191\u03bc (support \u2191((fun n => -FinStronglyMeasurable.approx hf n) n)) < \u22a4 ** suffices \u03bc (Function.support fun x => -(hf.approx n) x) < \u221e by convert this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : TopologicalAddGroup \u03b2 hf : FinStronglyMeasurable f \u03bc n : \u2115 \u22a2 \u2191\u2191\u03bc (support fun x => -\u2191(FinStronglyMeasurable.approx hf n) x) < \u22a4 ** rw [Function.support_neg (hf.approx n)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : TopologicalAddGroup \u03b2 hf : FinStronglyMeasurable f \u03bc n : \u2115 \u22a2 \u2191\u2191\u03bc (support \u2191(FinStronglyMeasurable.approx hf n)) < \u22a4 ** exact hf.fin_support_approx n ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : Countable \u03b9 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : AddGroup \u03b2 inst\u271d : TopologicalAddGroup \u03b2 hf : FinStronglyMeasurable f \u03bc n : \u2115 this : \u2191\u2191\u03bc (support fun x => -\u2191(FinStronglyMeasurable.approx hf n) x) < \u22a4 \u22a2 \u2191\u2191\u03bc (support \u2191((fun n => -FinStronglyMeasurable.approx hf n) n)) < \u22a4 ** convert this ** Qed",
        "informal": ""
    },
    {
        "formal": "Basis.addHaar_eq_iff ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9\u2070 : Fintype \u03b9 inst\u271d\u2079 : Fintype \u03b9' inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : SecondCountableTopology E b : Basis \u03b9 \u211d E \u03bc : Measure E inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : IsAddLeftInvariant \u03bc \u22a2 addHaar b = \u03bc \u2194 \u2191\u2191\u03bc \u2191(parallelepiped b) = 1 ** rw [Basis.addHaar_def] ** \u03b9 : Type u_1 \u03b9' : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9\u2070 : Fintype \u03b9 inst\u271d\u2079 : Fintype \u03b9' inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : BorelSpace E inst\u271d\u00b2 : SecondCountableTopology E b : Basis \u03b9 \u211d E \u03bc : Measure E inst\u271d\u00b9 : SigmaFinite \u03bc inst\u271d : IsAddLeftInvariant \u03bc \u22a2 addHaarMeasure (parallelepiped b) = \u03bc \u2194 \u2191\u2191\u03bc \u2191(parallelepiped b) = 1 ** exact addHaarMeasure_eq_iff b.parallelepiped \u03bc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.InnerRegular.isCompact_isClosed ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X \u22a2 InnerRegular \u03bc IsCompact IsClosed ** intro F hF r hr ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc F \u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K ** set B : \u2115 \u2192 Set X := compactCovering X ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc F B : \u2115 \u2192 Set X := compactCovering X \u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K ** have hBc : \u2200 n, IsCompact (F \u2229 B n) := fun n => (isCompact_compactCovering X n).inter_left hF ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc F B : \u2115 \u2192 Set X := compactCovering X hBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n) \u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K ** have hBU : \u22c3 n, F \u2229 B n = F := by rw [\u2190 inter_iUnion, iUnion_compactCovering, Set.inter_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc F B : \u2115 \u2192 Set X := compactCovering X hBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n) hBU : \u22c3 n, F \u2229 B n = F \u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K ** have : \u03bc F = \u2a06 n, \u03bc (F \u2229 B n) := by\n  rw [\u2190 measure_iUnion_eq_iSup, hBU]\n  exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc F B : \u2115 \u2192 Set X := compactCovering X hBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n) hBU : \u22c3 n, F \u2229 B n = F this : \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n) \u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K ** rw [this] at hr ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e B : \u2115 \u2192 Set X := compactCovering X hr : r < \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n) hBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n) hBU : \u22c3 n, F \u2229 B n = F this : \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n) \u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K ** rcases lt_iSup_iff.1 hr with \u27e8n, hn\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e B : \u2115 \u2192 Set X := compactCovering X hr : r < \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n) hBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n) hBU : \u22c3 n, F \u2229 B n = F this : \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n) n : \u2115 hn : r < \u2191\u2191\u03bc (F \u2229 B n) \u22a2 \u2203 K, K \u2286 F \u2227 IsCompact K \u2227 r < \u2191\u2191\u03bc K ** exact \u27e8_, inter_subset_left _ _, hBc n, hn\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc F B : \u2115 \u2192 Set X := compactCovering X hBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n) \u22a2 \u22c3 n, F \u2229 B n = F ** rw [\u2190 inter_iUnion, iUnion_compactCovering, Set.inter_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc F B : \u2115 \u2192 Set X := compactCovering X hBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n) hBU : \u22c3 n, F \u2229 B n = F \u22a2 \u2191\u2191\u03bc F = \u2a06 n, \u2191\u2191\u03bc (F \u2229 B n) ** rw [\u2190 measure_iUnion_eq_iSup, hBU] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03bc\u271d : Measure \u03b1 p q : Set \u03b1 \u2192 Prop U s : Set \u03b1 \u03b5 r\u271d : \u211d\u22650\u221e X : Type u_3 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : SigmaCompactSpace X inst\u271d : MeasurableSpace X \u03bc : Measure X F : Set X hF : IsClosed F r : \u211d\u22650\u221e hr : r < \u2191\u2191\u03bc F B : \u2115 \u2192 Set X := compactCovering X hBc : \u2200 (n : \u2115), IsCompact (F \u2229 B n) hBU : \u22c3 n, F \u2229 B n = F \u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => F \u2229 B n ** exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h) ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.valMinAbs_natCast_eq_self ** n a : \u2115 inst\u271d : NeZero n \u22a2 valMinAbs \u2191a = \u2191a \u2194 a \u2264 n / 2 ** refine' \u27e8fun ha => _, valMinAbs_natCast_of_le_half\u27e9 ** n a : \u2115 inst\u271d : NeZero n ha : valMinAbs \u2191a = \u2191a \u22a2 a \u2264 n / 2 ** rw [\u2190 Int.natAbs_ofNat a, \u2190 ha] ** n a : \u2115 inst\u271d : NeZero n ha : valMinAbs \u2191a = \u2191a \u22a2 Int.natAbs (valMinAbs \u2191a) \u2264 n / 2 ** exact natAbs_valMinAbs_le a ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.tr_respects_aux\u2083 ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : Stmt\u2082 v : \u03c3 L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n : \u2115 \u22a2 Reaches\u2080 (TM1.step (tr M))     { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }     { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) } ** induction' n with n IH ** case succ K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : Stmt\u2082 v : \u03c3 L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n : \u2115 IH :   Reaches\u2080 (TM1.step (tr M))     { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }     { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) } \u22a2 Reaches\u2080 (TM1.step (tr M))     { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) }     { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) } ** refine' Reaches\u2080.head _ IH ** case succ K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : Stmt\u2082 v : \u03c3 L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n : \u2115 IH :   Reaches\u2080 (TM1.step (tr M))     { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }     { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) } \u22a2 { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } \u2208     TM1.step (tr M)       { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) } ** simp only [Option.mem_def, TM1.step] ** case succ K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : Stmt\u2082 v : \u03c3 L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n : \u2115 IH :   Reaches\u2080 (TM1.step (tr M))     { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }     { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) } \u22a2 some (TM1.stepAux (tr M (ret q)) v ((Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)))) =     some { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } ** rw [Option.some_inj, tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat,\n  addBottom_nth_succ_fst, TM1.stepAux, iterate_succ', Function.comp_apply, Tape.move_right_left] ** case succ K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : Stmt\u2082 v : \u03c3 L : ListBlank ((k : K) \u2192 Option (\u0393 k)) n : \u2115 IH :   Reaches\u2080 (TM1.step (tr M))     { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }     { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) } \u22a2 (bif false then       TM1.stepAux (trNormal q) v (Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))))     else TM1.stepAux (goto fun x x => ret q) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))) =     { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } ** rfl ** case zero K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : Stmt\u2082 v : \u03c3 L : ListBlank ((k : K) \u2192 Option (\u0393 k)) \u22a2 Reaches\u2080 (TM1.step (tr M))     { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.zero] (Tape.mk' \u2205 (addBottom L)) }     { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) } ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.elim_preimage_pi ** \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i j : \u03b9 l\u271d : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 l : List \u03b9 hnd : Nodup l h : \u2200 (i : \u03b9), i \u2208 l t : (i : \u03b9) \u2192 Set (\u03b1 i) \u22a2 TProd.elim' h \u207b\u00b9' pi univ t = Set.tprod l t ** have h2 : { i | i \u2208 l } = univ := by\n  ext i\n  simp [h] ** \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i j : \u03b9 l\u271d : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 l : List \u03b9 hnd : Nodup l h : \u2200 (i : \u03b9), i \u2208 l t : (i : \u03b9) \u2192 Set (\u03b1 i) h2 : {i | i \u2208 l} = univ \u22a2 TProd.elim' h \u207b\u00b9' pi univ t = Set.tprod l t ** rw [\u2190 h2, \u2190 mk_preimage_tprod, preimage_preimage] ** \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i j : \u03b9 l\u271d : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 l : List \u03b9 hnd : Nodup l h : \u2200 (i : \u03b9), i \u2208 l t : (i : \u03b9) \u2192 Set (\u03b1 i) h2 : {i | i \u2208 l} = univ \u22a2 (fun x => TProd.mk l (TProd.elim' h x)) \u207b\u00b9' Set.tprod l t = Set.tprod l t ** simp only [TProd.mk_elim hnd h] ** \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i j : \u03b9 l\u271d : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 l : List \u03b9 hnd : Nodup l h : \u2200 (i : \u03b9), i \u2208 l t : (i : \u03b9) \u2192 Set (\u03b1 i) h2 : {i | i \u2208 l} = univ \u22a2 (fun x => x) \u207b\u00b9' Set.tprod l t = Set.tprod l t ** dsimp ** \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i j : \u03b9 l\u271d : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 l : List \u03b9 hnd : Nodup l h : \u2200 (i : \u03b9), i \u2208 l t : (i : \u03b9) \u2192 Set (\u03b1 i) h2 : {i | i \u2208 l} = univ \u22a2 Set.tprod l t = Set.tprod l t ** rfl ** \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i j : \u03b9 l\u271d : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 l : List \u03b9 hnd : Nodup l h : \u2200 (i : \u03b9), i \u2208 l t : (i : \u03b9) \u2192 Set (\u03b1 i) \u22a2 {i | i \u2208 l} = univ ** ext i ** case h \u03b9 : Type u \u03b1 : \u03b9 \u2192 Type v i\u271d j : \u03b9 l\u271d : List \u03b9 f : (i : \u03b9) \u2192 \u03b1 i inst\u271d : DecidableEq \u03b9 l : List \u03b9 hnd : Nodup l h : \u2200 (i : \u03b9), i \u2208 l t : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 \u22a2 i \u2208 {i | i \u2208 l} \u2194 i \u2208 univ ** simp [h] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.measurable_range_sup'' ** M : Type u_1 inst\u271d\u00b3 : MeasurableSpace M \u03b1 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 M \u03b4 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : MeasurableSup\u2082 \u03b1 f : \u2115 \u2192 \u03b4 \u2192 \u03b1 n : \u2115 hf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k) \u22a2 Measurable fun x => sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x ** convert Finset.measurable_range_sup' hf using 1 ** case h.e'_5 M : Type u_1 inst\u271d\u00b3 : MeasurableSpace M \u03b1 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 M \u03b4 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : MeasurableSup\u2082 \u03b1 f : \u2115 \u2192 \u03b4 \u2192 \u03b1 n : \u2115 hf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k) \u22a2 (fun x => sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x) =     sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k ** ext x ** case h.e'_5.h M : Type u_1 inst\u271d\u00b3 : MeasurableSpace M \u03b1 : Type u_2 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g : \u03b1 \u2192 M \u03b4 : Type u_3 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : MeasurableSup\u2082 \u03b1 f : \u2115 \u2192 \u03b4 \u2192 \u03b1 n : \u2115 hf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k) x : \u03b4 \u22a2 (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x) =     sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) (fun k => f k) x ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "String.Pos.zero_addString_byteIdx ** s : String \u22a2 (0 + s).byteIdx = utf8ByteSize s ** simp only [addString_byteIdx, byteIdx_zero, Nat.zero_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.withDensity_kernel_sum ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d : Countable \u03b9 \u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i) f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e \u22a2 withDensity (kernel.sum \u03ba) f = kernel.sum fun i => withDensity (\u03ba i) f ** by_cases hf : Measurable (Function.uncurry f) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d : Countable \u03b9 \u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i) f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (Function.uncurry f) \u22a2 withDensity (kernel.sum \u03ba) f = kernel.sum fun i => withDensity (\u03ba i) f ** ext1 a ** case pos.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d : Countable \u03b9 \u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i) f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable (Function.uncurry f) a : \u03b1 \u22a2 \u2191(withDensity (kernel.sum \u03ba) f) a = \u2191(kernel.sum fun i => withDensity (\u03ba i) f) a ** simp_rw [sum_apply, kernel.withDensity_apply _ hf, sum_apply,\n  withDensity_sum (fun n => \u03ba n a) (f a)] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d : Countable \u03b9 \u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i) f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u00acMeasurable (Function.uncurry f) \u22a2 withDensity (kernel.sum \u03ba) f = kernel.sum fun i => withDensity (\u03ba i) f ** simp_rw [withDensity_of_not_measurable _ hf] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d : Countable \u03b9 \u03ba : \u03b9 \u2192 { x // x \u2208 kernel \u03b1 \u03b2 } h\u03ba : \u2200 (i : \u03b9), IsSFiniteKernel (\u03ba i) f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u00acMeasurable (Function.uncurry f) \u22a2 0 = kernel.sum fun i => 0 ** exact sum_zero.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Multiset.noncommProd_map ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 s : Multiset \u03b1 comm : Set.Pairwise {x | x \u2208 s} Commute f : F \u22a2 \u2191f (noncommProd s comm) = noncommProd (map (\u2191f) s) (_ : Set.Pairwise {x | x \u2208 map (\u2191f) s} Commute) ** induction s using Quotient.inductionOn ** case h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b2 : Monoid \u03b1 inst\u271d\u00b9 : Monoid \u03b2 inst\u271d : MonoidHomClass F \u03b1 \u03b2 f : F a\u271d : List \u03b1 comm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute \u22a2 \u2191f (noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm) =     noncommProd (map (\u2191f) (Quotient.mk (List.isSetoid \u03b1) a\u271d))       (_ : Set.Pairwise {x | x \u2208 map (\u2191f) (Quotient.mk (List.isSetoid \u03b1) a\u271d)} Commute) ** simpa using map_list_prod f _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.get_set_eq_if ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 v : Vector \u03b1 n i j : Fin n a : \u03b1 \u22a2 get (set v i a) j = if i = j then a else get v j ** split_ifs <;> (try simp [*]) ** case neg n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 v : Vector \u03b1 n i j : Fin n a : \u03b1 h\u271d : \u00aci = j \u22a2 get (set v i a) j = get v j ** rwa [get_set_of_ne] ** case pos n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 v : Vector \u03b1 n i j : Fin n a : \u03b1 h\u271d : i = j \u22a2 get (set v i a) j = a ** try simp [*] ** case pos n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 v : Vector \u03b1 n i j : Fin n a : \u03b1 h\u271d : i = j \u22a2 get (set v i a) j = a ** simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_union\u2080_aux ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hs : NullMeasurableSet s ht : NullMeasurableSet t hd : AEDisjoint \u03bc s t \u22a2 \u2191\u2191\u03bc (s \u222a t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t ** rw [union_eq_iUnion, measure_iUnion\u2080, tsum_fintype, Fintype.sum_bool, cond, cond] ** case hd \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hs : NullMeasurableSet s ht : NullMeasurableSet t hd : AEDisjoint \u03bc s t \u22a2 Pairwise (AEDisjoint \u03bc on fun b => bif b then s else t)  case h \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hs : NullMeasurableSet s ht : NullMeasurableSet t hd : AEDisjoint \u03bc s t \u22a2 \u2200 (i : Bool), NullMeasurableSet (bif i then s else t) ** exacts [(pairwise_on_bool AEDisjoint.symmetric).2 hd, fun b => Bool.casesOn b ht hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.lintegral_condKernelReal_mem ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 s} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 s ** apply MeasurableSpace.induction_on_inter generateFrom_prod.symm isPiSystem_prod _ _ _ _ hs ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u2205} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 \u2205 ** simp only [mem_empty_iff_false, setOf_false, measure_empty, lintegral_const,\n  zero_mul] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u211d)),     t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u2192       \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t ** intro t ht ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : t \u2208 image2 (fun x x_1 => x \u00d7\u02e2 x_1) {s | MeasurableSet s} {t | MeasurableSet t} \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t ** rw [mem_image2] at ht ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : \u2203 a b, a \u2208 {s | MeasurableSet s} \u2227 b \u2208 {t | MeasurableSet t} \u2227 a \u00d7\u02e2 b = t \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t ** obtain \u27e8t\u2081, t\u2082, ht\u2081, ht\u2082, rfl\u27e9 := ht ** case intro.intro.intro.intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082) ** have h_prod_eq_snd : \u2200 a \u2208 t\u2081, {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 := by\n  intro a ha\n  simp only [ha, prod_mk_mem_set_prod_eq, true_and_iff, setOf_mem_eq] ** case intro.intro.intro.intro \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082) ** cases' eq_empty_or_nonempty t\u2082 with h h ** case intro.intro.intro.intro.inr \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082) ** rw [\u2190 lintegral_add_compl _ ht\u2081] ** case intro.intro.intro.intro.inr \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 \u22a2 \u222b\u207b (x : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 +       \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082) ** have h_eq1 : \u222b\u207b a in t\u2081, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202\u03c1.fst =\n    \u222b\u207b a in t\u2081, condKernelReal \u03c1 a t\u2082 \u2202\u03c1.fst := by\n  refine' set_lintegral_congr_fun ht\u2081 (eventually_of_forall fun a ha => _)\n  rw [h_prod_eq_snd a ha] ** case intro.intro.intro.intro.inr \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 h_eq1 :   \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 h_eq2 : \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0 \u22a2 \u222b\u207b (x : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 +       \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) x) {x_1 | (x, x_1) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082) ** rw [h_eq1, h_eq2, add_zero] ** case intro.intro.intro.intro.inr \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 h_eq1 :   \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 h_eq2 : \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0 \u22a2 \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082) ** exact set_lintegral_condKernelReal_prod \u03c1 ht\u2081 ht\u2082 ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} \u22a2 \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 ** intro a ha ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} a : \u03b1 ha : a \u2208 t\u2081 \u22a2 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 ** simp only [ha, prod_mk_mem_set_prod_eq, true_and_iff, setOf_mem_eq] ** case intro.intro.intro.intro.inl \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : t\u2082 = \u2205 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (t\u2081 \u00d7\u02e2 t\u2082) ** simp only [h, prod_empty, mem_empty_iff_false, setOf_false, measure_empty, lintegral_const,\n  zero_mul] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 \u22a2 \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 ** refine' set_lintegral_congr_fun ht\u2081 (eventually_of_forall fun a ha => _) ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 a : \u03b1 ha : a \u2208 t\u2081 \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 ** rw [h_prod_eq_snd a ha] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 h_eq1 :   \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 \u22a2 \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0 ** suffices h_eq_zero : \u2200 a \u2208 t\u2081\u1d9c, condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0 ** case h_eq_zero \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 h_eq1 :   \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 \u22a2 \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0 ** intro a hat\u2081 ** case h_eq_zero \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 h_eq1 :   \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 a : \u03b1 hat\u2081 : a \u2208 t\u2081\u1d9c \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0 ** rw [mem_compl_iff] at hat\u2081 ** case h_eq_zero \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 h_eq1 :   \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 a : \u03b1 hat\u2081 : \u00aca \u2208 t\u2081 \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0 ** simp only [hat\u2081, prod_mk_mem_set_prod_eq, false_and_iff, setOf_false, measure_empty] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 h_eq1 :   \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 h_eq_zero : \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0 \u22a2 \u222b\u207b (a : \u03b1) in t\u2081\u1d9c, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 = 0 ** rw [set_lintegral_congr_fun ht\u2081.compl (eventually_of_forall h_eq_zero)] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t\u2081 : Set \u03b1 t\u2082 : Set \u211d ht\u2081 : t\u2081 \u2208 {s | MeasurableSet s} ht\u2082 : t\u2082 \u2208 {t | MeasurableSet t} h_prod_eq_snd : \u2200 (a : \u03b1), a \u2208 t\u2081 \u2192 {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = t\u2082 h : Set.Nonempty t\u2082 h_eq1 :   \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1) in t\u2081, \u2191\u2191(\u2191(condKernelReal \u03c1) a) t\u2082 \u2202Measure.fst \u03c1 h_eq_zero : \u2200 (a : \u03b1), a \u2208 t\u2081\u1d9c \u2192 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u2081 \u00d7\u02e2 t\u2082} = 0 \u22a2 \u222b\u207b (x : \u03b1) in t\u2081\u1d9c, 0 \u2202Measure.fst \u03c1 = 0 ** simp only [lintegral_const, zero_mul] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u211d)),     MeasurableSet t \u2192       \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t \u2192         \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t\u1d9c} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t\u1d9c ** intro t ht ht_eq ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : MeasurableSet t ht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\u1d9c \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 ** congr with a : 1 ** case e_f.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : MeasurableSet t ht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t a : \u03b1 \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}\u1d9c =     \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} ** exact measure_compl (measurable_prod_mk_left ht) (measure_ne_top (condKernelReal \u03c1 a) _) ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : MeasurableSet t ht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -       \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 ** have h_le : (fun a => condKernelReal \u03c1 a {x : \u211d | (a, x) \u2208 t}) \u2264\u1d50[\u03c1.fst] fun a =>\n    condKernelReal \u03c1 a univ := eventually_of_forall fun a => measure_mono (subset_univ _) ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : MeasurableSet t ht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t h_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ - \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -       \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 ** rw [lintegral_sub _ _ h_le] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : MeasurableSet t ht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t h_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 \u2260 \u22a4 ** refine' ((lintegral_mono_ae h_le).trans_lt _).ne ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : MeasurableSet t ht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t h_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u22a2 \u222b\u207b (a : \u03b1), (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ) a \u2202Measure.fst \u03c1 < \u22a4 ** rw [lintegral_condKernelReal_univ] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : MeasurableSet t ht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t h_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u22a2 \u2191\u2191\u03c1 univ < \u22a4 ** exact measure_lt_top \u03c1 univ ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : MeasurableSet t ht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t h_le : (fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t}) \u2264\u1d50[Measure.fst \u03c1] fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u22a2 Measurable fun a => \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} ** exact kernel.measurable_kernel_prod_mk_left ht ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s t : Set (\u03b1 \u00d7 \u211d) ht : MeasurableSet t ht_eq : \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 t \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) univ \u2202Measure.fst \u03c1 -       \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 t} \u2202Measure.fst \u03c1 =     \u2191\u2191\u03c1 univ - \u2191\u2191\u03c1 t ** rw [ht_eq, lintegral_condKernelReal_univ] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s \u22a2 \u2200 (f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d)),     Pairwise (Disjoint on f) \u2192       (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192         (\u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i)) \u2192           \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 i, f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i) ** intro f hf_disj hf_meas hf_eq ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 i, f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i) ** have h_eq : \u2200 a, {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} := by\n  intro a\n  ext1 x\n  simp only [mem_iUnion, mem_setOf_eq] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 \u22c3 i, f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i) ** simp_rw [h_eq] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i) ** have h_disj : \u2200 a, Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i}) := by\n  intro a i j hij\n  have h_disj := hf_disj hij\n  rw [Function.onFun, disjoint_iff_inter_eq_empty] at h_disj \u22a2\n  ext1 x\n  simp only [mem_inter_iff, mem_setOf_eq, mem_empty_iff_false, iff_false_iff]\n  intro h_mem_both\n  suffices (a, x) \u2208 \u2205 by rwa [mem_empty_iff_false] at this\n  rwa [\u2190 h_disj, mem_inter_iff] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} h_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i}) \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (\u22c3 i, f i) ** calc\n  \u222b\u207b a, condKernelReal \u03c1 a (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202\u03c1.fst =\n      \u222b\u207b a, \u2211' i, condKernelReal \u03c1 a {x | (a, x) \u2208 f i} \u2202\u03c1.fst := by\n    congr with a : 1\n    rw [measure_iUnion (h_disj a) fun i => measurable_prod_mk_left (hf_meas i)]\n  _ = \u2211' i, \u222b\u207b a, condKernelReal \u03c1 a {x | (a, x) \u2208 f i} \u2202\u03c1.fst :=\n    (lintegral_tsum fun i => (kernel.measurable_kernel_prod_mk_left (hf_meas i)).aemeasurable)\n  _ = \u2211' i, \u03c1 (f i) := by simp_rw [hf_eq]\n  _ = \u03c1 (iUnion f) := (measure_iUnion hf_disj hf_meas).symm ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) \u22a2 \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} ** intro a ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) a : \u03b1 \u22a2 {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} ** ext1 x ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) a : \u03b1 x : \u211d \u22a2 x \u2208 {x | (a, x) \u2208 \u22c3 i, f i} \u2194 x \u2208 \u22c3 i, {x | (a, x) \u2208 f i} ** simp only [mem_iUnion, mem_setOf_eq] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} \u22a2 \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i}) ** intro a i j hij ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} a : \u03b1 i j : \u2115 hij : i \u2260 j \u22a2 (Disjoint on fun i => {x | (a, x) \u2208 f i}) i j ** have h_disj := hf_disj hij ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} a : \u03b1 i j : \u2115 hij : i \u2260 j h_disj : (Disjoint on f) i j \u22a2 (Disjoint on fun i => {x | (a, x) \u2208 f i}) i j ** rw [Function.onFun, disjoint_iff_inter_eq_empty] at h_disj \u22a2 ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} a : \u03b1 i j : \u2115 hij : i \u2260 j h_disj : f i \u2229 f j = \u2205 \u22a2 {x | (a, x) \u2208 f i} \u2229 {x | (a, x) \u2208 f j} = \u2205 ** ext1 x ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} a : \u03b1 i j : \u2115 hij : i \u2260 j h_disj : f i \u2229 f j = \u2205 x : \u211d \u22a2 x \u2208 {x | (a, x) \u2208 f i} \u2229 {x | (a, x) \u2208 f j} \u2194 x \u2208 \u2205 ** simp only [mem_inter_iff, mem_setOf_eq, mem_empty_iff_false, iff_false_iff] ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} a : \u03b1 i j : \u2115 hij : i \u2260 j h_disj : f i \u2229 f j = \u2205 x : \u211d \u22a2 \u00ac((a, x) \u2208 f i \u2227 (a, x) \u2208 f j) ** intro h_mem_both ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} a : \u03b1 i j : \u2115 hij : i \u2260 j h_disj : f i \u2229 f j = \u2205 x : \u211d h_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j \u22a2 False ** suffices (a, x) \u2208 \u2205 by rwa [mem_empty_iff_false] at this ** case h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} a : \u03b1 i j : \u2115 hij : i \u2260 j h_disj : f i \u2229 f j = \u2205 x : \u211d h_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j \u22a2 (a, x) \u2208 \u2205 ** rwa [\u2190 h_disj, mem_inter_iff] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} a : \u03b1 i j : \u2115 hij : i \u2260 j h_disj : f i \u2229 f j = \u2205 x : \u211d h_mem_both : (a, x) \u2208 f i \u2227 (a, x) \u2208 f j this : (a, x) \u2208 \u2205 \u22a2 False ** rwa [mem_empty_iff_false] at this ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} h_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i}) \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) \u2202Measure.fst \u03c1 =     \u222b\u207b (a : \u03b1), \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 ** congr with a : 1 ** case e_f.h \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} h_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i}) a : \u03b1 \u22a2 \u2191\u2191(\u2191(condKernelReal \u03c1) a) (\u22c3 i, {x | (a, x) \u2208 f i}) = \u2211' (i : \u2115), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} ** rw [measure_iUnion (h_disj a) fun i => measurable_prod_mk_left (hf_meas i)] ** \u03b1 : Type u_1 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) inst\u271d : IsFiniteMeasure \u03c1 s : Set (\u03b1 \u00d7 \u211d) hs : MeasurableSet s f : \u2115 \u2192 Set (\u03b1 \u00d7 \u211d) hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) hf_eq : \u2200 (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2191\u2191\u03c1 (f i) h_eq : \u2200 (a : \u03b1), {x | (a, x) \u2208 \u22c3 i, f i} = \u22c3 i, {x | (a, x) \u2208 f i} h_disj : \u2200 (a : \u03b1), Pairwise (Disjoint on fun i => {x | (a, x) \u2208 f i}) \u22a2 \u2211' (i : \u2115), \u222b\u207b (a : \u03b1), \u2191\u2191(\u2191(condKernelReal \u03c1) a) {x | (a, x) \u2208 f i} \u2202Measure.fst \u03c1 = \u2211' (i : \u2115), \u2191\u2191\u03c1 (f i) ** simp_rw [hf_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.data_takeWhile ** p : Char \u2192 Bool s : String \u22a2 (takeWhile s p).data = List.takeWhile p s.data ** rw [takeWhile_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.stoppedProcess_eq_of_mem_finset ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u22a2 stoppedProcess u \u03c4 n =     Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i) ** ext \u03c9 ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 \u22a2 stoppedProcess u \u03c4 n \u03c9 =     (Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)) \u03c9 ** rw [Pi.add_apply, Finset.sum_apply] ** case h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 \u22a2 stoppedProcess u \u03c4 n \u03c9 =     Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9 ** cases' le_or_lt n (\u03c4 \u03c9) with h h ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 \u22a2 stoppedProcess u \u03c4 n \u03c9 =     Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9 ** rw [stoppedProcess_eq_of_le h, Set.indicator_of_mem, Finset.sum_eq_zero, add_zero] ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 \u22a2 \u2200 (x : \u03b9), x \u2208 Finset.filter (fun x => x < n) s \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = x} (u x) \u03c9 = 0 ** intro m hm ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 m : \u03b9 hm : m \u2208 Finset.filter (fun x => x < n) s \u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = m} (u m) \u03c9 = 0 ** refine' Set.indicator_of_not_mem _ _ ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 m : \u03b9 hm : m \u2208 Finset.filter (fun x => x < n) s \u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m} ** rw [Finset.mem_filter] at hm ** case h.inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u271d : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 m : \u03b9 hm : m \u2208 s \u2227 m < n \u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m} ** exact (hm.2.trans_le h).ne' ** case h.inl.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : n \u2264 \u03c4 \u03c9 \u22a2 \u03c9 \u2208 {a | n \u2264 \u03c4 a} ** exact h ** case h.inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 stoppedProcess u \u03c4 n \u03c9 =     Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9 ** rw [stoppedProcess_eq_of_ge (le_of_lt h), Finset.sum_eq_single_of_mem (\u03c4 \u03c9)] ** case h.inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 u (\u03c4 \u03c9) \u03c9 = Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + Set.indicator {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9} (u (\u03c4 \u03c9)) \u03c9 ** rw [Set.indicator_of_not_mem, zero_add, Set.indicator_of_mem] ** case h.inr.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 \u03c9 \u2208 {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9} ** exact rfl ** case h.inr.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 \u00ac\u03c9 \u2208 {a | n \u2264 \u03c4 a} ** exact not_le.2 h ** case h.inr.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 \u03c4 \u03c9 \u2208 Finset.filter (fun x => x < n) s ** rw [Finset.mem_filter] ** case h.inr.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 \u03c4 \u03c9 \u2208 s \u2227 \u03c4 \u03c9 < n ** exact \u27e8hbdd \u03c9 h, h\u27e9 ** case h.inr.h\u2080 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n \u22a2 \u2200 (b : \u03b9), b \u2208 Finset.filter (fun x => x < n) s \u2192 b \u2260 \u03c4 \u03c9 \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0 ** intro b _ hneq ** case h.inr.h\u2080 \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n b : \u03b9 a\u271d : b \u2208 Finset.filter (fun x => x < n) s hneq : b \u2260 \u03c4 \u03c9 \u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0 ** rw [Set.indicator_of_not_mem] ** case h.inr.h\u2080.h \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u03c4 \u03c3 : \u03a9 \u2192 \u03b9 E : Type u_4 p : \u211d\u22650\u221e u : \u03b9 \u2192 \u03a9 \u2192 E inst\u271d\u00b9 : LinearOrder \u03b9 inst\u271d : AddCommMonoid E s : Finset \u03b9 n : \u03b9 hbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s \u03c9 : \u03a9 h : \u03c4 \u03c9 < n b : \u03b9 a\u271d : b \u2208 Finset.filter (fun x => x < n) s hneq : b \u2260 \u03c4 \u03c9 \u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = b} ** exact hneq.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.eventually_mul_right_iff ** \ud835\udd5c : Type u_1 G : Type u_2 H : Type u_3 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : MeasurableSpace H inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MeasurableMul G \u03bc : Measure G inst\u271d : IsMulRightInvariant \u03bc t : G p : G \u2192 Prop \u22a2 (\u2200\u1d50 (x : G) \u2202\u03bc, p (x * t)) \u2194 \u2200\u1d50 (x : G) \u2202\u03bc, p x ** conv_rhs => rw [Filter.Eventually, \u2190 map_mul_right_ae \u03bc t]; rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.toOuterMeasure_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d : MeasurableSpace \u03b1 \u22a2 \u2191\u22a4 = \u22a4 ** rw [\u2190 OuterMeasure.toMeasure_top, toMeasure_toOuterMeasure, OuterMeasure.trim_top] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.preimage_inr_range_inl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 f : \u03b9 \u2192 \u03b1 s t : Set \u03b1 \u22a2 Sum.inr \u207b\u00b9' range Sum.inl = \u2205 ** rw [\u2190 image_univ, preimage_inr_image_inl] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_smul ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F hT : DominatedFinMeasAdditive \u03bc T C h_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x c : \ud835\udd5c f : \u03b1 \u2192 E \u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F hT : DominatedFinMeasAdditive \u03bc T C h_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x c : \ud835\udd5c f : \u03b1 \u2192 E hf : Integrable f \u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f ** rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',\n  L1.setToL1_smul hT h_smul c _] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F hT : DominatedFinMeasAdditive \u03bc T C h_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x c : \ud835\udd5c f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f ** by_cases hr : c = 0 ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F hT : DominatedFinMeasAdditive \u03bc T C h_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x c : \ud835\udd5c f : \u03b1 \u2192 E hf : \u00acIntegrable f hr : c = 0 \u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f ** rw [hr] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F hT : DominatedFinMeasAdditive \u03bc T C h_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x c : \ud835\udd5c f : \u03b1 \u2192 E hf : \u00acIntegrable f hr : c = 0 \u22a2 setToFun \u03bc T hT (0 \u2022 f) = 0 \u2022 setToFun \u03bc T hT f ** simp ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F hT : DominatedFinMeasAdditive \u03bc T C h_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x c : \ud835\udd5c f : \u03b1 \u2192 E hf : \u00acIntegrable f hr : \u00acc = 0 \u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f ** have hf' : \u00acIntegrable (c \u2022 f) \u03bc := by rwa [integrable_smul_iff hr f] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F hT : DominatedFinMeasAdditive \u03bc T C h_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x c : \ud835\udd5c f : \u03b1 \u2192 E hf : \u00acIntegrable f hr : \u00acc = 0 hf' : \u00acIntegrable (c \u2022 f) \u22a2 setToFun \u03bc T hT (c \u2022 f) = c \u2022 setToFun \u03bc T hT f ** rw [setToFun_undef hT hf, setToFun_undef hT hf', smul_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedSpace \u211d E inst\u271d\u2078 : NormedAddCommGroup F inst\u271d\u2077 : NormedSpace \u211d F inst\u271d\u2076 : NormedAddCommGroup F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E inst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9 : NormedSpace \ud835\udd5c E inst\u271d : NormedSpace \ud835\udd5c F hT : DominatedFinMeasAdditive \u03bc T C h_smul : \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : E), \u2191(T s) (c \u2022 x) = c \u2022 \u2191(T s) x c : \ud835\udd5c f : \u03b1 \u2192 E hf : \u00acIntegrable f hr : \u00acc = 0 \u22a2 \u00acIntegrable (c \u2022 f) ** rwa [integrable_smul_iff hr f] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm'_add_le_of_le_one ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0 : 0 \u2264 q hq1 : q \u2264 1 \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264     (\u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc) ^ (1 / q) ** refine' ENNReal.rpow_le_rpow _ (by simp [hq0] : 0 \u2264 1 / q) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0 : 0 \u2264 q hq1 : q \u2264 1 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(f + g) a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ^ q \u2202\u03bc ** refine' lintegral_mono fun a => ENNReal.rpow_le_rpow _ hq0 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0 : 0 \u2264 q hq1 : q \u2264 1 a : \u03b1 \u22a2 \u2191\u2016(f + g) a\u2016\u208a \u2264 ((fun a => \u2191\u2016f a\u2016\u208a) + fun a => \u2191\u2016g a\u2016\u208a) a ** simp only [Pi.add_apply, \u2190 ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc hq0 : 0 \u2264 q hq1 : q \u2264 1 \u22a2 0 \u2264 1 / q ** simp [hq0] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pairwise_insert ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 r p q : \u03b1 \u2192 \u03b1 \u2192 Prop f g : \u03b9 \u2192 \u03b1 s t u : Set \u03b1 a b : \u03b1 \u22a2 Set.Pairwise (insert a s) r \u2194 Set.Pairwise s r \u2227 \u2200 (b : \u03b1), b \u2208 s \u2192 a \u2260 b \u2192 r a b \u2227 r b a ** simp only [insert_eq, pairwise_union, pairwise_singleton, true_and_iff, mem_singleton_iff,\n  forall_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "Holor.mul_assoc0 ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Semigroup \u03b1 x : Holor \u03b1 ds\u2081 y : Holor \u03b1 ds\u2082 z : Holor \u03b1 ds\u2083 t : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) \u22a2 (x \u2297 y \u2297 z) t = assocLeft (x \u2297 (y \u2297 z)) t ** rw [assocLeft] ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Semigroup \u03b1 x : Holor \u03b1 ds\u2081 y : Holor \u03b1 ds\u2082 z : Holor \u03b1 ds\u2083 t : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) \u22a2 (x \u2297 y \u2297 z) t = cast (_ : Holor \u03b1 (ds\u2081 ++ (ds\u2082 ++ ds\u2083)) = Holor \u03b1 (ds\u2081 ++ ds\u2082 ++ ds\u2083)) (x \u2297 (y \u2297 z)) t ** unfold mul ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Semigroup \u03b1 x : Holor \u03b1 ds\u2081 y : Holor \u03b1 ds\u2082 z : Holor \u03b1 ds\u2083 t : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) \u22a2 x (HolorIndex.take (HolorIndex.take t)) * y (HolorIndex.drop (HolorIndex.take t)) * z (HolorIndex.drop t) =     cast (_ : Holor \u03b1 (ds\u2081 ++ (ds\u2082 ++ ds\u2083)) = Holor \u03b1 (ds\u2081 ++ ds\u2082 ++ ds\u2083))       (fun t =>         x (HolorIndex.take t) * (y (HolorIndex.take (HolorIndex.drop t)) * z (HolorIndex.drop (HolorIndex.drop t))))       t ** rw [mul_assoc, \u2190HolorIndex.take_take, \u2190HolorIndex.drop_take, \u2190HolorIndex.drop_drop, cast_type] ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Semigroup \u03b1 x : Holor \u03b1 ds\u2081 y : Holor \u03b1 ds\u2082 z : Holor \u03b1 ds\u2083 t : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) \u22a2 x (HolorIndex.take (HolorIndex.assocRight t)) *       (y (HolorIndex.take (HolorIndex.drop (HolorIndex.assocRight t))) *         z (HolorIndex.drop (HolorIndex.drop (HolorIndex.assocRight t)))) =     (fun t =>         x (HolorIndex.take (cast (_ : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) = HolorIndex (ds\u2081 ++ (ds\u2082 ++ ds\u2083))) t)) *           (y               (HolorIndex.take                 (HolorIndex.drop (cast (_ : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) = HolorIndex (ds\u2081 ++ (ds\u2082 ++ ds\u2083))) t))) *             z               (HolorIndex.drop                 (HolorIndex.drop (cast (_ : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) = HolorIndex (ds\u2081 ++ (ds\u2082 ++ ds\u2083))) t)))))       t  case eq \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Semigroup \u03b1 x : Holor \u03b1 ds\u2081 y : Holor \u03b1 ds\u2082 z : Holor \u03b1 ds\u2083 t : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) \u22a2 ds\u2081 ++ (ds\u2082 ++ ds\u2083) = ds\u2081 ++ ds\u2082 ++ ds\u2083 ** rfl ** case eq \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Semigroup \u03b1 x : Holor \u03b1 ds\u2081 y : Holor \u03b1 ds\u2082 z : Holor \u03b1 ds\u2083 t : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083) \u22a2 ds\u2081 ++ (ds\u2082 ++ ds\u2083) = ds\u2081 ++ ds\u2082 ++ ds\u2083 ** rw [append_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_integrableOn ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : MetrizableSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsOpenPosMeasure \u03bc hs : IsCompact s c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 closure (interior s) hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** have : x\u2080 \u2208 s := by rw [\u2190 hs.isClosed.closure_eq]; exact closure_mono interior_subset h\u2080 ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : MetrizableSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsOpenPosMeasure \u03bc hs : IsCompact s c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 closure (interior s) hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 this : x\u2080 \u2208 s \u22a2 Tendsto (fun n => (\u222b (x : \u03b1) in s, c x ^ n \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : \u03b1) in s, c x ^ n \u2022 g x \u2202\u03bc) atTop (\ud835\udcdd (g x\u2080)) ** apply\n  tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos hs _ hc\n    h'c hnc hnc\u2080 this hmg hcg ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : MetrizableSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsOpenPosMeasure \u03bc hs : IsCompact s c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 closure (interior s) hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 this : x\u2080 \u2208 s \u22a2 \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 0 < \u2191\u2191\u03bc (u \u2229 s) ** intro u u_open x\u2080_u ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : MetrizableSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsOpenPosMeasure \u03bc hs : IsCompact s c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 closure (interior s) hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 this : x\u2080 \u2208 s u : Set \u03b1 u_open : IsOpen u x\u2080_u : x\u2080 \u2208 u \u22a2 0 < \u2191\u2191\u03bc (u \u2229 s) ** calc\n  0 < \u03bc (u \u2229 interior s) :=\n    (u_open.inter isOpen_interior).measure_pos \u03bc (_root_.mem_closure_iff.1 h\u2080 u u_open x\u2080_u)\n  _ \u2264 \u03bc (u \u2229 s) := measure_mono (inter_subset_inter_right _ interior_subset) ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : MetrizableSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsOpenPosMeasure \u03bc hs : IsCompact s c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 closure (interior s) hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u22a2 x\u2080 \u2208 s ** rw [\u2190 hs.isClosed.closure_eq] ** \u03b1 : Type u_1 E : Type u_2 \u03b9 : Type u_3 hm : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2077 : TopologicalSpace \u03b1 inst\u271d\u2076 : BorelSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E g : \u03b1 \u2192 E l : Filter \u03b9 x\u2080 : \u03b1 s : Set \u03b1 \u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : MetrizableSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc inst\u271d : IsOpenPosMeasure \u03bc hs : IsCompact s c : \u03b1 \u2192 \u211d hc : ContinuousOn c s h'c : \u2200 (y : \u03b1), y \u2208 s \u2192 y \u2260 x\u2080 \u2192 c y < c x\u2080 hnc : \u2200 (x : \u03b1), x \u2208 s \u2192 0 \u2264 c x hnc\u2080 : 0 < c x\u2080 h\u2080 : x\u2080 \u2208 closure (interior s) hmg : IntegrableOn g s hcg : ContinuousWithinAt g s x\u2080 \u22a2 x\u2080 \u2208 closure s ** exact closure_mono interior_subset h\u2080 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsStoppingTime.measurableSpace_le ** \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b2 : Preorder \u03b9\u271d f\u271d : Filtration \u03b9\u271d m \u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d \u03b9 : Type u_4 inst\u271d\u00b9 : SemilatticeSup \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : IsCountablyGenerated atTop h\u03c4 : IsStoppingTime f \u03c4 \u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m ** cases isEmpty_or_nonempty \u03b9 ** case inr \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b2 : Preorder \u03b9\u271d f\u271d : Filtration \u03b9\u271d m \u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d \u03b9 : Type u_4 inst\u271d\u00b9 : SemilatticeSup \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : IsCountablyGenerated atTop h\u03c4 : IsStoppingTime f \u03c4 h\u271d : Nonempty \u03b9 \u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m ** exact measurableSpace_le' h\u03c4 ** case inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b2 : Preorder \u03b9\u271d f\u271d : Filtration \u03b9\u271d m \u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d \u03b9 : Type u_4 inst\u271d\u00b9 : SemilatticeSup \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : IsCountablyGenerated atTop h\u03c4 : IsStoppingTime f \u03c4 h\u271d : IsEmpty \u03b9 \u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m ** haveI : IsEmpty \u03a9 := \u27e8fun \u03c9 => IsEmpty.false (\u03c4 \u03c9)\u27e9 ** case inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b2 : Preorder \u03b9\u271d f\u271d : Filtration \u03b9\u271d m \u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d \u03b9 : Type u_4 inst\u271d\u00b9 : SemilatticeSup \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : IsCountablyGenerated atTop h\u03c4 : IsStoppingTime f \u03c4 h\u271d : IsEmpty \u03b9 this : IsEmpty \u03a9 \u22a2 IsStoppingTime.measurableSpace h\u03c4 \u2264 m ** intro s _ ** case inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b2 : Preorder \u03b9\u271d f\u271d : Filtration \u03b9\u271d m \u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d \u03b9 : Type u_4 inst\u271d\u00b9 : SemilatticeSup \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : IsCountablyGenerated atTop h\u03c4 : IsStoppingTime f \u03c4 h\u271d : IsEmpty \u03b9 this : IsEmpty \u03a9 s : Set \u03a9 a\u271d : MeasurableSet s \u22a2 MeasurableSet s ** suffices hs : s = \u2205 ** case hs \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b2 : Preorder \u03b9\u271d f\u271d : Filtration \u03b9\u271d m \u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d \u03b9 : Type u_4 inst\u271d\u00b9 : SemilatticeSup \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : IsCountablyGenerated atTop h\u03c4 : IsStoppingTime f \u03c4 h\u271d : IsEmpty \u03b9 this : IsEmpty \u03a9 s : Set \u03a9 a\u271d : MeasurableSet s \u22a2 s = \u2205 ** haveI : Unique (Set \u03a9) := Set.uniqueEmpty ** case hs \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b2 : Preorder \u03b9\u271d f\u271d : Filtration \u03b9\u271d m \u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d \u03b9 : Type u_4 inst\u271d\u00b9 : SemilatticeSup \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : IsCountablyGenerated atTop h\u03c4 : IsStoppingTime f \u03c4 h\u271d : IsEmpty \u03b9 this\u271d : IsEmpty \u03a9 s : Set \u03a9 a\u271d : MeasurableSet s this : Unique (Set \u03a9) \u22a2 s = \u2205 ** rw [Unique.eq_default s, Unique.eq_default \u2205] ** case inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b2 : Preorder \u03b9\u271d f\u271d : Filtration \u03b9\u271d m \u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d \u03b9 : Type u_4 inst\u271d\u00b9 : SemilatticeSup \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : IsCountablyGenerated atTop h\u03c4 : IsStoppingTime f \u03c4 h\u271d : IsEmpty \u03b9 this : IsEmpty \u03a9 s : Set \u03a9 a\u271d : MeasurableSet s hs : s = \u2205 \u22a2 MeasurableSet s ** rw [hs] ** case inl \u03a9 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 m : MeasurableSpace \u03a9 inst\u271d\u00b2 : Preorder \u03b9\u271d f\u271d : Filtration \u03b9\u271d m \u03c4\u271d \u03c0 : \u03a9 \u2192 \u03b9\u271d \u03b9 : Type u_4 inst\u271d\u00b9 : SemilatticeSup \u03b9 f : Filtration \u03b9 m \u03c4 : \u03a9 \u2192 \u03b9 inst\u271d : IsCountablyGenerated atTop h\u03c4 : IsStoppingTime f \u03c4 h\u271d : IsEmpty \u03b9 this : IsEmpty \u03a9 s : Set \u03a9 a\u271d : MeasurableSet s hs : s = \u2205 \u22a2 MeasurableSet \u2205 ** exact MeasurableSet.empty ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AECover.integral_tendsto_of_countably_generated ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 l : Filter \u03b9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E inst\u271d : IsCountablyGenerated l \u03c6 : \u03b9 \u2192 Set \u03b1 h\u03c6 : AECover \u03bc l \u03c6 f : \u03b1 \u2192 E hfi : Integrable f h : Tendsto (fun i => \u222b (x : \u03b1), indicator (\u03c6 i) f x \u2202\u03bc) l (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc)) \u22a2 Tendsto (fun i => \u222b (x : \u03b1) in \u03c6 i, f x \u2202\u03bc) l (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc)) ** convert h using 2 ** case h.e'_3.h \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 inst\u271d\u2074 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 l : Filter \u03b9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E inst\u271d : IsCountablyGenerated l \u03c6 : \u03b9 \u2192 Set \u03b1 h\u03c6 : AECover \u03bc l \u03c6 f : \u03b1 \u2192 E hfi : Integrable f h : Tendsto (fun i => \u222b (x : \u03b1), indicator (\u03c6 i) f x \u2202\u03bc) l (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc)) x\u271d : \u03b9 \u22a2 \u222b (x : \u03b1) in \u03c6 x\u271d, f x \u2202\u03bc = \u222b (x : \u03b1), indicator (\u03c6 x\u271d) f x \u2202\u03bc ** rw [integral_indicator (h\u03c6.measurableSet _)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_comp_rpow_Ioi ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 \u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y ** let S := Ioi (0 : \u211d) ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 \u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y ** have a1 : \u2200 x : \u211d, x \u2208 S \u2192 HasDerivWithinAt (fun t : \u211d => t ^ p) (p * x ^ (p - 1)) S x :=\n  fun x hx => (hasDerivAt_rpow_const (Or.inl (mem_Ioi.mp hx).ne')).hasDerivWithinAt ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S \u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y ** have := integral_image_eq_integral_abs_deriv_smul measurableSet_Ioi a1 a2 g ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : \u222b (x : \u211d) in (fun t => t ^ p) '' Ioi 0, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p) \u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y ** rw [a3] at this ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p) \u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (y : \u211d) in Ioi 0, g y ** rw [this] ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p) \u22a2 \u222b (x : \u211d) in Ioi 0, (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p) ** refine' set_integral_congr measurableSet_Ioi _ ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p) \u22a2 EqOn (fun x => (|p| * x ^ (p - 1)) \u2022 g (x ^ p)) (fun x => |p * x ^ (p - 1)| \u2022 g (x ^ p)) (Ioi 0) ** intro x hx ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p) x : \u211d hx : x \u2208 Ioi 0 \u22a2 (fun x => (|p| * x ^ (p - 1)) \u2022 g (x ^ p)) x = (fun x => |p * x ^ (p - 1)| \u2022 g (x ^ p)) x ** dsimp only ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S a3 : (fun t => t ^ p) '' S = S this : \u222b (x : \u211d) in S, g x = \u222b (x : \u211d) in Ioi 0, |p * x ^ (p - 1)| \u2022 g (x ^ p) x : \u211d hx : x \u2208 Ioi 0 \u22a2 (|p| * x ^ (p - 1)) \u2022 g (x ^ p) = |p * x ^ (p - 1)| \u2022 g (x ^ p) ** rw [abs_mul, abs_of_nonneg (rpow_nonneg_of_nonneg (le_of_lt hx) _)] ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x \u22a2 InjOn (fun x => x ^ p) S ** rcases lt_or_gt_of_ne hp with (h | h) ** case inr E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p > 0 \u22a2 InjOn (fun x => x ^ p) S ** exact StrictMonoOn.injOn fun x hx y _ hxy => rpow_lt_rpow (mem_Ioi.mp hx).le hxy h ** case inl E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p < 0 \u22a2 InjOn (fun x => x ^ p) S ** apply StrictAntiOn.injOn ** case inl.H E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p < 0 \u22a2 StrictAntiOn (fun x => x ^ p) S ** intro x hx y hy hxy ** case inl.H E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p < 0 x : \u211d hx : x \u2208 S y : \u211d hy : y \u2208 S hxy : x < y \u22a2 (fun x => x ^ p) y < (fun x => x ^ p) x ** rw [\u2190 inv_lt_inv (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), \u2190 rpow_neg (le_of_lt hx),\n  \u2190 rpow_neg (le_of_lt hy)] ** case inl.H E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x h : p < 0 x : \u211d hx : x \u2208 S y : \u211d hy : y \u2208 S hxy : x < y \u22a2 x ^ (-p) < y ^ (-p) ** exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h) ** E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S \u22a2 (fun t => t ^ p) '' S = S ** ext1 x ** case h E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d \u22a2 x \u2208 (fun t => t ^ p) '' S \u2194 x \u2208 S ** rw [mem_image] ** case h E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d \u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2194 x \u2208 S ** constructor ** case h.mp E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d \u22a2 (\u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x) \u2192 x \u2208 S ** rintro \u27e8y, hy, rfl\u27e9 ** case h.mp.intro.intro E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S y : \u211d hy : y \u2208 S \u22a2 y ^ p \u2208 S ** exact rpow_pos_of_pos hy p ** case h.mpr E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d \u22a2 x \u2208 S \u2192 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x ** intro hx ** case h.mpr E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d hx : x \u2208 S \u22a2 \u2203 x_1, x_1 \u2208 S \u2227 x_1 ^ p = x ** refine' \u27e8x ^ (1 / p), rpow_pos_of_pos hx _, _\u27e9 ** case h.mpr E : Type u_1 f : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E g : \u211d \u2192 E p : \u211d hp : p \u2260 0 S : Set \u211d := Ioi 0 a1 : \u2200 (x : \u211d), x \u2208 S \u2192 HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x a2 : InjOn (fun x => x ^ p) S x : \u211d hx : x \u2208 S \u22a2 (x ^ (1 / p)) ^ p = x ** rw [\u2190 rpow_mul (le_of_lt hx), one_div_mul_cancel hp, rpow_one] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.Tape.mk'_nth_nat ** \u0393 : Type u_1 inst\u271d : Inhabited \u0393 L R : ListBlank \u0393 n : \u2115 \u22a2 nth (mk' L R) \u2191n = ListBlank.nth R n ** rw [\u2190 Tape.right\u2080_nth, Tape.mk'_right\u2080] ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.isEmpty ** l m r : List Char x\u271d : Substring h : ValidFor l m r x\u271d \u22a2 Substring.isEmpty x\u271d = true \u2194 m = [] ** simp [Substring.isEmpty, h.bsize] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.exists_of_set ** \u03b1 : Type u_1 n : Nat a' : \u03b1 l : List \u03b1 h : n < length l \u22a2 \u2203 l\u2081 a l\u2082, l = l\u2081 ++ a :: l\u2082 \u2227 length l\u2081 = n \u2227 set l n a' = l\u2081 ++ a' :: l\u2082 ** rw [set_eq_modifyNth] ** \u03b1 : Type u_1 n : Nat a' : \u03b1 l : List \u03b1 h : n < length l \u22a2 \u2203 l\u2081 a l\u2082, l = l\u2081 ++ a :: l\u2082 \u2227 length l\u2081 = n \u2227 modifyNth (fun x => a') n l = l\u2081 ++ a' :: l\u2082 ** exact exists_of_modifyNth _ h ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.toListAppend_eq ** \u03b1 : Type u_1 arr : Array \u03b1 l : List \u03b1 \u22a2 toListAppend arr l = arr.data ++ l ** simp [toListAppend, foldr_eq_foldr_data] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_finset_sum ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) \u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc ** by_cases hG : CompleteSpace G ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) hG : CompleteSpace G \u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc ** simp only [integral, hG, L1.integral] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) hG : CompleteSpace G \u22a2 (if h : True then       if hf : Integrable fun a => \u2211 i in s, f i a then \u2191L1.integralCLM (Integrable.toL1 (fun a => \u2211 i in s, f i a) hf)       else 0     else 0) =     \u2211 x in s,       if h : True then         if hf : Integrable fun a => f x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => f x a) hf) else 0       else 0 ** exact setToFun_finset_sum (dominatedFinMeasAdditive_weightedSMul _) s hf ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) hG : \u00acCompleteSpace G \u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc ** simp [integral, hG] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.ModEq.cancel_left_div_gcd ** m n a b c d : \u2124 hm : 0 < m h : c * a \u2261 c * b [ZMOD m] \u22a2 a * c \u2261 b * c [ZMOD m] ** simpa [mul_comm] using h ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.aestronglyMeasurable_exp_mul_sum ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d X : \u03b9 \u2192 \u03a9 \u2192 \u211d s : Finset \u03b9 h_int : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc \u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc ** induction' s using Finset.induction_on with i s hi_notin_s h_rec h_int ** case empty \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d X : \u03b9 \u2192 \u03a9 \u2192 \u211d s : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc h_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc \u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum \u2205 (fun i => X i) \u03c9)) \u03bc ** simp only [Pi.zero_apply, sum_apply, sum_empty, mul_zero, exp_zero] ** case empty \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d X : \u03b9 \u2192 \u03a9 \u2192 \u211d s : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc h_int : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc \u22a2 AEStronglyMeasurable (fun \u03c9 => 1) \u03bc ** exact aestronglyMeasurable_const ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_rec :   (\u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc) \u2192     AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc h_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc \u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)) \u03bc ** have : \u2200 i : \u03b9, i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 : \u03a9 => exp (t * X i \u03c9)) \u03bc := fun i hi =>\n  h_int i (mem_insert_of_mem hi) ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_rec :   (\u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc) \u2192     AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc h_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc this : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc \u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)) \u03bc ** specialize h_rec this ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc this : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc h_rec : AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc \u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum (insert i s) (fun i => X i) \u03c9)) \u03bc ** rw [sum_insert hi_notin_s] ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 h_int\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_int : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i_1 \u03c9)) \u03bc this : \u2200 (i : \u03b9), i \u2208 s \u2192 AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc h_rec : AEStronglyMeasurable (fun \u03c9 => rexp (t * Finset.sum s (fun i => X i) \u03c9)) \u03bc \u22a2 AEStronglyMeasurable (fun \u03c9 => rexp (t * (X i + \u2211 x in s, X x) \u03c9)) \u03bc ** apply aestronglyMeasurable_exp_mul_add (h_int i (mem_insert_self _ _)) h_rec ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBSet.mem_toList_insert ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp \u22a2 v' \u2208 toList (insert t v) \u2194 v' \u2208 toList t \u2227 find? t v \u2260 some v' \u2228 v' = v ** let \u27e8ht\u2081, _, _, ht\u2082\u27e9 := t.2.out ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp ht\u2081 : RBNode.Ordered cmp t.val w\u271d\u00b9 : RBColor w\u271d : Nat ht\u2082 : RBNode.Balanced t.val w\u271d\u00b9 w\u271d \u22a2 v' \u2208 toList (insert t v) \u2194 v' \u2208 toList t \u2227 find? t v \u2260 some v' \u2228 v' = v ** simpa [mem_toList] using RBNode.mem_insert ht\u2082 ht\u2081 ** Qed",
        "informal": ""
    },
    {
        "formal": "measurableSet_graph ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 \u211d s : Set \u03b1 hf : Measurable f \u22a2 MeasurableSet {p | p.2 = f p.1} ** simpa using measurableSet_region_between_cc hf hf MeasurableSet.univ ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_div ** \u03b9 : Type u_1 \ud835\udd5c\u271d : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E a b : \u211d f\u271d g : \u211d \u2192 E \u03bc : Measure \u211d \ud835\udd5c : Type u_6 inst\u271d : IsROrC \ud835\udd5c r : \ud835\udd5c f : \u211d \u2192 \ud835\udd5c \u22a2 \u222b (x : \u211d) in a..b, f x / r \u2202\u03bc = (\u222b (x : \u211d) in a..b, f x \u2202\u03bc) / r ** simpa only [div_eq_mul_inv] using integral_mul_const r\u207b\u00b9 f ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.ofNat'_succ ** \u03b1 : Type u_1 \u22a2 ofNat' (0 + 1) = ofNat' 0 + 1 ** simp ** \u03b1 : Type u_1 b : Bool n : \u2115 ih : ofNat' (n + 1) = ofNat' n + 1 \u22a2 ofNat' (Nat.bit b n + 1) = ofNat' (Nat.bit b n) + 1 ** cases b ** case false \u03b1 : Type u_1 n : \u2115 ih : ofNat' (n + 1) = ofNat' n + 1 \u22a2 ofNat' (Nat.bit false n + 1) = ofNat' (Nat.bit false n) + 1 ** erw [ofNat'_bit true n, ofNat'_bit] ** case false \u03b1 : Type u_1 n : \u2115 ih : ofNat' (n + 1) = ofNat' n + 1 \u22a2 cond true Num.bit1 Num.bit0 (ofNat' n) = cond false Num.bit1 Num.bit0 (ofNat' n) + 1 ** simp only [\u2190 bit1_of_bit1, \u2190 bit0_of_bit0, cond, _root_.bit1] ** case true \u03b1 : Type u_1 n : \u2115 ih : ofNat' (n + 1) = ofNat' n + 1 \u22a2 ofNat' (Nat.bit true n + 1) = ofNat' (Nat.bit true n) + 1 ** erw [show n.bit true + 1 = (n + 1).bit false by\n  simp [Nat.bit, _root_.bit1, _root_.bit0]; exact Nat.add_left_comm n 1 1,\n  ofNat'_bit, ofNat'_bit, ih] ** case true \u03b1 : Type u_1 n : \u2115 ih : ofNat' (n + 1) = ofNat' n + 1 \u22a2 cond false Num.bit1 Num.bit0 (ofNat' n + 1) = cond true Num.bit1 Num.bit0 (ofNat' n) + 1 ** simp only [cond, add_one, bit1_succ] ** \u03b1 : Type u_1 n : \u2115 ih : ofNat' (n + 1) = ofNat' n + 1 \u22a2 Nat.bit true n + 1 = Nat.bit false (n + 1) ** simp [Nat.bit, _root_.bit1, _root_.bit0] ** \u03b1 : Type u_1 n : \u2115 ih : ofNat' (n + 1) = ofNat' n + 1 \u22a2 n + (1 + 1) = 1 + (n + 1) ** exact Nat.add_left_comm n 1 1 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.continuousOn_of_dominated ** \u03b1 : Type u_1 E : Type u_2 F\u271d : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F\u271d inst\u271d\u2075 : NormedSpace \u211d F\u271d inst\u271d\u2074 : CompleteSpace F\u271d G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X F : X \u2192 \u03b1 \u2192 G bound : \u03b1 \u2192 \u211d s : Set X hF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc h_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a bound_integrable : Integrable bound h_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s \u22a2 ContinuousOn (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) s ** by_cases hG : CompleteSpace G ** case pos \u03b1 : Type u_1 E : Type u_2 F\u271d : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F\u271d inst\u271d\u2075 : NormedSpace \u211d F\u271d inst\u271d\u2074 : CompleteSpace F\u271d G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X F : X \u2192 \u03b1 \u2192 G bound : \u03b1 \u2192 \u211d s : Set X hF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc h_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a bound_integrable : Integrable bound h_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s hG : CompleteSpace G \u22a2 ContinuousOn (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) s ** simp only [integral, hG, L1.integral] ** case pos \u03b1 : Type u_1 E : Type u_2 F\u271d : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F\u271d inst\u271d\u2075 : NormedSpace \u211d F\u271d inst\u271d\u2074 : CompleteSpace F\u271d G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X F : X \u2192 \u03b1 \u2192 G bound : \u03b1 \u2192 \u211d s : Set X hF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc h_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a bound_integrable : Integrable bound h_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s hG : CompleteSpace G \u22a2 ContinuousOn     (fun x =>       if h : True then         if hf : Integrable fun a => F x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => F x a) hf) else 0       else 0)     s ** exact continuousOn_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul \u03bc)\n  hF_meas h_bound bound_integrable h_cont ** case neg \u03b1 : Type u_1 E : Type u_2 F\u271d : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F\u271d inst\u271d\u2075 : NormedSpace \u211d F\u271d inst\u271d\u2074 : CompleteSpace F\u271d G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X F : X \u2192 \u03b1 \u2192 G bound : \u03b1 \u2192 \u211d s : Set X hF_meas : \u2200 (x : X), x \u2208 s \u2192 AEStronglyMeasurable (F x) \u03bc h_bound : \u2200 (x : X), x \u2208 s \u2192 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a bound_integrable : Integrable bound h_cont : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, ContinuousOn (fun x => F x a) s hG : \u00acCompleteSpace G \u22a2 ContinuousOn (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) s ** simp [integral, hG, continuousOn_const] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.measure_lt_top_of_mem\u2112p_indicator ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 c : E hc : c \u2260 0 s : Set \u03b1 hs : MeasurableSet s hcs : Mem\u2112p (\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0))) p \u22a2 \u2191\u2191\u03bc s < \u22a4 ** have : Function.support (const \u03b1 c) = Set.univ := Function.support_const hc ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e hp_pos : p \u2260 0 hp_ne_top : p \u2260 \u22a4 c : E hc : c \u2260 0 s : Set \u03b1 hs : MeasurableSet s hcs : Mem\u2112p (\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0))) p this : support \u2191(const \u03b1 c) = Set.univ \u22a2 \u2191\u2191\u03bc s < \u22a4 ** simpa only [mem\u2112p_iff_finMeasSupp hp_pos hp_ne_top, finMeasSupp_iff_support,\n  support_indicator, Set.inter_univ, this] using hcs ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.uIcc_eq_union ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 \u22a2 \u2191[[a, b]] = \u2191(Icc a b \u222a Icc b a) ** push_cast ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 \u22a2 Set.uIcc a b = Set.Icc a b \u222a Set.Icc b a ** exact Set.uIcc_eq_union ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_interval_sub_interval_comm' ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d hab : IntervalIntegrable f \u03bc a b hcd : IntervalIntegrable f \u03bc c d hac : IntervalIntegrable f \u03bc a c \u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc - \u222b (x : \u211d) in c..d, f x \u2202\u03bc = \u222b (x : \u211d) in d..b, f x \u2202\u03bc - \u222b (x : \u211d) in c..a, f x \u2202\u03bc ** rw [integral_interval_sub_interval_comm hab hcd hac, integral_symm b d, integral_symm a c,\n  sub_neg_eq_add, sub_eq_neg_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "String.valid_next ** s : String p : Pos h : Pos.Valid s p h\u2082 : p < endPos s \u22a2 Pos.Valid s (next s p) ** match s, p, h with\n| \u27e8_\u27e9, \u27e8_\u27e9, \u27e8cs, [], rfl, rfl\u27e9 => simp at h\u2082\n| \u27e8_\u27e9, \u27e8_\u27e9, \u27e8cs, c::cs', rfl, rfl\u27e9 =>\n  rw [utf8ByteSize.go_eq, next_of_valid]\n  simpa using Pos.Valid.mk (cs ++ [c]) cs' ** s : String p : Pos h : Pos.Valid s p cs : List Char h\u2082 : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ [] } \u22a2 Pos.Valid { data := cs ++ [] } (next { data := cs ++ [] } { byteIdx := utf8ByteSize.go cs }) ** simp at h\u2082 ** s : String p : Pos h : Pos.Valid s p cs : List Char c : Char cs' : List Char h\u2082 : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ c :: cs' } \u22a2 Pos.Valid { data := cs ++ c :: cs' } (next { data := cs ++ c :: cs' } { byteIdx := utf8ByteSize.go cs }) ** rw [utf8ByteSize.go_eq, next_of_valid] ** s : String p : Pos h : Pos.Valid s p cs : List Char c : Char cs' : List Char h\u2082 : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ c :: cs' } \u22a2 Pos.Valid { data := cs ++ c :: cs' } { byteIdx := utf8Len cs + csize c } ** simpa using Pos.Valid.mk (cs ++ [c]) cs' ** Qed",
        "informal": ""
    },
    {
        "formal": "Complex.abs_circleTransformBoundingFunction_le ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z : \u2102 \u22a2 \u2203 x,     \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),       \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x) ** have cts := continuousOn_abs_circleTransformBoundingFunction hr z ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z : \u2102 cts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) \u22a2 \u2203 x,     \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),       \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x) ** have comp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) := by\n  apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z : \u2102 cts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) comp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) \u22a2 \u2203 x,     \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),       \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x) ** have none : (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]).Nonempty :=\n  (nonempty_closedBall.2 hr').prod nonempty_uIcc ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z : \u2102 cts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) comp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) none : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) \u22a2 \u2203 x,     \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),       \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x) ** have := IsCompact.exists_isMaxOn comp none (cts.mono\n  (by intro z; simp only [mem_prod, mem_closedBall, mem_univ, and_true_iff, and_imp]; tauto)) ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z : \u2102 cts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) comp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) none : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) this :   \u2203 x,     x \u2208 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2227       IsMaxOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) x \u22a2 \u2203 x,     \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),       \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x) ** simp only [IsMaxOn, IsMaxFilter] at this ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z : \u2102 cts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) comp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) none : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) this :   \u2203 x,     x \u2208 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2227       \u2200\u1da0 (x_1 : \u2102 \u00d7 \u211d) in \ud835\udcdf (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]),         (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) x_1 \u2264           (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) x \u22a2 \u2203 x,     \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),       \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x) ** simpa [SetCoe.forall, Subtype.coe_mk, SetCoe.exists] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z : \u2102 cts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) \u22a2 IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) ** apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z : \u2102 cts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) comp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) none : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) \u22a2 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2286 closedBall z r \u00d7\u02e2 univ ** intro z ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d\u00b9 w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z\u271d : \u2102 cts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z\u271d t) (closedBall z\u271d r \u00d7\u02e2 univ) comp : IsCompact (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]]) none : Set.Nonempty (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]]) z : \u2102 \u00d7 \u211d \u22a2 z \u2208 closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2192 z \u2208 closedBall z\u271d r \u00d7\u02e2 univ ** simp only [mem_prod, mem_closedBall, mem_univ, and_true_iff, and_imp] ** E : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \u2102 E R\u271d : \u211d z\u271d\u00b9 w : \u2102 R r : \u211d hr : r < R hr' : 0 \u2264 r z\u271d : \u2102 cts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z\u271d t) (closedBall z\u271d r \u00d7\u02e2 univ) comp : IsCompact (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]]) none : Set.Nonempty (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]]) z : \u2102 \u00d7 \u211d \u22a2 dist z.1 z\u271d \u2264 r \u2192 z.2 \u2208 [[0, 2 * \u03c0]] \u2192 dist z.1 z\u271d \u2264 r ** tauto ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pairwise_disjoint_Ioc_add_int_cast ** \u03b1 : Type u_1 inst\u271d : OrderedRing \u03b1 a : \u03b1 \u22a2 Pairwise (Disjoint on fun n => Ioc (a + \u2191n) (a + \u2191n + 1)) ** simpa only [zsmul_one, Int.cast_add, Int.cast_one, \u2190 add_assoc] using\n  pairwise_disjoint_Ioc_add_zsmul a (1 : \u03b1) ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.merge' ** f g : \u2115 \u2192. \u2115 hf : Partrec f hg : Partrec g \u22a2 \u2203 h, Partrec h \u2227 \u2200 (a : \u2115), (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((h a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) ** obtain \u27e8cf, rfl\u27e9 := Code.exists_code.1 hf ** case intro g : \u2115 \u2192. \u2115 hg : Partrec g cf : Code hf : Partrec (Code.eval cf) \u22a2 \u2203 h,     Partrec h \u2227       \u2200 (a : \u2115), (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 g a) \u2227 ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (g a).Dom) ** obtain \u27e8cg, rfl\u27e9 := Code.exists_code.1 hg ** case intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) \u22a2 \u2203 h,     Partrec h \u2227       \u2200 (a : \u2115),         (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 Code.eval cg a) \u2227           ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (Code.eval cg a).Dom) ** have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=\n  Partrec.nat_iff.1\n    (Partrec.rfindOpt <|\n      Primrec.option_orElse.to_comp.comp\n        (Code.evaln_prim.to_comp.comp <| (snd.pair (const cf)).pair fst)\n        (Code.evaln_prim.to_comp.comp <| (snd.pair (const cg)).pair fst)) ** case intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n \u22a2 \u2203 h,     Partrec h \u2227       \u2200 (a : \u2115),         (\u2200 (x : \u2115), x \u2208 h a \u2192 x \u2208 Code.eval cf a \u2228 x \u2208 Code.eval cg a) \u2227           ((h a).Dom \u2194 (Code.eval cf a).Dom \u2228 (Code.eval cg a).Dom) ** refine' \u27e8_, this, fun n => _\u27e9 ** case intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 \u22a2 (\u2200 (x : \u2115),       (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192         x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n) \u2227     ((rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom \u2194       (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom) ** suffices ** case intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this : ?m.104784 \u22a2 (\u2200 (x : \u2115),       (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192         x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n) \u2227     ((rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom \u2194       (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom)  case this cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 \u22a2 ?m.104784 ** refine' \u27e8this, \u27e8fun h => (this _ \u27e8h, rfl\u27e9).imp Exists.fst Exists.fst, _\u27e9\u27e9 ** case this cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 \u22a2 \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n ** intro x h ** case this cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n x : \u2115 h : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n \u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n ** obtain \u27e8k, e\u27e9 := Nat.rfindOpt_spec h ** case this.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n x : \u2115 h : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n k : \u2115 e : x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n \u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n ** revert e ** case this.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n x : \u2115 h : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n k : \u2115 \u22a2 (x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n ** simp only [Option.mem_def] ** case this.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n x : \u2115 h : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n k : \u2115 \u22a2 (HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) = Option.some x \u2192     x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n ** cases' e' : cf.evaln k n with y <;> simp <;> intro e ** case intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n \u22a2 (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom \u2192     (rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom ** intro h ** case intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n h : (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom \u22a2 (rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n).Dom ** rw [Nat.rfindOpt_dom] ** case intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n h : (Code.eval cf n).Dom \u2228 (Code.eval cg n).Dom \u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n ** simp only [dom_iff_mem, Code.evaln_complete, Option.mem_def] at h ** case intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n h : (\u2203 y k, Code.evaln k cf n = Option.some y) \u2228 \u2203 y k, Code.evaln k cg n = Option.some y \u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n ** obtain \u27e8x, k, e\u27e9 | \u27e8x, k, e\u27e9 := h ** case intro.intro.inl.intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n x k : \u2115 e : Code.evaln k cf n = Option.some x \u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n ** refine' \u27e8k, x, _\u27e9 ** case intro.intro.inl.intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n x k : \u2115 e : Code.evaln k cf n = Option.some x \u22a2 x \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n ** simp only [e, Option.some_orElse, Option.mem_def] ** case intro.intro.inr.intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n x k : \u2115 e : Code.evaln k cg n = Option.some x \u22a2 \u2203 n_1 a, a \u2208 HOrElse.hOrElse (Code.evaln n_1 cf n) fun x => Code.evaln n_1 cg n ** refine' \u27e8k, _\u27e9 ** case intro.intro.inr.intro.intro cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n x k : \u2115 e : Code.evaln k cg n = Option.some x \u22a2 \u2203 a, a \u2208 HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n ** cases' cf.evaln k n with y ** case intro.intro.inr.intro.intro.none cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n x k : \u2115 e : Code.evaln k cg n = Option.some x \u22a2 \u2203 a, a \u2208 HOrElse.hOrElse Option.none fun x => Code.evaln k cg n ** exact \u27e8x, by simp only [e, Option.mem_def, Option.none_orElse]\u27e9 ** cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n x k : \u2115 e : Code.evaln k cg n = Option.some x \u22a2 x \u2208 HOrElse.hOrElse Option.none fun x => Code.evaln k cg n ** simp only [e, Option.mem_def, Option.none_orElse] ** case intro.intro.inr.intro.intro.some cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n x k : \u2115 e : Code.evaln k cg n = Option.some x y : \u2115 \u22a2 \u2203 a, a \u2208 HOrElse.hOrElse (Option.some y) fun x => Code.evaln k cg n ** exact \u27e8y, by simp only [Option.some_orElse, Option.mem_def]\u27e9 ** cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this\u271d : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n : \u2115 this :   \u2200 (x : \u2115),     (x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n) \u2192       x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n x k : \u2115 e : Code.evaln k cg n = Option.some x y : \u2115 \u22a2 y \u2208 HOrElse.hOrElse (Option.some y) fun x => Code.evaln k cg n ** simp only [Option.some_orElse, Option.mem_def] ** case this.intro.none cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n x : \u2115 h : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n k : \u2115 e' : Code.evaln k cf n = Option.none e : Code.evaln k cg n = Option.some x \u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n ** exact Or.inr (Code.evaln_sound e) ** case this.intro.some cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n x : \u2115 h : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n k y : \u2115 e' : Code.evaln k cf n = Option.some y e : y = x \u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n ** subst y ** case this.intro.some cf : Code hf : Partrec (Code.eval cf) cg : Code hg : Partrec (Code.eval cg) this : Partrec fun n => rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n n x : \u2115 h : x \u2208 rfindOpt fun k => HOrElse.hOrElse (Code.evaln k cf n) fun x => Code.evaln k cg n k : \u2115 e' : Code.evaln k cf n = Option.some x \u22a2 x \u2208 Code.eval cf n \u2228 x \u2208 Code.eval cg n ** exact Or.inl (Code.evaln_sound e') ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.piecewise_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : \u03b1 \u2192 Sort u_4 s : Finset \u03b1 f g : (i : \u03b1) \u2192 \u03b4 i inst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s) inst\u271d : DecidableEq \u03b1 i : \u03b1 \u22a2 piecewise {i} f g = update g i (f i) ** rw [\u2190 insert_emptyc_eq, piecewise_insert, piecewise_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.IndepFun.variance_sum ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s : Finset \u03b9 hs : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j) \u22a2 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 ** induction' s using Finset.induction_on with k s ks IH ** case insert \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 variance (\u2211 i in insert k s, X i) \u2119 = \u2211 i in insert k s, variance (X i) \u2119 ** rw [variance_def' (mem\u2112p_finset_sum' _ hs), sum_insert ks, sum_insert ks] ** case insert \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 (\u222b (a : \u03a9), ((X k + \u2211 x in s, X x) ^ 2) a) - (\u222b (a : \u03a9), (X k + \u2211 x in s, X x) a) ^ 2 =     variance (X k) \u2119 + \u2211 x in s, variance (X x) \u2119 ** simp only [add_sq'] ** case empty \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j) hs : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191\u2205 fun i j => IndepFun (X i) (X j) \u22a2 variance (\u2211 i in \u2205, X i) \u2119 = \u2211 i in \u2205, variance (X i) \u2119 ** simp only [Finset.sum_empty, variance_zero] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 (\u222b (a : \u03a9), (X k ^ 2 + (\u2211 i in s, X i) ^ 2 + 2 * X k * \u2211 i in s, X i) a) - (\u222b (a : \u03a9), (X k + \u2211 i in s, X i) a) ^ 2 =     (((\u222b (a : \u03a9), (X k ^ 2) a) + \u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) + \u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) -       ((\u222b (a : \u03a9), X k a) + \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2 ** rw [integral_add', integral_add', integral_add'] ** case hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable (X k) ** exact Mem\u2112p.integrable one_le_two (hs _ (mem_insert_self _ _)) ** case hg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable (\u2211 i in s, X i) ** apply integrable_finset_sum' _ fun i hi => ?_ ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s \u22a2 Integrable (X i) ** exact Mem\u2112p.integrable one_le_two (hs _ (mem_insert_of_mem hi)) ** case hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable (X k ^ 2) ** exact Mem\u2112p.integrable_sq (hs _ (mem_insert_self _ _)) ** case hg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable ((\u2211 i in s, X i) ^ 2) ** apply Mem\u2112p.integrable_sq ** case hg.h \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Mem\u2112p (fun x => Finset.sum s (fun i => X i) x) 2 ** exact mem\u2112p_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi) ** case hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable (X k ^ 2 + (\u2211 i in s, X i) ^ 2) ** apply Integrable.add ** case hf.hf \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable (X k ^ 2) ** exact Mem\u2112p.integrable_sq (hs _ (mem_insert_self _ _)) ** case hf.hg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable ((\u2211 i in s, X i) ^ 2) ** apply Mem\u2112p.integrable_sq ** case hf.hg.h \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Mem\u2112p (fun x => Finset.sum s (fun i => X i) x) 2 ** exact mem\u2112p_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi) ** case hg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable (2 * X k * \u2211 i in s, X i) ** rw [mul_assoc] ** case hg \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable (2 * (X k * \u2211 i in s, X i)) ** apply Integrable.const_mul _ (2 : \u211d) ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable fun x => (X k * \u2211 i in s, X i) x ** simp only [mul_sum, sum_apply, Pi.mul_apply] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 Integrable fun x => \u2211 x_1 in s, X k x * X x_1 x ** apply integrable_finset_sum _ fun i hi => ?_ ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s \u22a2 Integrable fun a => X k a * X i a ** apply IndepFun.integrable_mul _ (Mem\u2112p.integrable one_le_two (hs _ (mem_insert_self _ _)))\n  (Mem\u2112p.integrable one_le_two (hs _ (mem_insert_of_mem hi))) ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s \u22a2 IndepFun (X k) (X i) ** apply h (mem_insert_self _ _) (mem_insert_of_mem hi) ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s \u22a2 k \u2260 i ** exact fun hki => ks (hki.symm \u25b8 hi) ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 (((\u222b (a : \u03a9), (X k ^ 2) a) + \u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) + \u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) -       ((\u222b (a : \u03a9), X k a) + \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2 =     variance (X k) \u2119 + variance (\u2211 i in s, X i) \u2119 +       ((\u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) - (2 * \u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ** rw [variance_def' (hs _ (mem_insert_self _ _)),\n  variance_def' (mem\u2112p_finset_sum' _ fun i hi => hs _ (mem_insert_of_mem hi))] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 (((\u222b (a : \u03a9), (X k ^ 2) a) + \u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) + \u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) -       ((\u222b (a : \u03a9), X k a) + \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2 =     (\u222b (a : \u03a9), (X k ^ 2) a) - (\u222b (a : \u03a9), X k a) ^ 2 +         ((\u222b (a : \u03a9), ((\u2211 i in s, X i) ^ 2) a) - (\u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ^ 2) +       ((\u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) - (2 * \u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) ** ring ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 variance (X k) \u2119 + variance (\u2211 i in s, X i) \u2119 +       ((\u222b (a : \u03a9), (2 * X k * \u2211 i in s, X i) a) - (2 * \u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), Finset.sum s (fun i => X i) a) =     variance (X k) \u2119 + variance (\u2211 i in s, X i) \u2119 ** simp only [mul_assoc, integral_mul_left, Pi.mul_apply, Pi.one_apply, sum_apply,\n  add_right_eq_self, mul_sum] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 (\u222b (a : \u03a9), \u2211 x in s, OfNat.ofNat 2 a * (X k a * X x a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), \u2211 c in s, X c a) = 0 ** rw [integral_finset_sum s fun i hi => ?_] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 (\u2211 i in s, \u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), \u2211 c in s, X c a) = 0  \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s \u22a2 Integrable fun a => OfNat.ofNat 2 a * (X k a * X i a) ** swap ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 (\u2211 i in s, \u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), \u2211 c in s, X c a) = 0 ** rw [integral_finset_sum s fun i hi =>\n    Mem\u2112p.integrable one_le_two (hs _ (mem_insert_of_mem hi)),\n  mul_sum, mul_sum, \u2190 sum_sub_distrib] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 \u2211 x in s, ((\u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X x a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X x a)) = 0 ** apply Finset.sum_eq_zero fun i hi => ?_ ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s \u22a2 (\u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X i a) = 0 ** have : \u2200 (a : \u03a9), @OfNat.ofNat (\u03a9 \u2192 \u211d) 2 instOfNat a = (2 : \u211d) := fun a => rfl ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s this : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2 \u22a2 (\u222b (a : \u03a9), OfNat.ofNat 2 a * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X i a) = 0 ** conv_lhs => enter [1, 2, a]; rw [this] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s this : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2 \u22a2 (\u222b (a : \u03a9), 2 * (X k a * X i a)) - 2 * ((\u222b (a : \u03a9), X k a) * \u222b (a : \u03a9), X i a) = 0 ** rw [integral_mul_left, IndepFun.integral_mul', sub_self] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s \u22a2 Integrable fun a => OfNat.ofNat 2 a * (X k a * X i a) ** apply Integrable.const_mul _ (2 : \u211d) ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s \u22a2 Integrable fun x => X k x * X i x ** apply IndepFun.integrable_mul _ (Mem\u2112p.integrable one_le_two (hs _ (mem_insert_self _ _)))\n  (Mem\u2112p.integrable one_le_two (hs _ (mem_insert_of_mem hi))) ** case hXY \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s this : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2 \u22a2 IndepFun (fun a => X k a) fun a => X i a ** apply h (mem_insert_self _ _) (mem_insert_of_mem hi) ** case hXY \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s this : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2 \u22a2 k \u2260 i ** exact fun hki => ks (hki.symm \u25b8 hi) ** case hX \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s this : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2 \u22a2 AEStronglyMeasurable (fun a => X k a) \u2119 ** exact Mem\u2112p.aestronglyMeasurable (hs _ (mem_insert_self _ _)) ** case hY \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) i : \u03b9 hi : i \u2208 s this : \u2200 (a : \u03a9), OfNat.ofNat 2 a = 2 \u22a2 AEStronglyMeasurable (fun a => X i a) \u2119 ** exact Mem\u2112p.aestronglyMeasurable (hs _ (mem_insert_of_mem hi)) ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 variance (X k) \u2119 + variance (\u2211 i in s, X i) \u2119 = variance (X k) \u2119 + \u2211 i in s, variance (X i) \u2119 ** rw [IH (fun i hi => hs i (mem_insert_of_mem hi))\n    (h.mono (by simp only [coe_insert, Set.subset_insert]))] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 \u03b9 : Type u_2 X : \u03b9 \u2192 \u03a9 \u2192 \u211d s\u271d : Finset \u03b9 hs\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 Mem\u2112p (X i) 2 h\u271d : Set.Pairwise \u2191s\u271d fun i j => IndepFun (X i) (X j) k : \u03b9 s : Finset \u03b9 ks : \u00ack \u2208 s IH :   (\u2200 (i : \u03b9), i \u2208 s \u2192 Mem\u2112p (X i) 2) \u2192     (Set.Pairwise \u2191s fun i j => IndepFun (X i) (X j)) \u2192 variance (\u2211 i in s, X i) \u2119 = \u2211 i in s, variance (X i) \u2119 hs : \u2200 (i : \u03b9), i \u2208 insert k s \u2192 Mem\u2112p (X i) 2 h : Set.Pairwise \u2191(insert k s) fun i j => IndepFun (X i) (X j) \u22a2 \u2191s \u2286 \u2191(insert k s) ** simp only [coe_insert, Set.subset_insert] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.pdf.ae_lt_top ** \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc\u271d : Measure E inst\u271d : IsFiniteMeasure \u2119 \u03bc : Measure E X : \u03a9 \u2192 E \u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4 ** by_cases hpdf : HasPDF X \u2119 \u03bc ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc\u271d : Measure E inst\u271d : IsFiniteMeasure \u2119 \u03bc : Measure E X : \u03a9 \u2192 E hpdf : HasPDF X \u2119 \u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4 ** haveI := hpdf ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc\u271d : Measure E inst\u271d : IsFiniteMeasure \u2119 \u03bc : Measure E X : \u03a9 \u2192 E hpdf this : HasPDF X \u2119 \u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4 ** refine' ae_lt_top (measurable_pdf X \u2119 \u03bc) _ ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc\u271d : Measure E inst\u271d : IsFiniteMeasure \u2119 \u03bc : Measure E X : \u03a9 \u2192 E hpdf this : HasPDF X \u2119 \u22a2 \u222b\u207b (x : E), pdf X \u2119 x \u2202\u03bc \u2260 \u22a4 ** rw [lintegral_eq_measure_univ] ** case pos \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc\u271d : Measure E inst\u271d : IsFiniteMeasure \u2119 \u03bc : Measure E X : \u03a9 \u2192 E hpdf this : HasPDF X \u2119 \u22a2 \u2191\u2191\u2119 Set.univ \u2260 \u22a4 ** exact (measure_lt_top _ _).ne ** case neg \u03a9 : Type u_1 E : Type u_2 inst\u271d\u00b9 : MeasurableSpace E m : MeasurableSpace \u03a9 \u2119 : Measure \u03a9 \u03bc\u271d : Measure E inst\u271d : IsFiniteMeasure \u2119 \u03bc : Measure E X : \u03a9 \u2192 E hpdf : \u00acHasPDF X \u2119 \u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, pdf X \u2119 x < \u22a4 ** simp [pdf, hpdf] ** Qed",
        "informal": ""
    },
    {
        "formal": "FinEnum.mem_toList ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : FinEnum \u03b1 x : \u03b1 \u22a2 x \u2208 toList \u03b1 ** simp [toList] ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : FinEnum \u03b1 x : \u03b1 \u22a2 \u2203 a, \u2191equiv.symm a = x ** exists equiv x ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : FinEnum \u03b1 x : \u03b1 \u22a2 \u2191equiv.symm (\u2191equiv x) = x ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.hasSum_integral_measure ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f \u22a2 HasSum (fun i => \u222b (a : \u03b1), f a \u2202\u03bc i) (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) ** have hfi : \u2200 i, Integrable f (\u03bc i) := fun i => hf.mono_measure (Measure.le_sum _ _) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u22a2 HasSum (fun i => \u222b (a : \u03b1), f a \u2202\u03bc i) (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) ** simp only [HasSum, \u2190 integral_finset_sum_measure fun i _ => hfi i] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u22a2 Tendsto (fun s => \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i) atTop (\ud835\udcdd (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc)) ** refine' Metric.nhds_basis_ball.tendsto_right_iff.mpr fun \u03b5 \u03b50 => _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d \u03b50 : 0 < \u03b5 \u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u03b5 ** lift \u03b5 to \u211d\u22650 using \u03b50.le ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 \u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5 ** have hf_lt : (\u222b\u207b x, \u2016f x\u2016\u208a \u2202Measure.sum \u03bc) < \u221e := hf.2 ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 \u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5 ** have hmem : \u2200\u1da0 y in \ud835\udcdd (\u222b\u207b x, \u2016f x\u2016\u208a \u2202Measure.sum \u03bc), (\u222b\u207b x, \u2016f x\u2016\u208a \u2202Measure.sum \u03bc) < y + \u03b5 := by\n  refine' tendsto_id.add tendsto_const_nhds (lt_mem_nhds (\u03b1 := \u211d\u22650\u221e) <| ENNReal.lt_add_right _ _)\n  exacts [hf_lt.ne, ENNReal.coe_ne_zero.2 (NNReal.coe_ne_zero.1 \u03b50.ne')] ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 hmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 \u22a2 \u2200\u1da0 (x : Finset \u03b9) in atTop, \u222b (a : \u03b1), f a \u2202\u2211 i in x, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5 ** refine' ((hasSum_lintegral_measure (fun x => \u2016f x\u2016\u208a) \u03bc).eventually hmem).mono fun s hs => _ ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 hmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 s : Finset \u03b9 hs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5 \u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5 ** obtain \u27e8\u03bd, h\u03bd\u27e9 : \u2203 \u03bd, (\u2211 i in s, \u03bc i) + \u03bd = Measure.sum \u03bc := by\n  refine' \u27e8Measure.sum fun i : \u21a5(s\u1d9c : Set \u03b9) => \u03bc i, _\u27e9\n  simpa only [\u2190 Measure.sum_coe_finset] using Measure.sum_add_sum_compl (s : Set \u03b9) \u03bc ** case intro.intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd\u271d : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 hmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 s : Finset \u03b9 hs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5 \u03bd : Measure \u03b1 h\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc \u22a2 \u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i \u2208 Metric.ball (\u222b (a : \u03b1), f a \u2202Measure.sum \u03bc) \u2191\u03b5 ** rw [Metric.mem_ball, \u2190 coe_nndist, NNReal.coe_lt_coe, \u2190 ENNReal.coe_lt_coe, \u2190 h\u03bd] ** case intro.intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd\u271d : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 hmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 s : Finset \u03b9 hs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5 \u03bd : Measure \u03b1 h\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc \u22a2 \u2191(nndist (\u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i) (\u222b (a : \u03b1), f a \u2202(\u2211 i in s, \u03bc i + \u03bd))) < \u2191\u03b5 ** rw [\u2190 h\u03bd, integrable_add_measure] at hf ** case intro.intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd\u271d : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 hmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 s : Finset \u03b9 hs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5 \u03bd : Measure \u03b1 hf : Integrable f \u2227 Integrable f h\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc \u22a2 \u2191(nndist (\u222b (a : \u03b1), f a \u2202\u2211 i in s, \u03bc i) (\u222b (a : \u03b1), f a \u2202(\u2211 i in s, \u03bc i + \u03bd))) < \u2191\u03b5 ** refine' (nndist_integral_add_measure_le_lintegral hf.1 hf.2).trans_lt _ ** case intro.intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd\u271d : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 hmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 s : Finset \u03b9 hs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5 \u03bd : Measure \u03b1 hf : Integrable f \u2227 Integrable f h\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bd < \u2191\u03b5 ** rw [\u2190 h\u03bd, lintegral_add_measure, lintegral_finset_sum_measure] at hs ** case intro.intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd\u271d : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 hmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 s : Finset \u03b9 \u03bd : Measure \u03b1 hs : \u2211 i in s, \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i + \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bd < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5 hf : Integrable f \u2227 Integrable f h\u03bd : \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202\u03bd < \u2191\u03b5 ** exact lt_of_add_lt_add_left hs ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 \u22a2 \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 ** refine' tendsto_id.add tendsto_const_nhds (lt_mem_nhds (\u03b1 := \u211d\u22650\u221e) <| ENNReal.lt_add_right _ _) ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc \u2260 \u22a4  case refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 \u22a2 \u2191\u03b5 \u2260 0 ** exacts [hf_lt.ne, ENNReal.coe_ne_zero.2 (NNReal.coe_ne_zero.1 \u03b50.ne')] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 hmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 s : Finset \u03b9 hs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5 \u22a2 \u2203 \u03bd, \u2211 i in s, \u03bc i + \u03bd = Measure.sum \u03bc ** refine' \u27e8Measure.sum fun i : \u21a5(s\u1d9c : Set \u03b9) => \u03bc i, _\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X \u03bd : Measure \u03b1 \u03b9 : Type u_7 m : MeasurableSpace \u03b1 f : \u03b1 \u2192 G \u03bc : \u03b9 \u2192 Measure \u03b1 hf : Integrable f hfi : \u2200 (i : \u03b9), Integrable f \u03b5 : \u211d\u22650 \u03b50 : 0 < \u2191\u03b5 hf_lt : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u22a4 hmem : \u2200\u1da0 (y : \u211d\u22650\u221e) in \ud835\udcdd (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc), \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < y + \u2191\u03b5 s : Finset \u03b9 hs : \u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a \u2202Measure.sum \u03bc < \u2211 b in s, (fun i => \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc i) b + \u2191\u03b5 \u22a2 (\u2211 i in s, \u03bc i + Measure.sum fun i => \u03bc \u2191i) = Measure.sum \u03bc ** simpa only [\u2190 Measure.sum_coe_finset] using Measure.sum_add_sum_compl (s : Set \u03b9) \u03bc ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.QuasiMeasurePreserving.image_zpow_ae_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm k : \u2124 hs : \u2191e '' s =\u1d50[\u03bc] s \u22a2 \u2191(e ^ k) '' s =\u1d50[\u03bc] s ** rw [Equiv.image_eq_preimage] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm k : \u2124 hs : \u2191e '' s =\u1d50[\u03bc] s \u22a2 \u2191(e ^ k).symm \u207b\u00b9' s =\u1d50[\u03bc] s ** obtain \u27e8k, rfl | rfl\u27e9 := k.eq_nat_or_neg ** case intro.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm hs : \u2191e '' s =\u1d50[\u03bc] s k : \u2115 \u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s ** replace hs : (\u21d1e\u207b\u00b9) \u207b\u00b9' s =\u1d50[\u03bc] s ** case intro.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm k : \u2115 hs : \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s \u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s ** replace he' : (\u21d1e\u207b\u00b9)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s := he'.preimage_iterate_ae_eq k hs ** case intro.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e k : \u2115 hs : \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s he' : (\u2191e\u207b\u00b9)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s \u22a2 \u2191(e ^ \u2191k).symm \u207b\u00b9' s =\u1d50[\u03bc] s ** rwa [Equiv.Perm.iterate_eq_pow e\u207b\u00b9 k, inv_pow e k] at he' ** case hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm hs : \u2191e '' s =\u1d50[\u03bc] s k : \u2115 \u22a2 \u2191e\u207b\u00b9 \u207b\u00b9' s =\u1d50[\u03bc] s ** rwa [Equiv.image_eq_preimage] at hs ** case intro.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm hs : \u2191e '' s =\u1d50[\u03bc] s k : \u2115 \u22a2 \u2191(e ^ (-\u2191k)).symm \u207b\u00b9' s =\u1d50[\u03bc] s ** rw [zpow_neg, zpow_ofNat] ** case intro.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm hs : \u2191e '' s =\u1d50[\u03bc] s k : \u2115 \u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s ** replace hs : e \u207b\u00b9' s =\u1d50[\u03bc] s ** case intro.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm k : \u2115 hs : \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s \u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s ** replace he : (\u21d1e)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s := he.preimage_iterate_ae_eq k hs ** case intro.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he' : QuasiMeasurePreserving \u2191e.symm k : \u2115 hs : \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s he : (\u2191e)^[k] \u207b\u00b9' s =\u1d50[\u03bc] s \u22a2 \u2191(e ^ k)\u207b\u00b9.symm \u207b\u00b9' s =\u1d50[\u03bc] s ** rwa [Equiv.Perm.iterate_eq_pow e k] at he ** case hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm hs : \u2191e '' s =\u1d50[\u03bc] s k : \u2115 \u22a2 \u2191e \u207b\u00b9' s =\u1d50[\u03bc] s ** convert he.preimage_ae_eq hs.symm ** case h.e'_5 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t : Set \u03b1 \u03bca \u03bca' : Measure \u03b1 \u03bcb \u03bcb' : Measure \u03b2 \u03bcc : Measure \u03b3 f : \u03b1 \u2192 \u03b2 s : Set \u03b1 e : \u03b1 \u2243 \u03b1 he : QuasiMeasurePreserving \u2191e he' : QuasiMeasurePreserving \u2191e.symm hs : \u2191e '' s =\u1d50[\u03bc] s k : \u2115 \u22a2 s = \u2191e \u207b\u00b9' (\u2191e '' s) ** rw [Equiv.preimage_image] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ProbabilityMeasure.coeFn_univ_ne_zero ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bd : ProbabilityMeasure \u03a9 \u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) univ \u2260 0 ** simp only [coeFn_univ, Ne.def, one_ne_zero, not_false_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SignedMeasure.someExistsOneDivLT_spec ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u00acrestrict s i \u2264 restrict 0 i \u22a2 MeasureTheory.SignedMeasure.someExistsOneDivLT s i \u2286 i \u2227     MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) \u2227       1 / (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s i) + 1) <         \u2191s (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) ** rw [someExistsOneDivLT, dif_pos hi] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 hi : \u00acrestrict s i \u2264 restrict 0 i \u22a2 Classical.choose         (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)) \u2286       i \u2227     MeasurableSet         (Classical.choose           (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i))) \u2227       1 / (\u2191(MeasureTheory.SignedMeasure.findExistsOneDivLT s i) + 1) <         \u2191s           (Classical.choose             (_ : MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i))) ** exact Classical.choose_spec (findExistsOneDivLT_spec hi) ** Qed",
        "informal": ""
    },
    {
        "formal": "essInf_antitone_measure ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : CompleteLattice \u03b2 f : \u03b1 \u2192 \u03b2 h\u03bc\u03bd : \u03bc \u226a \u03bd \u22a2 essInf f \u03bd \u2264 essInf f \u03bc ** refine' liminf_le_liminf_of_le (Measure.ae_le_iff_absolutelyContinuous.mpr h\u03bc\u03bd) _ _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : CompleteLattice \u03b2 f : \u03b1 \u2192 \u03b2 h\u03bc\u03bd : \u03bc \u226a \u03bd \u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bd) f  case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : CompleteLattice \u03b2 f : \u03b1 \u2192 \u03b2 h\u03bc\u03bd : \u03bc \u226a \u03bd \u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f ** all_goals isBoundedDefault ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d : CompleteLattice \u03b2 f : \u03b1 \u2192 \u03b2 h\u03bc\u03bd : \u03bc \u226a \u03bd \u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) (Measure.ae \u03bc) f ** isBoundedDefault ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.Infinite.encard_eq ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 h : Set.Infinite s \u22a2 encard s = \u22a4 ** have := h.to_subtype ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 h : Set.Infinite s this : Infinite \u2191s \u22a2 encard s = \u22a4 ** rw [encard, \u2190PartENat.withTopEquiv.symm.injective.eq_iff, Equiv.symm_apply_apply,\n  PartENat.withTopEquiv_symm_top, PartENat.card_eq_top_of_infinite] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.AEEqFun.coeFn_comp\u2082 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 inst\u271d : TopologicalSpace \u03b4 g : \u03b2 \u2192 \u03b3 \u2192 \u03b4 hg : Continuous (uncurry g) f\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2 f\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3 \u22a2 \u2191(comp\u2082 g hg f\u2081 f\u2082) =\u1d50[\u03bc] fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a) ** rw [comp\u2082_eq_mk] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : TopologicalSpace \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b3 inst\u271d : TopologicalSpace \u03b4 g : \u03b2 \u2192 \u03b3 \u2192 \u03b4 hg : Continuous (uncurry g) f\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2 f\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3 \u22a2 \u2191(mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (fun x => uncurry g (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)) =\u1d50[\u03bc] fun a =>     g (\u2191f\u2081 a) (\u2191f\u2082 a) ** apply coeFn_mk ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.subNatNat_add_negSucc ** m n k : Nat \u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k) ** have h := Nat.lt_or_ge m n ** m n k : Nat h : m < n \u2228 m \u2265 n \u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k) ** cases h with\n| inr h' =>\n  rw [subNatNat_of_le h']\n  simp\n  rw [subNatNat_sub h', Nat.add_comm]\n| inl h' =>\n  have h\u2082 : m < n + succ k := Nat.lt_of_lt_of_le h' (le_add_right _ _)\n  have h\u2083 : m \u2264 n + k := le_of_succ_le_succ h\u2082\n  rw [subNatNat_of_lt h', subNatNat_of_lt h\u2082]\n  simp [Nat.add_comm]\n  rw [\u2190 add_succ, succ_pred_eq_of_pos (Nat.sub_pos_of_lt h'), add_succ, succ_sub h\u2083,\n    Nat.pred_succ]\n  rw [Nat.add_comm n, Nat.add_sub_assoc (Nat.le_of_lt h')] ** case inr m n k : Nat h' : m \u2265 n \u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k) ** rw [subNatNat_of_le h'] ** case inr m n k : Nat h' : m \u2265 n \u22a2 \u2191(m - n) + -[k+1] = subNatNat m (n + succ k) ** simp ** case inr m n k : Nat h' : m \u2265 n \u22a2 subNatNat (m - n) (succ k) = subNatNat m (n + succ k) ** rw [subNatNat_sub h', Nat.add_comm] ** case inl m n k : Nat h' : m < n \u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k) ** have h\u2082 : m < n + succ k := Nat.lt_of_lt_of_le h' (le_add_right _ _) ** case inl m n k : Nat h' : m < n h\u2082 : m < n + succ k \u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k) ** have h\u2083 : m \u2264 n + k := le_of_succ_le_succ h\u2082 ** case inl m n k : Nat h' : m < n h\u2082 : m < n + succ k h\u2083 : m \u2264 n + k \u22a2 subNatNat m n + -[k+1] = subNatNat m (n + succ k) ** rw [subNatNat_of_lt h', subNatNat_of_lt h\u2082] ** case inl m n k : Nat h' : m < n h\u2082 : m < n + succ k h\u2083 : m \u2264 n + k \u22a2 -[pred (n - m)+1] + -[k+1] = -[pred (n + succ k - m)+1] ** simp [Nat.add_comm] ** case inl m n k : Nat h' : m < n h\u2082 : m < n + succ k h\u2083 : m \u2264 n + k \u22a2 succ (k + pred (n - m)) = pred (n + succ k - m) ** rw [\u2190 add_succ, succ_pred_eq_of_pos (Nat.sub_pos_of_lt h'), add_succ, succ_sub h\u2083,\n  Nat.pred_succ] ** case inl m n k : Nat h' : m < n h\u2082 : m < n + succ k h\u2083 : m \u2264 n + k \u22a2 k + (n - m) = n + k - m ** rw [Nat.add_comm n, Nat.add_sub_assoc (Nat.le_of_lt h')] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.tendstoUniformlyOn_of_ae_tendsto' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u2075 : MetricSpace \u03b2 \u03bc : Measure \u03b1 inst\u271d\u2074 : SemilatticeSup \u03b9 inst\u271d\u00b3 : Nonempty \u03b9 inst\u271d\u00b2 : Countable \u03b9 \u03b3 : Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2203 t, MeasurableSet t \u2227 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c ** have \u27e8t, _, ht, htendsto\u27e9 := tendstoUniformlyOn_of_ae_tendsto hf hg MeasurableSet.univ\n  (measure_ne_top \u03bc Set.univ) (by filter_upwards [hfg] with _ htendsto _ using htendsto) h\u03b5 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u2075 : MetricSpace \u03b2 \u03bc : Measure \u03b1 inst\u271d\u2074 : SemilatticeSup \u03b9 inst\u271d\u00b3 : Nonempty \u03b9 inst\u271d\u00b2 : Countable \u03b9 \u03b3 : Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 t : Set \u03b1 w\u271d : t \u2286 univ ht : MeasurableSet t htendsto : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn (fun n => f n) g atTop (univ \\ t) \u22a2 \u2203 t, MeasurableSet t \u2227 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c ** refine' \u27e8_, ht, _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u2075 : MetricSpace \u03b2 \u03bc : Measure \u03b1 inst\u271d\u2074 : SemilatticeSup \u03b9 inst\u271d\u00b3 : Nonempty \u03b9 inst\u271d\u00b2 : Countable \u03b9 \u03b3 : Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 t : Set \u03b1 w\u271d : t \u2286 univ ht : MeasurableSet t htendsto : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn (fun n => f n) g atTop (univ \\ t) \u22a2 \u2191\u2191\u03bc t \u2264 ENNReal.ofReal \u03b5 \u2227 TendstoUniformlyOn f g atTop t\u1d9c ** rwa [Set.compl_eq_univ_diff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 inst\u271d\u2075 : MetricSpace \u03b2 \u03bc : Measure \u03b1 inst\u271d\u2074 : SemilatticeSup \u03b9 inst\u271d\u00b3 : Nonempty \u03b9 inst\u271d\u00b2 : Countable \u03b9 \u03b3 : Type u_4 inst\u271d\u00b9 : TopologicalSpace \u03b3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 s : Set \u03b1 inst\u271d : IsFiniteMeasure \u03bc hf : \u2200 (n : \u03b9), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) \u03b5 : \u211d h\u03b5 : 0 < \u03b5 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 univ \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) ** filter_upwards [hfg] with _ htendsto _ using htendsto ** Qed",
        "informal": ""
    },
    {
        "formal": "Multiset.noncommFoldr_coe ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 l : List \u03b1 comm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b) b : \u03b2 \u22a2 noncommFoldr f (\u2191l) comm b = List.foldr f b l ** simp only [noncommFoldr, coe_foldr, coe_attach, List.attach, Function.comp] ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 l : List \u03b1 comm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b) b : \u03b2 \u22a2 List.foldr (fun x => f \u2191x) b (List.pmap Subtype.mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l)) = List.foldr f b l ** rw [\u2190 List.foldr_map] ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 l : List \u03b1 comm : Set.Pairwise {x | x \u2208 \u2191l} fun x y => \u2200 (b : \u03b2), f x (f y b) = f y (f x b) b : \u03b2 \u22a2 List.foldr f b (List.map (fun x => \u2191x) (List.pmap Subtype.mk l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l))) = List.foldr f b l ** simp [List.map_pmap, List.pmap_eq_map] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.exists_ne_of_one_lt_encard ** \u03b1 : Type u_1 s t : Set \u03b1 h : 1 < encard s a : \u03b1 \u22a2 \u2203 b, b \u2208 s \u2227 b \u2260 a ** by_contra' h' ** \u03b1 : Type u_1 s t : Set \u03b1 h : 1 < encard s a : \u03b1 h' : \u2200 (b : \u03b1), b \u2208 s \u2192 b = a \u22a2 False ** obtain \u27e8b, b', hb, hb', hne\u27e9 := one_lt_encard_iff.1 h ** case intro.intro.intro.intro \u03b1 : Type u_1 s t : Set \u03b1 h : 1 < encard s a : \u03b1 h' : \u2200 (b : \u03b1), b \u2208 s \u2192 b = a b b' : \u03b1 hb : b \u2208 s hb' : b' \u2208 s hne : b \u2260 b' \u22a2 False ** apply hne ** case intro.intro.intro.intro \u03b1 : Type u_1 s t : Set \u03b1 h : 1 < encard s a : \u03b1 h' : \u2200 (b : \u03b1), b \u2208 s \u2192 b = a b b' : \u03b1 hb : b \u2208 s hb' : b' \u2208 s hne : b \u2260 b' \u22a2 b = b' ** rw [h' b hb, h' b' hb'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.Ico_union_Ico_eq_Ico ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d b\u271d a b c : \u03b1 hab : a \u2264 b hbc : b \u2264 c \u22a2 Ico a b \u222a Ico b c = Ico a c ** rw [\u2190 coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico_eq_Ico hab hbc] ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.coe_essSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f\u271d : \u03b1 \u2192 \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650 hf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f r : \u211d\u22650\u221e \u22a2 r \u2264 \u2a05 a \u2208 fun x => sets (map f (Measure.ae \u03bc)) {x_1 | (fun x_2 => (fun x x_3 => x \u2264 x_3) x_2 x) x_1}, \u2191a \u2194     r \u2264 essSup (fun x => \u2191(f x)) \u03bc ** simp [essSup, limsup, limsSup, eventually_map, ENNReal.forall_ennreal] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f\u271d : \u03b1 \u2192 \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650 hf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) (Measure.ae \u03bc) f r : \u211d\u22650\u221e \u22a2 (\u2200 (i : \u211d\u22650), (i \u2208 fun x => sets (map f (Measure.ae \u03bc)) {x_1 | x_1 \u2264 x}) \u2192 r \u2264 \u2191i) \u2194     \u2200 (r_1 : \u211d\u22650), (\u2200\u1d50 (a : \u03b1) \u2202\u03bc, f a \u2264 r_1) \u2192 r \u2264 \u2191r_1 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Supermartingale.smul_nonneg ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc \u22a2 Supermartingale (c \u2022 f) \u2131 \u03bc ** refine' \u27e8hf.1.smul c, fun i j hij => _, fun i => (hf.2.2 i).smul c\u27e9 ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j \u22a2 \u03bc[(c \u2022 f) j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i ** refine' (condexp_smul c (f j)).le.trans _ ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j \u22a2 c \u2022 \u03bc[f j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i ** filter_upwards [hf.2.1 i j hij] with _ hle ** case h \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j a\u271d : \u03a9 hle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d \u22a2 (c \u2022 \u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 (c \u2022 f) i a\u271d ** simp_rw [Pi.smul_apply] ** case h \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j a\u271d : \u03a9 hle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d \u22a2 c \u2022 (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 c \u2022 f i a\u271d ** exact smul_le_smul_of_nonneg hle hc ** Qed",
        "informal": ""
    },
    {
        "formal": "Bundle.TotalSpace.mk_inj ** B : Type u_1 F : Type u_2 E : B \u2192 Type u_3 b : B y y' : E b \u22a2 mk' F b y = mk' F b y' \u2194 y = y' ** simp [TotalSpace.ext_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "circleIntegral.norm_integral_lt_of_norm_le_const_of_lt ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 < R hc : ContinuousOn f (sphere c R) hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C hlt : \u2203 z, z \u2208 sphere c R \u2227 \u2016f z\u2016 < C \u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C ** rw [\u2190 _root_.abs_of_pos hR, \u2190 image_circleMap_Ioc] at hlt ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 < R hc : ContinuousOn f (sphere c R) hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C hlt : \u2203 z, z \u2208 circleMap c R '' Ioc 0 (2 * \u03c0) \u2227 \u2016f z\u2016 < C \u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C ** rcases hlt with \u27e8_, \u27e8\u03b8\u2080, hmem, rfl\u27e9, hlt\u27e9 ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 < R hc : ContinuousOn f (sphere c R) hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C \u03b8\u2080 : \u211d hmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0) hlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C \u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016deriv (circleMap c R) \u03b8 \u2022 f (circleMap c R \u03b8)\u2016 < \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C ** simp only [norm_smul, deriv_circleMap, norm_eq_abs, map_mul, abs_I, mul_one,\n  abs_circleMap_zero, abs_of_pos hR] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 < R hc : ContinuousOn f (sphere c R) hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C \u03b8\u2080 : \u211d hmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0) hlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C \u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, R * \u2016f (circleMap c R \u03b8)\u2016 < \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C ** refine' intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt\n    Real.two_pi_pos _ continuousOn_const (fun \u03b8 _ => _) \u27e8\u03b8\u2080, Ioc_subset_Icc_self hmem, _\u27e9 ** case refine'_1 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 < R hc : ContinuousOn f (sphere c R) hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C \u03b8\u2080 : \u211d hmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0) hlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C \u22a2 ContinuousOn (fun \u03b8 => R * \u2016f (circleMap c R \u03b8)\u2016) (Icc 0 (2 * \u03c0)) ** exact continuousOn_const.mul (hc.comp (continuous_circleMap _ _).continuousOn fun \u03b8 _ =>\n  circleMap_mem_sphere _ hR.le _).norm ** case refine'_2 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 < R hc : ContinuousOn f (sphere c R) hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C \u03b8\u2080 : \u211d hmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0) hlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C \u03b8 : \u211d x\u271d : \u03b8 \u2208 Ioc 0 (2 * \u03c0) \u22a2 R * \u2016f (circleMap c R \u03b8)\u2016 \u2264 R * C ** exact mul_le_mul_of_nonneg_left (hf _ <| circleMap_mem_sphere _ hR.le _) hR.le ** case refine'_3 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 < R hc : ContinuousOn f (sphere c R) hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C \u03b8\u2080 : \u211d hmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0) hlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C \u22a2 R * \u2016f (circleMap c R \u03b8\u2080)\u2016 < R * C ** exact (mul_lt_mul_left hR).2 hlt ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 < R hc : ContinuousOn f (sphere c R) hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C \u03b8\u2080 : \u211d hmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0) hlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C \u22a2 \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C = 2 * \u03c0 * R * C ** simp [mul_assoc] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E f : \u2102 \u2192 E c : \u2102 R C : \u211d hR : 0 < R hc : ContinuousOn f (sphere c R) hf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C \u03b8\u2080 : \u211d hmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0) hlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C \u22a2 R * (2 * (\u03c0 * C)) = 2 * (\u03c0 * (R * C)) ** ring ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.mapM_eq_mapM_data ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 m \u03b2 arr : Array \u03b1 \u22a2 mapM f arr = do     let __do_lift \u2190 List.mapM f arr.data     pure { data := __do_lift } ** rw [mapM_eq_foldlM, foldlM_eq_foldlM_data, \u2190 List.foldrM_reverse] ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 m \u03b2 arr : Array \u03b1 \u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] (List.reverse arr.data) = do     let __do_lift \u2190 List.mapM f arr.data     pure { data := __do_lift } ** conv => rhs; rw [\u2190 List.reverse_reverse arr.data] ** m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 m \u03b2 arr : Array \u03b1 \u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] (List.reverse arr.data) = do     let __do_lift \u2190 List.mapM f (List.reverse (List.reverse arr.data))     pure { data := __do_lift } ** induction arr.data.reverse with\n| nil => simp; rfl\n| cons a l ih => simp [ih]; simp [map_eq_pure_bind, push] ** case nil m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 m \u03b2 arr : Array \u03b1 \u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] [] = do     let __do_lift \u2190 List.mapM f (List.reverse [])     pure { data := __do_lift } ** simp ** case nil m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 m \u03b2 arr : Array \u03b1 \u22a2 pure #[] = pure { data := [] } ** rfl ** case cons m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 m \u03b2 arr : Array \u03b1 a : \u03b1 l : List \u03b1 ih :   List.foldrM (fun a bs => push bs <$> f a) #[] l = do     let __do_lift \u2190 List.mapM f (List.reverse l)     pure { data := __do_lift } \u22a2 List.foldrM (fun a bs => push bs <$> f a) #[] (a :: l) = do     let __do_lift \u2190 List.mapM f (List.reverse (a :: l))     pure { data := __do_lift } ** simp [ih] ** case cons m : Type u_1 \u2192 Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_1 inst\u271d\u00b9 : Monad m inst\u271d : LawfulMonad m f : \u03b1 \u2192 m \u03b2 arr : Array \u03b1 a : \u03b1 l : List \u03b1 ih :   List.foldrM (fun a bs => push bs <$> f a) #[] l = do     let __do_lift \u2190 List.mapM f (List.reverse l)     pure { data := __do_lift } \u22a2 (do       let x \u2190 List.mapM f (List.reverse l)       push { data := x } <$> f a) =     do     let x \u2190 List.mapM f (List.reverse l)     let x_1 \u2190 f a     pure { data := x ++ [x_1] } ** simp [map_eq_pure_bind, push] ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.dom_fintype ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Primcodable \u03b2 inst\u271d\u00b3 : Primcodable \u03b3 inst\u271d\u00b2 : Primcodable \u03b4 inst\u271d\u00b9 : Primcodable \u03c3 inst\u271d : Fintype \u03b1 f : \u03b1 \u2192 \u03c3 l : List \u03b1 left\u271d : List.Nodup l m : \u2200 (x : \u03b1), x \u2208 l \u22a2 Primrec fun a => some (f a) ** haveI := decidableEqOfEncodable \u03b1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Primcodable \u03b2 inst\u271d\u00b3 : Primcodable \u03b3 inst\u271d\u00b2 : Primcodable \u03b4 inst\u271d\u00b9 : Primcodable \u03c3 inst\u271d : Fintype \u03b1 f : \u03b1 \u2192 \u03c3 l : List \u03b1 left\u271d : List.Nodup l m : \u2200 (x : \u03b1), x \u2208 l this : DecidableEq \u03b1 \u22a2 Primrec fun a => some (f a) ** refine ((list_get?\u2081 (l.map f)).comp (list_indexOf\u2081 l)).of_eq fun a => ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Primcodable \u03b2 inst\u271d\u00b3 : Primcodable \u03b3 inst\u271d\u00b2 : Primcodable \u03b4 inst\u271d\u00b9 : Primcodable \u03c3 inst\u271d : Fintype \u03b1 f : \u03b1 \u2192 \u03c3 l : List \u03b1 left\u271d : List.Nodup l m : \u2200 (x : \u03b1), x \u2208 l this : DecidableEq \u03b1 a : \u03b1 \u22a2 List.get? (List.map f l) (List.indexOf a l) = some (f a) ** rw [List.get?_map, List.indexOf_get? (m a), Option.map_some'] ** Qed",
        "informal": ""
    },
    {
        "formal": "list_casesOn' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03b2 \u00d7 List \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this\u271d : Primcodable (List \u03b2) := prim H this : Primrec fun a => Option.map (fun o => Option.casesOn o (g a) (h a)) (decode (encode (f a))) a : \u03b1 \u22a2 Option.map (fun o => Option.casesOn o (g a) (h a)) (decode (encode (f a))) =     some (List.casesOn (f a) (g a) fun b l => h a (b, l)) ** cases' f a with b l <;> simp [encodek] ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec.beq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03c3 : Type u_5 inst\u271d\u2075 : Primcodable \u03b1 inst\u271d\u2074 : Primcodable \u03b2 inst\u271d\u00b3 : Primcodable \u03b3 inst\u271d\u00b2 : Primcodable \u03b4 inst\u271d\u00b9 : Primcodable \u03c3 inst\u271d : DecidableEq \u03b1 a : \u2115 \u00d7 \u2115 \u22a2 (fun x x_1 => x \u2264 x_1) a.1 a.2 \u2227 (fun x x_1 => x \u2264 x_1) a.2 a.1 \u2194 (fun a b => a = b) a.1 a.2 ** simp [le_antisymm_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.Partrec.Code.eval_prec_zero ** cf cg : Code a : \u2115 \u22a2 eval (prec cf cg) (Nat.pair a 0) = eval cf a ** rw [eval, Nat.unpaired, Nat.unpair_pair] ** cf cg : Code a : \u2115 \u22a2 Nat.rec (eval cf (a, 0).1)       (fun y IH => do         let i \u2190 IH         eval cg (Nat.pair (a, 0).1 (Nat.pair y i)))       (a, 0).2 =     eval cf a ** simp (config := { Lean.Meta.Simp.neutralConfig with proj := true }) only [] ** cf cg : Code a : \u2115 \u22a2 Nat.rec (eval cf a)       (fun y IH => do         let i \u2190 IH         eval cg (Nat.pair a (Nat.pair y i)))       0 =     eval cf a ** rw [Nat.rec_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_of_tendsto_metrizable' ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) \u22a2 Measurable g ** letI : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 \u22a2 Measurable g ** apply measurable_of_is_closed' ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 \u22a2 \u2200 (s : Set \u03b2), IsClosed s \u2192 Set.Nonempty s \u2192 s \u2260 Set.univ \u2192 MeasurableSet (g \u207b\u00b9' s) ** intro s h1s h2s h3s ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ \u22a2 MeasurableSet (g \u207b\u00b9' s) ** have : Measurable fun x => infNndist (g x) s := by\n  suffices : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s)\n  exact measurable_of_tendsto_nnreal' u (fun i => (hf i).infNndist) this\n  rw [tendsto_pi_nhds] at lim \u22a2\n  intro x\n  exact ((continuous_infNndist_pt s).tendsto (g x)).comp (lim x) ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ this : Measurable fun x => infNndist (g x) s \u22a2 MeasurableSet (g \u207b\u00b9' s) ** have h4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0} := by\n  ext x\n  simp [h1s, \u2190 h1s.mem_iff_infDist_zero h2s, \u2190 NNReal.coe_eq_zero] ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ this : Measurable fun x => infNndist (g x) s h4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0} \u22a2 MeasurableSet (g \u207b\u00b9' s) ** rw [h4s] ** case hf \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ this : Measurable fun x => infNndist (g x) s h4s : g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0} \u22a2 MeasurableSet ((fun x => infNndist (g x) s) \u207b\u00b9' {0}) ** exact this (measurableSet_singleton 0) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ \u22a2 Measurable fun x => infNndist (g x) s ** suffices : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ this : Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s) \u22a2 Measurable fun x => infNndist (g x) s  case this \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ \u22a2 Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s) ** exact measurable_of_tendsto_nnreal' u (fun i => (hf i).infNndist) this ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ \u22a2 Tendsto (fun i x => infNndist (f i x) s) u (\ud835\udcdd fun x => infNndist (g x) s) ** rw [tendsto_pi_nhds] at lim \u22a2 ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x)) this : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ \u22a2 \u2200 (x : \u03b1), Tendsto (fun i => infNndist (f i x) s) u (\ud835\udcdd (infNndist (g x) s)) ** intro x ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : \u2200 (x : \u03b1), Tendsto (fun i => f i x) u (\ud835\udcdd (g x)) this : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ x : \u03b1 \u22a2 Tendsto (fun i => infNndist (f i x) s) u (\ud835\udcdd (infNndist (g x) s)) ** exact ((continuous_infNndist_pt s).tendsto (g x)).comp (lim x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ this : Measurable fun x => infNndist (g x) s \u22a2 g \u207b\u00b9' s = (fun x => infNndist (g x) s) \u207b\u00b9' {0} ** ext x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : TopologicalSpace \u03b2 inst\u271d\u2074 : PseudoMetrizableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : BorelSpace \u03b2 \u03b9 : Type u_3 f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b2 u : Filter \u03b9 inst\u271d\u00b9 : NeBot u inst\u271d : IsCountablyGenerated u hf : \u2200 (i : \u03b9), Measurable (f i) lim : Tendsto f u (\ud835\udcdd g) this\u271d : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2 s : Set \u03b2 h1s : IsClosed s h2s : Set.Nonempty s h3s : s \u2260 Set.univ this : Measurable fun x => infNndist (g x) s x : \u03b1 \u22a2 x \u2208 g \u207b\u00b9' s \u2194 x \u2208 (fun x => infNndist (g x) s) \u207b\u00b9' {0} ** simp [h1s, \u2190 h1s.mem_iff_infDist_zero h2s, \u2190 NNReal.coe_eq_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexp_mono ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** by_cases hm : m \u2264 m0 ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm) ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]  case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** swap ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm ** case pos \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm this : SigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** exact (condexp_ae_eq_condexpL1 hm _).trans_le\n  ((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm) ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : \u00acm \u2264 m0 \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** simp_rw [condexp_of_not_le hm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : \u00acm \u2264 m0 \u22a2 0 \u2264\u1d50[\u03bc] 0 ** rfl ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m] ** simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm] ** case neg \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F inst\u271d\u2077 : NormedAddCommGroup F' inst\u271d\u2076 : NormedSpace \ud835\udd5c F' inst\u271d\u2075 : NormedSpace \u211d F' inst\u271d\u2074 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d g\u271d : \u03b1 \u2192 F' s : Set \u03b1 E : Type u_5 inst\u271d\u00b3 : NormedLatticeAddCommGroup E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : OrderedSMul \u211d E f g : \u03b1 \u2192 E hf : Integrable f hg : Integrable g hfg : f \u2264\u1d50[\u03bc] g hm : m \u2264 m0 h\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm) \u22a2 0 \u2264\u1d50[\u03bc] 0 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.MeasurePreserving.skew_product ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd \u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2) ** have : Measurable fun p : \u03b1 \u00d7 \u03b3 => (f p.1, g p.1 p.2) := (hf.1.comp measurable_fst).prod_mk hgm ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) \u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2) ** rcases eq_or_ne \u03bca 0 with (rfl | ha) ** case inr \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 \u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2) ** have sf : SigmaFinite \u03bcc := by\n  rcases (ae_neBot.2 ha).nonempty_of_mem hg with \u27e8x, hx : map (g x) \u03bcc = \u03bcd\u27e9\n  exact SigmaFinite.of_map _ hgm.of_uncurry_left.aemeasurable (by rwa [hx]) ** case inr \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc \u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2) ** refine' \u27e8this, (prod_eq fun s t hs ht => _).symm\u27e9 ** case inr \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t \u22a2 \u2191\u2191(map (fun p => (f p.1, g p.1 p.2)) (Measure.prod \u03bca \u03bcc)) (s \u00d7\u02e2 t) = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t ** rw [map_apply this (hs.prod ht)] ** case inr \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t \u22a2 \u2191\u2191(Measure.prod \u03bca \u03bcc) ((fun p => (f p.1, g p.1 p.2)) \u207b\u00b9' s \u00d7\u02e2 t) = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t ** refine' (prod_apply (this <| hs.prod ht)).trans _ ** case inr \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc (Prod.mk x \u207b\u00b9' ((fun p => (f p.1, g p.1 p.2)) \u207b\u00b9' s \u00d7\u02e2 t)) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t ** have : \u2200\u1d50 x \u2202\u03bca,\n    \u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun _ => \u03bcd t) x := by\n  refine' hg.mono fun x hx => _\n  subst hx\n  simp only [mk_preimage_prod_right_fn_eq_if, indicator_apply, mem_preimage]\n  split_ifs\n  exacts [(map_apply hgm.of_uncurry_left ht).symm, measure_empty] ** case inr \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this\u271d : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t this : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc (Prod.mk x \u207b\u00b9' ((fun p => (f p.1, g p.1 p.2)) \u207b\u00b9' s \u00d7\u02e2 t)) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t ** simp only [preimage_preimage] ** case inr \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this\u271d : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t this : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x \u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191\u03bcc ((fun x_1 => (f x, g x x_1)) \u207b\u00b9' s \u00d7\u02e2 t) \u2202\u03bca = \u2191\u2191\u03bcb s * \u2191\u2191\u03bcd t ** rw [lintegral_congr_ae this, lintegral_indicator _ (hf.1 hs), set_lintegral_const,\n  hf.measure_preimage hs, mul_comm] ** case inl \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) this : Measurable fun p => (f p.1, g p.1 p.2) hf : MeasurePreserving f hg : \u2200\u1d50 (x : \u03b1) \u22020, map (g x) \u03bcc = \u03bcd \u22a2 MeasurePreserving fun p => (f p.1, g p.1 p.2) ** exact \u27e8this, by simp only [Measure.map_zero]\u27e9 ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) this : Measurable fun p => (f p.1, g p.1 p.2) hf : MeasurePreserving f hg : \u2200\u1d50 (x : \u03b1) \u22020, map (g x) \u03bcc = \u03bcd \u22a2 map (fun p => (f p.1, g p.1 p.2)) 0 = 0 ** simp only [Measure.map_zero] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 \u22a2 SigmaFinite \u03bcc ** rcases (ae_neBot.2 ha).nonempty_of_mem hg with \u27e8x, hx : map (g x) \u03bcc = \u03bcd\u27e9 ** case intro \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 x : \u03b1 hx : map (g x) \u03bcc = \u03bcd \u22a2 SigmaFinite \u03bcc ** exact SigmaFinite.of_map _ hgm.of_uncurry_left.aemeasurable (by rwa [hx]) ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 x : \u03b1 hx : map (g x) \u03bcc = \u03bcd \u22a2 SigmaFinite (map (g ?m.119880) \u03bcc) ** rwa [hx] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bca, \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x ** refine' hg.mono fun x hx => _ ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 \u03bcd : Measure \u03b4 inst\u271d\u00b9 : SigmaFinite \u03bcb inst\u271d : SigmaFinite \u03bcd f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) hg : \u2200\u1d50 (x : \u03b1) \u2202\u03bca, map (g x) \u03bcc = \u03bcd this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t x : \u03b1 hx : map (g x) \u03bcc = \u03bcd \u22a2 \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x => \u2191\u2191\u03bcd t) x ** subst hx ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 inst\u271d\u00b9 : SigmaFinite \u03bcb f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t x : \u03b1 inst\u271d : SigmaFinite (map (g x) \u03bcc) hg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc \u22a2 \u2191\u2191\u03bcc ((fun y => (f x, g x y)) \u207b\u00b9' s \u00d7\u02e2 t) = indicator (f \u207b\u00b9' s) (fun x_1 => \u2191\u2191(map (g x) \u03bcc) t) x ** simp only [mk_preimage_prod_right_fn_eq_if, indicator_apply, mem_preimage] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 inst\u271d\u00b9 : SigmaFinite \u03bcb f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t x : \u03b1 inst\u271d : SigmaFinite (map (g x) \u03bcc) hg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc \u22a2 \u2191\u2191\u03bcc (if f x \u2208 s then (fun b => g x b) \u207b\u00b9' t else \u2205) = if f x \u2208 s then \u2191\u2191(map (g x) \u03bcc) t else 0 ** split_ifs ** case pos \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 inst\u271d\u00b9 : SigmaFinite \u03bcb f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t x : \u03b1 inst\u271d : SigmaFinite (map (g x) \u03bcc) hg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc h\u271d : f x \u2208 s \u22a2 \u2191\u2191\u03bcc ((fun b => g x b) \u207b\u00b9' t) = \u2191\u2191(map (g x) \u03bcc) t  case neg \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2078 : MeasurableSpace \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1' inst\u271d\u2076 : MeasurableSpace \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2' inst\u271d\u2074 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b3 : NormedAddCommGroup E \u03b4 : Type u_7 inst\u271d\u00b2 : MeasurableSpace \u03b4 \u03bca : Measure \u03b1 \u03bcb : Measure \u03b2 \u03bcc : Measure \u03b3 inst\u271d\u00b9 : SigmaFinite \u03bcb f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f g : \u03b1 \u2192 \u03b3 \u2192 \u03b4 hgm : Measurable (uncurry g) this : Measurable fun p => (f p.1, g p.1 p.2) ha : \u03bca \u2260 0 sf : SigmaFinite \u03bcc s : Set \u03b2 t : Set \u03b4 hs : MeasurableSet s ht : MeasurableSet t x : \u03b1 inst\u271d : SigmaFinite (map (g x) \u03bcc) hg : \u2200\u1d50 (x_1 : \u03b1) \u2202\u03bca, map (g x_1) \u03bcc = map (g x) \u03bcc h\u271d : \u00acf x \u2208 s \u22a2 \u2191\u2191\u03bcc \u2205 = 0 ** exacts [(map_apply hgm.of_uncurry_left ht).symm, measure_empty] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_smul_average ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f\u271d g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc ** cases' eq_or_ne \u03bc 0 with h\u03bc h\u03bc ** case inl \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f\u271d g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E h\u03bc : \u03bc = 0 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc ** rw [h\u03bc, integral_zero_measure, average_zero_measure, smul_zero] ** case inr \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f\u271d g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E h\u03bc : \u03bc \u2260 0 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc = \u222b (x : \u03b1), f x \u2202\u03bc ** rw [average_eq, smul_inv_smul\u2080] ** case inr.hc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f\u271d g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E h\u03bc : \u03bc \u2260 0 \u22a2 ENNReal.toReal (\u2191\u2191\u03bc univ) \u2260 0 ** refine' (ENNReal.toReal_pos _ <| measure_ne_top _ _).ne' ** case inr.hc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f\u271d g : \u03b1 \u2192 E inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 E h\u03bc : \u03bc \u2260 0 \u22a2 \u2191\u2191\u03bc univ \u2260 0 ** rwa [Ne.def, measure_univ_eq_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "BoundedContinuousFunction.mem_Lp ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192\u1d47 E \u22a2 ContinuousMap.toAEEqFun \u03bc f.toContinuousMap \u2208 Lp E p ** refine' Lp.mem_Lp_of_ae_bound \u2016f\u2016 _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192\u1d47 E \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016\u2191(ContinuousMap.toAEEqFun \u03bc f.toContinuousMap) x\u2016 \u2264 \u2016f\u2016 ** filter_upwards [f.toContinuousMap.coeFn_toAEEqFun \u03bc] with x _ ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192\u1d47 E x : \u03b1 a\u271d : \u2191(ContinuousMap.toAEEqFun \u03bc f.toContinuousMap) x = \u2191f.toContinuousMap x \u22a2 \u2016\u2191(ContinuousMap.toAEEqFun \u03bc f.toContinuousMap) x\u2016 \u2264 \u2016f\u2016 ** convert f.norm_coe_le_norm x using 2 ** Qed",
        "informal": ""
    },
    {
        "formal": "EuclideanSpace.volume_preserving_measurableEquiv ** \u03b9\u271d : Type u_1 F : Type u_2 inst\u271d\u2076 : Fintype \u03b9\u271d inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F inst\u271d\u00b3 : FiniteDimensional \u211d F inst\u271d\u00b2 : MeasurableSpace F inst\u271d\u00b9 : BorelSpace F \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 \u22a2 MeasurePreserving \u2191(EuclideanSpace.measurableEquiv \u03b9) ** suffices volume = map (EuclideanSpace.measurableEquiv \u03b9).symm volume by\n  convert ((EuclideanSpace.measurableEquiv \u03b9).symm.measurable.measurePreserving _).symm ** \u03b9\u271d : Type u_1 F : Type u_2 inst\u271d\u2076 : Fintype \u03b9\u271d inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F inst\u271d\u00b3 : FiniteDimensional \u211d F inst\u271d\u00b2 : MeasurableSpace F inst\u271d\u00b9 : BorelSpace F \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 \u22a2 volume = map (\u2191(MeasurableEquiv.symm (EuclideanSpace.measurableEquiv \u03b9))) volume ** rw [\u2190 addHaarMeasure_eq_volume_pi, \u2190 Basis.parallelepiped_basisFun, \u2190 Basis.addHaar_def,\n  coe_measurableEquiv_symm, \u2190 PiLp.continuousLinearEquiv_symm_apply 2 \u211d, Basis.map_addHaar] ** \u03b9\u271d : Type u_1 F : Type u_2 inst\u271d\u2076 : Fintype \u03b9\u271d inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F inst\u271d\u00b3 : FiniteDimensional \u211d F inst\u271d\u00b2 : MeasurableSpace F inst\u271d\u00b9 : BorelSpace F \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 \u22a2 volume =     Basis.addHaar       (Basis.map (Pi.basisFun \u211d \u03b9)         (ContinuousLinearEquiv.symm (PiLp.continuousLinearEquiv 2 \u211d fun a => \u211d)).toLinearEquiv) ** exact (EuclideanSpace.basisFun _ _).addHaar_eq_volume.symm ** \u03b9\u271d : Type u_1 F : Type u_2 inst\u271d\u2076 : Fintype \u03b9\u271d inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F inst\u271d\u00b3 : FiniteDimensional \u211d F inst\u271d\u00b2 : MeasurableSpace F inst\u271d\u00b9 : BorelSpace F \u03b9 : Type u_3 inst\u271d : Fintype \u03b9 this : volume = map (\u2191(MeasurableEquiv.symm (EuclideanSpace.measurableEquiv \u03b9))) volume \u22a2 MeasurePreserving \u2191(EuclideanSpace.measurableEquiv \u03b9) ** convert ((EuclideanSpace.measurableEquiv \u03b9).symm.measurable.measurePreserving _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "WithBot.image_coe_Icc ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Icc a b = Icc \u2191a \u2191b ** rw [\u2190 preimage_coe_Icc, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left\n    (Subset.trans Icc_subset_Ici_self <| Ici_subset_Ioi.2 <| bot_lt_coe a)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.VectorMeasure.ext_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : TopologicalSpace M v w : VectorMeasure \u03b1 M \u22a2 v = w \u2194 \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i ** constructor ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : TopologicalSpace M v w : VectorMeasure \u03b1 M \u22a2 v = w \u2192 \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i ** rintro rfl _ _ ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : TopologicalSpace M v : VectorMeasure \u03b1 M i\u271d : Set \u03b1 a\u271d : MeasurableSet i\u271d \u22a2 \u2191v i\u271d = \u2191v i\u271d ** rfl ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : TopologicalSpace M v w : VectorMeasure \u03b1 M \u22a2 (\u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i) \u2192 v = w ** rw [ext_iff'] ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : TopologicalSpace M v w : VectorMeasure \u03b1 M \u22a2 (\u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i) \u2192 \u2200 (i : Set \u03b1), \u2191v i = \u2191w i ** intro h i ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : TopologicalSpace M v w : VectorMeasure \u03b1 M h : \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i i : Set \u03b1 \u22a2 \u2191v i = \u2191w i ** by_cases hi : MeasurableSet i ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : TopologicalSpace M v w : VectorMeasure \u03b1 M h : \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191v i = \u2191w i ** exact h i hi ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : TopologicalSpace M v w : VectorMeasure \u03b1 M h : \u2200 (i : Set \u03b1), MeasurableSet i \u2192 \u2191v i = \u2191w i i : Set \u03b1 hi : \u00acMeasurableSet i \u22a2 \u2191v i = \u2191w i ** simp_rw [not_measurable _ hi] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.min_eq_top ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s : Finset \u03b1 h : Finset.min s = \u22a4 H : Finset.Nonempty s \u22a2 s = \u2205 ** let \u27e8a, ha\u27e9 := min_of_nonempty H ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s : Finset \u03b1 h : Finset.min s = \u22a4 H : Finset.Nonempty s a : \u03b1 ha : Finset.min s = \u2191a \u22a2 s = \u2205 ** rw [h] at ha ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s : Finset \u03b1 h : Finset.min s = \u22a4 H : Finset.Nonempty s a : \u03b1 ha : \u22a4 = \u2191a \u22a2 s = \u2205 ** cases ha ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.map_linearMap_addHaar_pi_eq_smul_addHaar ** \u03b9 : Type u_1 inst\u271d\u00b9 : Finite \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsAddHaarMeasure \u03bc \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** cases nonempty_fintype \u03b9 ** case intro \u03b9 : Type u_1 inst\u271d\u00b9 : Finite \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsAddHaarMeasure \u03bc val\u271d : Fintype \u03b9 \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** have := addHaarMeasure_unique \u03bc (piIcc01 \u03b9) ** case intro \u03b9 : Type u_1 inst\u271d\u00b9 : Finite \u03b9 f : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d hf : \u2191LinearMap.det f \u2260 0 \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsAddHaarMeasure \u03bc val\u271d : Fintype \u03b9 this : \u03bc = \u2191\u2191\u03bc \u2191(piIcc01 \u03b9) \u2022 addHaarMeasure (piIcc01 \u03b9) \u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc ** rw [this, addHaarMeasure_eq_volume_pi, Measure.map_smul,\n  Real.map_linearMap_volume_pi_eq_smul_volume_pi hf, smul_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.IsFundamentalDomain.measure_fundamentalFrontier ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b3 : Countable G inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MulAction G \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : IsFundamentalDomain G s \u22a2 \u2191\u2191\u03bc (fundamentalFrontier G s) = 0 ** simpa only [fundamentalFrontier, iUnion\u2082_inter, measure_iUnion_null_iff', one_smul,\n  measure_iUnion_null_iff, inter_comm s, Function.onFun] using fun g (hg : g \u2260 1) =>\n  hs.aedisjoint hg ** Qed",
        "informal": ""
    },
    {
        "formal": "list_foldl' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h \u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a) ** letI := prim H ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H \u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a) ** let G (a : \u03b1) (IH : \u03c3 \u00d7 List \u03b2) : \u03c3 \u00d7 List \u03b2 := List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) \u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a) ** have hG : Primrec\u2082 G := list_casesOn' H (snd.comp snd) snd <|\n  to\u2082 <|\n  pair (hh.comp (fst.comp fst) <| pair ((fst.comp snd).comp fst) (fst.comp snd))\n    (snd.comp snd) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G \u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a) ** let F := fun (a : \u03b1) (n : \u2115) => (G a)^[n] (g a, f a) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) \u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a) ** have hF : Primrec fun a => (F a (encode (f a))).1 :=\n  (fst.comp <|\n    nat_iterate (encode_iff.2 hf) (pair hg hf) <|\n    hG) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 \u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a) ** suffices \u2200 a n, F a n = (((f a).take n).foldl (fun s b => h a (s, b)) (g a), (f a).drop n) by\n  refine hF.of_eq fun a => ?_\n  rw [this, List.take_all_of_le (length_le_encode _)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 \u22a2 \u2200 (a : \u03b1) (n : \u2115), F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a)) ** introv ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 a : \u03b1 n : \u2115 \u22a2 F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a)) ** dsimp only ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 a : \u03b1 n : \u2115 \u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (g a, f a) =     (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a)) ** generalize f a = l ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 a : \u03b1 n : \u2115 l : List \u03b2 \u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (g a, l) =     (List.foldl (fun s b => h a (s, b)) (g a) (List.take n l), List.drop n l) ** generalize g a = x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 a : \u03b1 n : \u2115 l : List \u03b2 x : \u03c3 \u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (x, l) =     (List.foldl (fun s b => h a (s, b)) x (List.take n l), List.drop n l) ** induction' n with n IH generalizing l x ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 a : \u03b1 l\u271d : List \u03b2 x\u271d : \u03c3 n : \u2115 IH :   \u2200 (l : List \u03b2) (x : \u03c3),     (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (x, l) =       (List.foldl (fun s b => h a (s, b)) x (List.take n l), List.drop n l) l : List \u03b2 x : \u03c3 \u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[Nat.succ n] (x, l) =     (List.foldl (fun s b => h a (s, b)) x (List.take (Nat.succ n) l), List.drop (Nat.succ n) l) ** simp only [iterate_succ, comp_apply] ** case succ \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 a : \u03b1 l\u271d : List \u03b2 x\u271d : \u03c3 n : \u2115 IH :   \u2200 (l : List \u03b2) (x : \u03c3),     (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n] (x, l) =       (List.foldl (fun s b => h a (s, b)) x (List.take n l), List.drop n l) l : List \u03b2 x : \u03c3 \u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[n]       (List.rec (x, l) (fun head tail tail_ih => (h a (x, head), tail)) l) =     (List.foldl (fun s b => h a (s, b)) x (List.take (Nat.succ n) l), List.drop (Nat.succ n) l) ** cases' l with b l <;> simp [IH] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this\u271d : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 this : \u2200 (a : \u03b1) (n : \u2115), F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a)) \u22a2 Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a) ** refine hF.of_eq fun a => ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this\u271d : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 this : \u2200 (a : \u03b1) (n : \u2115), F a n = (List.foldl (fun s b => h a (s, b)) (g a) (List.take n (f a)), List.drop n (f a)) a : \u03b1 \u22a2 (F a (encode (f a))).1 = List.foldl (fun s b => h a (s, b)) (g a) (f a) ** rw [this, List.take_all_of_le (length_le_encode _)] ** case zero \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) f : \u03b1 \u2192 List \u03b2 g : \u03b1 \u2192 \u03c3 h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3 hf : Primrec f hg : Primrec g hh : Primrec\u2082 h this : Primcodable (List \u03b2) := prim H G : \u03b1 \u2192 \u03c3 \u00d7 List \u03b2 \u2192 \u03c3 \u00d7 List \u03b2 := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l) hG : Primrec\u2082 G F : \u03b1 \u2192 \u2115 \u2192 \u03c3 \u00d7 List \u03b2 := fun a n => (G a)^[n] (g a, f a) hF : Primrec fun a => (F a (encode (f a))).1 a : \u03b1 l\u271d : List \u03b2 x\u271d : \u03c3 l : List \u03b2 x : \u03c3 \u22a2 (fun IH => List.rec IH (fun head tail tail_ih => (h a (IH.1, head), tail)) IH.2)^[Nat.zero] (x, l) =     (List.foldl (fun s b => h a (s, b)) x (List.take Nat.zero l), List.drop Nat.zero l) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.mem\u2112p_of_finite_measure_preimage ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 \u22a2 Mem\u2112p (\u2191f) p ** by_cases hp0 : p = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 \u22a2 Mem\u2112p (\u2191f) p ** by_cases hp_top : p = \u221e ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 Mem\u2112p (\u2191f) p ** refine' \u27e8f.aestronglyMeasurable, _\u27e9 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm (\u2191f) p \u03bc < \u22a4 ** rw [snorm_eq_snorm' hp0 hp_top, f.snorm'_eq] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 (\u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y})) ^ (1 / ENNReal.toReal p) < \u22a4 ** refine' ENNReal.rpow_lt_top_of_nonneg (by simp) (ENNReal.sum_lt_top_iff.mpr fun y _ => _).ne ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 y : E x\u271d : y \u2208 SimpleFunc.range f \u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** by_cases hy0 : y = 0 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : p = 0 \u22a2 Mem\u2112p (\u2191f) p ** rw [hp0, mem\u2112p_zero_iff_aestronglyMeasurable] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : p = 0 \u22a2 AEStronglyMeasurable (\u2191f) \u03bc ** exact f.aestronglyMeasurable ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 Mem\u2112p (\u2191f) p ** rw [hp_top] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 Mem\u2112p \u2191f \u22a4 ** exact mem\u2112p_top f \u03bc ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 0 \u2264 1 / ENNReal.toReal p ** simp ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 y : E x\u271d : y \u2208 SimpleFunc.range f hy0 : y = 0 \u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** simp [hy0, ENNReal.toReal_pos hp0 hp_top] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 y : E x\u271d : y \u2208 SimpleFunc.range f hy0 : \u00acy = 0 \u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ** refine' ENNReal.mul_lt_top _ (hf y hy0).ne ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p\u271d p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E hf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 y : E x\u271d : y \u2208 SimpleFunc.range f hy0 : \u00acy = 0 \u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p \u2260 \u22a4 ** exact (ENNReal.rpow_lt_top_of_nonneg ENNReal.toReal_nonneg ENNReal.coe_ne_top).ne ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero ** \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f : \u03b1 \u2192 E hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0 t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 f =\u1d50[Measure.restrict \u03bc t] 0 ** rcases (hf_int_finite t ht h\u03bct.lt_top).aestronglyMeasurable.isSeparable_ae_range with\n  \u27e8u, u_sep, hu\u27e9 ** case intro.intro \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f : \u03b1 \u2192 E hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0 t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 u : Set E u_sep : IsSeparable u hu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u \u22a2 f =\u1d50[Measure.restrict \u03bc t] 0 ** refine' ae_eq_zero_of_forall_dual_of_isSeparable \u211d u_sep (fun c => _) hu ** case intro.intro \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f : \u03b1 \u2192 E hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0 t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 u : Set E u_sep : IsSeparable u hu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u c : Dual \u211d E \u22a2 (fun x => \u2191c (f x)) =\u1d50[Measure.restrict \u03bc t] 0 ** refine' ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real _ _ ht h\u03bct ** case intro.intro.refine'_1 \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f : \u03b1 \u2192 E hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0 t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 u : Set E u_sep : IsSeparable u hu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u c : Dual \u211d E \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (fun x => \u2191c (f x)) s ** intro s hs h\u03bcs ** case intro.intro.refine'_1 \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t\u271d : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f : \u03b1 \u2192 E hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0 t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 u : Set E u_sep : IsSeparable u hu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u c : Dual \u211d E s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 IntegrableOn (fun x => \u2191c (f x)) s ** exact ContinuousLinearMap.integrable_comp c (hf_int_finite s hs h\u03bcs) ** case intro.intro.refine'_2 \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t\u271d : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f : \u03b1 \u2192 E hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0 t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 u : Set E u_sep : IsSeparable u hu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u c : Dual \u211d E \u22a2 \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, \u2191c (f x) \u2202\u03bc = 0 ** intro s hs h\u03bcs ** case intro.intro.refine'_2 \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t\u271d : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f : \u03b1 \u2192 E hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0 t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 u : Set E u_sep : IsSeparable u hu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u c : Dual \u211d E s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2191c (f x) \u2202\u03bc = 0 ** rw [ContinuousLinearMap.integral_comp_comm c (hf_int_finite s hs h\u03bcs), hf_zero s hs h\u03bcs] ** case intro.intro.refine'_2 \u03b1 : Type u_1 E : Type u_2 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s\u271d t\u271d : Set \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E p : \u211d\u22650\u221e f : \u03b1 \u2192 E hf_int_finite : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn f s hf_zero : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u222b (x : \u03b1) in s, f x \u2202\u03bc = 0 t : Set \u03b1 ht : MeasurableSet t h\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4 u : Set E u_sep : IsSeparable u hu : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc t, f x \u2208 u c : Dual \u211d E s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2191c 0 = 0 ** exact ContinuousLinearMap.map_zero _ ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.max_le ** a b c : Int x\u271d : a \u2264 c \u2227 b \u2264 c h\u2081 : a \u2264 c h\u2082 : b \u2264 c \u22a2 max a b \u2264 c ** rw [Int.max_def] ** a b c : Int x\u271d : a \u2264 c \u2227 b \u2264 c h\u2081 : a \u2264 c h\u2082 : b \u2264 c \u22a2 (if a \u2264 b then b else a) \u2264 c ** split <;> assumption ** Qed",
        "informal": ""
    },
    {
        "formal": "Vector.removeNth_insertNth' ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 \u22a2 removeNth (Fin.succAbove { val := j, isLt := hj } { val := i, isLt := hi }) (insertNth a { val := j, isLt := hj } v) =     insertNth a (Fin.predAbove { val := i, isLt := hi } { val := j, isLt := hj }) (removeNth { val := i, isLt := hi } v) ** dsimp [insertNth, removeNth, Fin.succAbove, Fin.predAbove] ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 \u22a2 {       val :=         List.removeNth (List.insertNth j a \u2191v)           \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }             else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }),       property :=         (_ :           List.length               (List.removeNth (List.insertNth j a \u2191v)                 \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }                   else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) })) =             n + 1 + 1 - 1) } =     {       val :=         List.insertNth           (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then               Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)             else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))           a           \u2191(match v with             | { val := l, property := p } =>               { val := List.removeNth l i,                 property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }),       property :=         (_ :           List.length               (List.insertNth                 (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then                     Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)                   else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))                 a                 \u2191(match v with                   | { val := l, property := p } =>                     { val := List.removeNth l i,                       property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) })) =             n + 1) } ** rw [Subtype.mk_eq_mk] ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 \u22a2 List.removeNth (List.insertNth j a \u2191v)       \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }         else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }) =     List.insertNth       (\u2191(if h : { val := i, isLt := (_ : i < Nat.succ (n + 1)) } < { val := j, isLt := hj } then           Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)         else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))       a       \u2191(match v with         | { val := l, property := p } =>           { val := List.removeNth l i,             property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }) ** simp only [Fin.lt_iff_val_lt_val] ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 \u22a2 List.removeNth (List.insertNth j a \u2191v)       \u2191(if i < j then { val := i, isLt := (_ : i < Nat.succ (n + 1)) }         else { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) }) =     List.insertNth       (\u2191(if h : i < j then Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0)         else { val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }))       a       \u2191(match v with         | { val := l, property := p } =>           { val := List.removeNth l i,             property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }) ** split_ifs with hij ** case pos n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 hij : i < j \u22a2 List.removeNth (List.insertNth j a \u2191v) \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } =     List.insertNth (\u2191(Fin.pred { val := j, isLt := hj } (_ : { val := j, isLt := hj } \u2260 0))) a       \u2191(match v with         | { val := l, property := p } =>           { val := List.removeNth l i,             property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }) ** rcases Nat.exists_eq_succ_of_ne_zero\n  (Nat.pos_iff_ne_zero.1 (lt_of_le_of_lt (Nat.zero_le _) hij)) with \u27e8j, rfl\u27e9 ** case pos.intro n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : Nat.succ j < n + 2 hij : i < Nat.succ j \u22a2 List.removeNth (List.insertNth (Nat.succ j) a \u2191v) \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } =     List.insertNth (\u2191(Fin.pred { val := Nat.succ j, isLt := hj } (_ : { val := Nat.succ j, isLt := hj } \u2260 0))) a       \u2191(match v with         | { val := l, property := p } =>           { val := List.removeNth l i,             property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }) ** rw [\u2190 List.insertNth_removeNth_of_ge] ** case pos.intro n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : Nat.succ j < n + 2 hij : i < Nat.succ j \u22a2 List.insertNth j a (List.removeNth \u2191v \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) }) =     List.insertNth (\u2191(Fin.pred { val := Nat.succ j, isLt := hj } (_ : { val := Nat.succ j, isLt := hj } \u2260 0))) a       \u2191(match v with         | { val := l, property := p } =>           { val := List.removeNth l i,             property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }) ** simp ** case pos.intro n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : Nat.succ j < n + 2 hij : i < Nat.succ j \u22a2 List.insertNth j a (List.removeNth (\u2191v) i) =     List.insertNth j a       \u2191(match v with         | { val := l, property := p } =>           { val := List.removeNth l i,             property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }) ** rfl ** case pos.intro.a n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : Nat.succ j < n + 2 hij : i < Nat.succ j \u22a2 \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } < List.length \u2191v ** simpa ** case pos.intro.a n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : Nat.succ j < n + 2 hij : i < Nat.succ j \u22a2 \u2191{ val := i, isLt := (_ : i < Nat.succ (n + 1)) } \u2264 j ** simpa [Nat.lt_succ_iff] using hij ** case neg n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 hij : \u00aci < j \u22a2 List.removeNth (List.insertNth j a \u2191v) \u2191{ val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) } =     List.insertNth (\u2191{ val := j, isLt := (_ : \u2191{ val := j, isLt := hj } < n + 1) }) a       \u2191(match v with         | { val := l, property := p } =>           { val := List.removeNth l i,             property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }) ** dsimp ** case neg n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 hij : \u00aci < j \u22a2 List.removeNth (List.insertNth j a \u2191v) (i + 1) =     List.insertNth j a       \u2191(match v with         | { val := l, property := p } =>           { val := List.removeNth l i,             property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }) ** rw [\u2190 List.insertNth_removeNth_of_le i j _ _ _] ** case neg n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 hij : \u00aci < j \u22a2 List.insertNth j a (List.removeNth (\u2191v) i) =     List.insertNth j a       \u2191(match v with         | { val := l, property := p } =>           { val := List.removeNth l i,             property := (_ : List.length (List.removeNth l \u2191{ val := i, isLt := hi }) = n + 1 - 1) }) ** rfl ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 hij : \u00aci < j \u22a2 i < List.length \u2191v ** simpa ** n : \u2115 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a : \u03b1 v : Vector \u03b1 (n + 1) i : \u2115 hi : i < n + 1 j : \u2115 hj : j < n + 2 hij : \u00aci < j \u22a2 j \u2264 i ** simpa [not_lt] using hij ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.neg_modEq_neg ** m n a b c d : \u2124 \u22a2 -a \u2261 -b [ZMOD n] \u2194 a \u2261 b [ZMOD n] ** simp [-sub_neg_eq_add, neg_sub_neg, modEq_iff_dvd, dvd_sub_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.integral_eq_norm_posPart_sub ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace E f : { x // x \u2208 Lp \u211d 1 } \u22a2 integral f = \u2016Lp.posPart f\u2016 - \u2016Lp.negPart f\u2016 ** refine' @isClosed_property _ _ _ ((\u2191) : (\u03b1 \u2192\u2081\u209b[\u03bc] \u211d) \u2192 \u03b1 \u2192\u2081[\u03bc] \u211d)\n    (fun f : \u03b1 \u2192\u2081[\u03bc] \u211d => integral f = \u2016Lp.posPart f\u2016 - \u2016Lp.negPart f\u2016)\n    (simpleFunc.denseRange one_ne_top) (isClosed_eq _ _) _ f ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace E f : { x // x \u2208 Lp \u211d 1 } \u22a2 Continuous fun x => integral x ** simp only [integral] ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace E f : { x // x \u2208 Lp \u211d 1 } \u22a2 Continuous fun x => \u2191integralCLM x ** exact cont _ ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace E f : { x // x \u2208 Lp \u211d 1 } \u22a2 Continuous fun x => \u2016Lp.posPart x\u2016 - \u2016Lp.negPart x\u2016 ** refine' Continuous.sub (continuous_norm.comp Lp.continuous_posPart)\n  (continuous_norm.comp Lp.continuous_negPart) ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace E f : { x // x \u2208 Lp \u211d 1 } \u22a2 \u2200 (a : { x // x \u2208 \u2191(simpleFunc \u211d 1 \u03bc) }), (fun f => integral f = \u2016Lp.posPart f\u2016 - \u2016Lp.negPart f\u2016) \u2191a ** intro s ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace E f : { x // x \u2208 Lp \u211d 1 } s : { x // x \u2208 \u2191(simpleFunc \u211d 1 \u03bc) } \u22a2 integral \u2191s = \u2016Lp.posPart \u2191s\u2016 - \u2016Lp.negPart \u2191s\u2016 ** norm_cast ** case refine'_3 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace E f : { x // x \u2208 Lp \u211d 1 } s : { x // x \u2208 \u2191(simpleFunc \u211d 1 \u03bc) } \u22a2 SimpleFunc.integral s = \u2016SimpleFunc.posPart s\u2016 - \u2016SimpleFunc.negPart s\u2016 ** exact SimpleFunc.integral_eq_norm_posPart_sub _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (i : \u03b9), MeasurableSet (s i) H : \u2191\u2191\u03bc univ < \u2211' (i : \u03b9), \u2191\u2191\u03bc (s i) \u22a2 \u2203 i j _h, Set.Nonempty (s i \u2229 s j) ** contrapose! H ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (i : \u03b9), MeasurableSet (s i) H : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j) \u22a2 \u2211' (i : \u03b9), \u2191\u2191\u03bc (s i) \u2264 \u2191\u2191\u03bc univ ** apply tsum_measure_le_measure_univ hs ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (i : \u03b9), MeasurableSet (s i) H : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j) \u22a2 Pairwise (Disjoint on fun i => s i) ** intro i j hij ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (i : \u03b9), MeasurableSet (s i) H : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j) i j : \u03b9 hij : i \u2260 j \u22a2 (Disjoint on fun i => s i) i j ** rw [Function.onFun, disjoint_iff_inf_le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m\u271d : MeasurableSpace \u03b1 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : \u03b9 \u2192 Set \u03b1 hs : \u2200 (i : \u03b9), MeasurableSet (s i) H : \u2200 (i j : \u03b9), i \u2260 j \u2192 \u00acSet.Nonempty (s i \u2229 s j) i j : \u03b9 hij : i \u2260 j \u22a2 s i \u2293 s j \u2264 \u22a5 ** exact fun x hx => H i j hij \u27e8x, hx\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_norm_eq_pos_sub_neg ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} \u22a2 \u222b (x : \u03b1), \u2016f x\u2016 \u2202\u03bc = \u222b (x : \u03b1) in {x | 0 \u2264 f x}, \u2016f x\u2016 \u2202\u03bc + \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc ** rw [\u2190 integral_add_compl\u2080 h_meas hfi.norm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, \u2016f x\u2016 \u2202\u03bc + \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc =     \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc + \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc ** congr 1 ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, \u2016f x\u2016 \u2202\u03bc = \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc ** refine' set_integral_congr\u2080 h_meas fun x hx => _ ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} x : \u03b1 hx : x \u2208 {x | 0 \u2264 f x} \u22a2 \u2016f x\u2016 = f x ** rw [Real.norm_eq_abs, abs_eq_self.mpr _] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} x : \u03b1 hx : x \u2208 {x | 0 \u2264 f x} \u22a2 0 \u2264 f x ** exact hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc + \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc =     \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc ** congr 1 ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc = -\u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc ** rw [\u2190 integral_neg] ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, \u2016f x\u2016 \u2202\u03bc = \u222b (a : \u03b1) in {x | 0 \u2264 f x}\u1d9c, -f a \u2202\u03bc ** refine' set_integral_congr\u2080 h_meas.compl fun x hx => _ ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} x : \u03b1 hx : x \u2208 {x | 0 \u2264 f x}\u1d9c \u22a2 \u2016f x\u2016 = -f x ** rw [Real.norm_eq_abs, abs_eq_neg_self.mpr _] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} x : \u03b1 hx : x \u2208 {x | 0 \u2264 f x}\u1d9c \u22a2 f x \u2264 0 ** rw [Set.mem_compl_iff, Set.nmem_setOf_iff] at hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} x : \u03b1 hx : \u00ac0 \u2264 f x \u22a2 f x \u2264 0 ** linarith ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc =     \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | f x \u2264 0}, f x \u2202\u03bc ** rw [\u2190 set_integral_neg_eq_set_integral_nonpos hfi.1] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} \u22a2 \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | 0 \u2264 f x}\u1d9c, f x \u2202\u03bc =     \u222b (x : \u03b1) in {x | 0 \u2264 f x}, f x \u2202\u03bc - \u222b (x : \u03b1) in {x | f x < 0}, f x \u2202\u03bc ** congr ** case e_a.e_\u03bc.e_s \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} \u22a2 {x | 0 \u2264 f x}\u1d9c = {x | f x < 0} ** ext1 x ** case e_a.e_\u03bc.e_s.h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f\u271d g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E f : \u03b1 \u2192 \u211d hfi : Integrable f h_meas : NullMeasurableSet {x | 0 \u2264 f x} x : \u03b1 \u22a2 x \u2208 {x | 0 \u2264 f x}\u1d9c \u2194 x \u2208 {x | f x < 0} ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "DomMulAct.dist_smul_Lp ** M : Type u_1 N : Type u_2 \u03b1 : Type u_3 E : Type u_4 inst\u271d\u2076 : MeasurableSpace M inst\u271d\u2075 : MeasurableSpace N inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : SMul M \u03b1 inst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc inst\u271d : MeasurableSMul M \u03b1 c : M\u1d48\u1d50\u1d43 f g : { x // x \u2208 Lp E p } \u22a2 dist (c \u2022 f) (c \u2022 g) = dist f g ** simp only [dist, \u2190 smul_Lp_sub, norm_smul_Lp] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.Regular.smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : Regular \u03bc x : \u211d\u22650\u221e hx : x \u2260 \u22a4 \u22a2 Regular (x \u2022 \u03bc) ** haveI := OuterRegular.smul \u03bc hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : Regular \u03bc x : \u211d\u22650\u221e hx : x \u2260 \u22a4 this : OuterRegular (x \u2022 \u03bc) \u22a2 Regular (x \u2022 \u03bc) ** haveI := IsFiniteMeasureOnCompacts.smul \u03bc hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : Regular \u03bc x : \u211d\u22650\u221e hx : x \u2260 \u22a4 this\u271d : OuterRegular (x \u2022 \u03bc) this : IsFiniteMeasureOnCompacts (x \u2022 \u03bc) \u22a2 Regular (x \u2022 \u03bc) ** exact \u27e8Regular.innerRegular.smul x\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hn : upcrossingsBefore a b f N \u03c9 < n \u22a2 stoppedValue f (upperCrossingTime a b f N (n + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N n) \u03c9 = 0 ** have : N \u2264 upperCrossingTime a b f N n \u03c9 := by\n  rw [upcrossingsBefore] at hn\n  rw [\u2190 not_lt]\n  exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h) ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hn : upcrossingsBefore a b f N \u03c9 < n this : N \u2264 upperCrossingTime a b f N n \u03c9 \u22a2 stoppedValue f (upperCrossingTime a b f N (n + 1)) \u03c9 - stoppedValue f (lowerCrossingTime a b f N n) \u03c9 = 0 ** simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this,\n  lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hn : upcrossingsBefore a b f N \u03c9 < n \u22a2 N \u2264 upperCrossingTime a b f N n \u03c9 ** rw [upcrossingsBefore] at hn ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hn : sSup {n | upperCrossingTime a b f N n \u03c9 < N} < n \u22a2 N \u2264 upperCrossingTime a b f N n \u03c9 ** rw [\u2190 not_lt] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hab : a < b hn : sSup {n | upperCrossingTime a b f N n \u03c9 < N} < n \u22a2 \u00acupperCrossingTime a b f N n \u03c9 < N ** exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasurableSpace.map_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u : Set \u03b1 m : MeasurableSpace \u03b1 b : \u03b2 s : Set \u03b2 x\u271d : MeasurableSet s \u22a2 MeasurableSet s ** by_cases b \u2208 s <;> simp [*, map_def] <;> rw [Set.preimage_id'] <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.pairwiseDisjoint_smul_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b9 : LeftCancelSemigroup \u03b1 inst\u271d : DecidableEq \u03b1 s\u271d t\u271d : Finset \u03b1 a : \u03b1 s : Set \u03b1 t : Finset \u03b1 \u22a2 (Set.PairwiseDisjoint s fun x => x \u2022 t) \u2194 Set.InjOn (fun p => p.1 * p.2) (s \u00d7\u02e2 \u2191t) ** simp_rw [\u2190 pairwiseDisjoint_coe, coe_smul_finset, Set.pairwiseDisjoint_smul_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.toMeasure_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d : MeasurableSpace \u03b1 \u22a2 toMeasure 0 (_ : inst\u271d \u2264 OuterMeasure.caratheodory 0) = 0 ** rw [\u2190 Measure.measure_univ_eq_zero, toMeasure_apply _ _ MeasurableSet.univ,\n  OuterMeasure.coe_zero, Pi.zero_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "Multiset.noncommProd_eq_pow_card ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : Monoid \u03b2 s : Multiset \u03b1 comm : Set.Pairwise {x | x \u2208 s} Commute m : \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2192 x = m \u22a2 noncommProd s comm = m ^ \u2191card s ** induction s using Quotient.inductionOn ** case h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : Monoid \u03b2 m : \u03b1 a\u271d : List \u03b1 comm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute h : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 x = m \u22a2 noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm = m ^ \u2191card (Quotient.mk (List.isSetoid \u03b1) a\u271d) ** simp only [quot_mk_to_coe, noncommProd_coe, coe_card, mem_coe] at * ** case h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : Monoid \u03b2 m : \u03b1 a\u271d : List \u03b1 comm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute h : \u2200 (x : \u03b1), x \u2208 a\u271d \u2192 x = m \u22a2 List.prod a\u271d = m ^ List.length a\u271d ** exact List.prod_eq_pow_card _ m h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.integrable_pair ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedAddCommGroup F \u03bc : Measure \u03b1 p : \u211d\u22650\u221e f : \u03b1 \u2192\u209b E g : \u03b1 \u2192\u209b F \u22a2 Integrable \u2191f \u2192 Integrable \u2191g \u2192 Integrable \u2191(pair f g) ** simpa only [integrable_iff_finMeasSupp] using FinMeasSupp.pair ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.norm_eq_sum_mul ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } \u22a2 \u2016f\u2016 = \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016 ** rw [norm_toSimpleFunc, snorm_one_eq_lintegral_nnnorm] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } \u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191\u2016\u2191(toSimpleFunc f) x\u2016\u208a \u2202\u03bc) =     \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016 ** have h_eq := SimpleFunc.map_apply (fun x => (\u2016x\u2016\u208a : \u211d\u22650\u221e)) (toSimpleFunc f) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a \u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191\u2016\u2191(toSimpleFunc f) x\u2016\u208a \u2202\u03bc) =     \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016 ** simp_rw [\u2190 h_eq] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a \u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) x \u2202\u03bc) =     \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016 ** rw [SimpleFunc.lintegral_eq_lintegral, SimpleFunc.map_lintegral, ENNReal.toReal_sum] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a \u22a2 \u2211 a in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {a})) =     \u2211 x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016 ** congr ** case e_f \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a \u22a2 (fun a => ENNReal.toReal (\u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {a}))) = fun x =>     ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016 ** ext1 x ** case e_f.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a x : G \u22a2 ENNReal.toReal (\u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) = ENNReal.toReal (\u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x})) * \u2016x\u2016 ** rw [ENNReal.toReal_mul, mul_comm, \u2190 ofReal_norm_eq_coe_nnnorm,\n  ENNReal.toReal_ofReal (norm_nonneg _)] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a \u22a2 \u2200 (a : G), a \u2208 SimpleFunc.range (toSimpleFunc f) \u2192 \u2191\u2016a\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {a}) \u2260 \u22a4 ** intro x _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a x : G a\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f) \u22a2 \u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x}) \u2260 \u22a4 ** by_cases hx0 : x = 0 ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a x : G a\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f) hx0 : x = 0 \u22a2 \u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x}) \u2260 \u22a4 ** rw [hx0] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a x : G a\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f) hx0 : x = 0 \u22a2 \u2191\u20160\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {0}) \u2260 \u22a4 ** simp ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : { x // x \u2208 simpleFunc G 1 \u03bc } h_eq : \u2200 (a : \u03b1), \u2191(SimpleFunc.map (fun x => \u2191\u2016x\u2016\u208a) (toSimpleFunc f)) a = \u2191\u2016\u2191(toSimpleFunc f) a\u2016\u208a x : G a\u271d : x \u2208 SimpleFunc.range (toSimpleFunc f) hx0 : \u00acx = 0 \u22a2 \u2191\u2016x\u2016\u208a * \u2191\u2191\u03bc (\u2191(toSimpleFunc f) \u207b\u00b9' {x}) \u2260 \u22a4 ** exact\n  ENNReal.mul_ne_top ENNReal.coe_ne_top\n    (SimpleFunc.measure_preimage_lt_top_of_integrable _ (SimpleFunc.integrable f) hx0).ne ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.quasiMeasurePreserving_mul_left ** G : Type u_1 inst\u271d\u2076 : MeasurableSpace G inst\u271d\u2075 : Group G inst\u271d\u2074 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b3 : SigmaFinite \u03bd inst\u271d\u00b2 : SigmaFinite \u03bc s : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulRightInvariant \u03bc g : G \u22a2 QuasiMeasurePreserving fun h => g * h ** have :=\n  (quasiMeasurePreserving_mul_right \u03bc.inv g\u207b\u00b9).mono (inv_absolutelyContinuous \u03bc.inv)\n    (absolutelyContinuous_inv \u03bc.inv) ** G : Type u_1 inst\u271d\u2076 : MeasurableSpace G inst\u271d\u2075 : Group G inst\u271d\u2074 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b3 : SigmaFinite \u03bd inst\u271d\u00b2 : SigmaFinite \u03bc s : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulRightInvariant \u03bc g : G this : QuasiMeasurePreserving fun h => h * g\u207b\u00b9 \u22a2 QuasiMeasurePreserving fun h => g * h ** have :=\n  (quasiMeasurePreserving_inv_of_right_invariant \u03bc).comp\n    (this.comp (quasiMeasurePreserving_inv_of_right_invariant \u03bc)) ** G : Type u_1 inst\u271d\u2076 : MeasurableSpace G inst\u271d\u2075 : Group G inst\u271d\u2074 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b3 : SigmaFinite \u03bd inst\u271d\u00b2 : SigmaFinite \u03bc s : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulRightInvariant \u03bc g : G this\u271d : QuasiMeasurePreserving fun h => h * g\u207b\u00b9 this : QuasiMeasurePreserving (Inv.inv \u2218 (fun h => h * g\u207b\u00b9) \u2218 Inv.inv) \u22a2 QuasiMeasurePreserving fun h => g * h ** simp_rw [Function.comp, mul_inv_rev, inv_inv] at this ** G : Type u_1 inst\u271d\u2076 : MeasurableSpace G inst\u271d\u2075 : Group G inst\u271d\u2074 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b3 : SigmaFinite \u03bd inst\u271d\u00b2 : SigmaFinite \u03bc s : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulRightInvariant \u03bc g : G this\u271d : QuasiMeasurePreserving fun h => h * g\u207b\u00b9 this : QuasiMeasurePreserving fun x => g * x \u22a2 QuasiMeasurePreserving fun h => g * h ** exact this ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.uniformIntegrable_finite ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u22a2 UniformIntegrable f p \u03bc ** cases nonempty_fintype \u03b9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), Mem\u2112p (f i) p val\u271d : Fintype \u03b9 \u22a2 UniformIntegrable f p \u03bc ** refine' \u27e8fun n => (hf n).1, unifIntegrable_finite \u03bc hp_one hp_top hf, _\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), Mem\u2112p (f i) p val\u271d : Fintype \u03b9 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C ** by_cases h\u03b9 : Nonempty \u03b9 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), Mem\u2112p (f i) p val\u271d : Fintype \u03b9 h\u03b9 : Nonempty \u03b9 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C ** choose _ hf using hf ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 val\u271d : Fintype \u03b9 h\u03b9 : Nonempty \u03b9 h\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C ** set C := (Finset.univ.image fun i : \u03b9 => snorm (f i) p \u03bc).max'\n  \u27e8snorm (f h\u03b9.some) p \u03bc, Finset.mem_image.2 \u27e8h\u03b9.some, Finset.mem_univ _, rfl\u27e9\u27e9 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 val\u271d : Fintype \u03b9 h\u03b9 : Nonempty \u03b9 h\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4 C : \u211d\u22650\u221e :=   Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)     (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ) \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C ** refine' \u27e8C.toNNReal, fun i => _\u27e9 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 val\u271d : Fintype \u03b9 h\u03b9 : Nonempty \u03b9 h\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4 C : \u211d\u22650\u221e :=   Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)     (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ) i : \u03b9 \u22a2 snorm (f i) p \u03bc \u2264 \u2191(ENNReal.toNNReal C) ** rw [ENNReal.coe_toNNReal] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 val\u271d : Fintype \u03b9 h\u03b9 : Nonempty \u03b9 h\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4 C : \u211d\u22650\u221e :=   Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)     (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ) i : \u03b9 \u22a2 snorm (f i) p \u03bc \u2264 C ** exact Finset.le_max' (\u03b1 := \u211d\u22650\u221e) _ _ (Finset.mem_image.2 \u27e8i, Finset.mem_univ _, rfl\u27e9) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 val\u271d : Fintype \u03b9 h\u03b9 : Nonempty \u03b9 h\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4 C : \u211d\u22650\u221e :=   Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)     (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ) i : \u03b9 \u22a2 C \u2260 \u22a4 ** refine' ne_of_lt ((Finset.max'_lt_iff _ _).2 fun y hy => _) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 val\u271d : Fintype \u03b9 h\u03b9 : Nonempty \u03b9 h\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4 C : \u211d\u22650\u221e :=   Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)     (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ) i : \u03b9 y : \u211d\u22650\u221e hy : y \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ \u22a2 y < \u22a4 ** rw [Finset.mem_image] at hy ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 val\u271d : Fintype \u03b9 h\u03b9 : Nonempty \u03b9 h\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4 C : \u211d\u22650\u221e :=   Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)     (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ) i : \u03b9 y : \u211d\u22650\u221e hy : \u2203 a, a \u2208 Finset.univ \u2227 snorm (f a) p \u03bc = y \u22a2 y < \u22a4 ** obtain \u27e8i, -, rfl\u27e9 := hy ** case pos.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 val\u271d : Fintype \u03b9 h\u03b9 : Nonempty \u03b9 h\u271d : \u2200 (i : \u03b9), AEStronglyMeasurable (f i) \u03bc hf : \u2200 (i : \u03b9), snorm (f i) p \u03bc < \u22a4 C : \u211d\u22650\u221e :=   Finset.max' (Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ)     (_ : \u2203 x, x \u2208 Finset.image (fun i => snorm (f i) p \u03bc) Finset.univ) i\u271d i : \u03b9 \u22a2 snorm (f i) p \u03bc < \u22a4 ** exact hf i ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 p : \u211d\u22650\u221e f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 inst\u271d : Finite \u03b9 hp_one : 1 \u2264 p hp_top : p \u2260 \u22a4 hf : \u2200 (i : \u03b9), Mem\u2112p (f i) p val\u271d : Fintype \u03b9 h\u03b9 : \u00acNonempty \u03b9 \u22a2 \u2203 C, \u2200 (i : \u03b9), snorm (f i) p \u03bc \u2264 \u2191C ** exact \u27e80, fun i => False.elim <| h\u03b9 <| Nonempty.intro i\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.eval_eq_eval_mv_eval' ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 s : Fin n \u2192 R y : R f : MvPolynomial (Fin (n + 1)) R \u22a2 \u2191(eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (eval s) (\u2191(finSuccEquiv R n) f)) ** let \u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=\n  { Polynomial.mapRingHom (eval s) with\n    commutes' := fun r => by\n      convert Polynomial.map_C (eval s)\n      exact (eval_C _).symm } ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 s : Fin n \u2192 R y : R f : MvPolynomial (Fin (n + 1)) R \u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=   let src := mapRingHom (eval s);   {     toRingHom :=       { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),         map_add' :=           (_ :             \u2200 (x y : (MvPolynomial (Fin n) R)[X]),               OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },     commutes' :=       (_ :         \u2200 (r : R),           OneHom.toFun               (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),                     map_add' :=                       (_ :                         \u2200 (x y : (MvPolynomial (Fin n) R)[X]),                           OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })               (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =             \u2191(algebraMap R R[X]) r) } \u22a2 \u2191(eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (eval s) (\u2191(finSuccEquiv R n) f)) ** show\n  aeval (Fin.cons y s : Fin (n + 1) \u2192 R) f =\n    (Polynomial.aeval y).comp (\u03c6.comp (finSuccEquiv R n).toAlgHom) f ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 s : Fin n \u2192 R y : R f : MvPolynomial (Fin (n + 1)) R \u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=   let src := mapRingHom (eval s);   {     toRingHom :=       { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),         map_add' :=           (_ :             \u2200 (x y : (MvPolynomial (Fin n) R)[X]),               OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },     commutes' :=       (_ :         \u2200 (r : R),           OneHom.toFun               (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),                     map_add' :=                       (_ :                         \u2200 (x y : (MvPolynomial (Fin n) R)[X]),                           OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })               (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =             \u2191(algebraMap R R[X]) r) } \u22a2 \u2191(aeval (Fin.cons y s)) f = \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) f ** congr 2 ** case e_a R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 s : Fin n \u2192 R y : R f : MvPolynomial (Fin (n + 1)) R \u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=   let src := mapRingHom (eval s);   {     toRingHom :=       { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),         map_add' :=           (_ :             \u2200 (x y : (MvPolynomial (Fin n) R)[X]),               OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },     commutes' :=       (_ :         \u2200 (r : R),           OneHom.toFun               (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),                     map_add' :=                       (_ :                         \u2200 (x y : (MvPolynomial (Fin n) R)[X]),                           OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })               (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =             \u2191(algebraMap R R[X]) r) } \u22a2 aeval (Fin.cons y s) = AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n)) ** apply MvPolynomial.algHom_ext ** case e_a.hf R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 s : Fin n \u2192 R y : R f : MvPolynomial (Fin (n + 1)) R \u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=   let src := mapRingHom (eval s);   {     toRingHom :=       { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),         map_add' :=           (_ :             \u2200 (x y : (MvPolynomial (Fin n) R)[X]),               OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },     commutes' :=       (_ :         \u2200 (r : R),           OneHom.toFun               (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),                     map_add' :=                       (_ :                         \u2200 (x y : (MvPolynomial (Fin n) R)[X]),                           OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })               (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =             \u2191(algebraMap R R[X]) r) } \u22a2 \u2200 (i : Fin (n + 1)),     \u2191(aeval (Fin.cons y s)) (X i) = \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) (X i) ** rw [Fin.forall_fin_succ] ** case e_a.hf R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 s : Fin n \u2192 R y : R f : MvPolynomial (Fin (n + 1)) R \u03c6 : (MvPolynomial (Fin n) R)[X] \u2192\u2090[R] R[X] :=   let src := mapRingHom (eval s);   {     toRingHom :=       { toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),         map_add' :=           (_ :             \u2200 (x y : (MvPolynomial (Fin n) R)[X]),               OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) },     commutes' :=       (_ :         \u2200 (r : R),           OneHom.toFun               (\u2191\u2191{ toMonoidHom := \u2191src, map_zero' := (_ : OneHom.toFun (\u2191\u2191src) 0 = 0),                     map_add' :=                       (_ :                         \u2200 (x y : (MvPolynomial (Fin n) R)[X]),                           OneHom.toFun (\u2191\u2191src) (x + y) = OneHom.toFun (\u2191\u2191src) x + OneHom.toFun (\u2191\u2191src) y) })               (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =             \u2191(algebraMap R R[X]) r) } \u22a2 \u2191(aeval (Fin.cons y s)) (X 0) = \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) (X 0) \u2227     \u2200 (i : Fin n),       \u2191(aeval (Fin.cons y s)) (X (Fin.succ i)) =         \u2191(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp \u03c6 \u2191(finSuccEquiv R n))) (X (Fin.succ i)) ** simp only [aeval_X, Fin.cons_zero, AlgEquiv.toAlgHom_eq_coe, AlgHom.coe_comp,\n  Polynomial.coe_aeval_eq_eval, Polynomial.map_C, AlgHom.coe_mk, RingHom.toFun_eq_coe,\n  Polynomial.coe_mapRingHom, comp_apply, finSuccEquiv_apply, eval\u2082Hom_X',\n  Fin.cases_zero, Polynomial.map_X, Polynomial.eval_X, Fin.cons_succ,\n  Fin.cases_succ, eval_X, Polynomial.eval_C,\n  RingHom.coe_mk, MonoidHom.coe_coe, AlgHom.coe_coe, implies_true, and_self,\n  RingHom.toMonoidHom_eq_coe] ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 s : Fin n \u2192 R y : R f : MvPolynomial (Fin (n + 1)) R src\u271d : (MvPolynomial (Fin n) R)[X] \u2192+* R[X] := mapRingHom (eval s) r : R \u22a2 OneHom.toFun       (\u2191\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),             map_add' :=               (_ :                 \u2200 (x y : (MvPolynomial (Fin n) R)[X]),                   OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) })       (\u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =     \u2191(algebraMap R R[X]) r ** convert Polynomial.map_C (eval s) ** case h.e'_3.h.e'_6 R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s\u271d : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 s : Fin n \u2192 R y : R f : MvPolynomial (Fin (n + 1)) R src\u271d : (MvPolynomial (Fin n) R)[X] \u2192+* R[X] := mapRingHom (eval s) r : R \u22a2 r = \u2191(eval s) (\u2191(algebraMap R (MvPolynomial (Fin n) R)) r) ** exact (eval_C _).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_measure_preimage_mul_right_lt_top ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200\u1d50 (x : G) \u2202\u03bc, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4 ** refine' ae_of_forall_measure_lt_top_ae_restrict' \u03bd.inv _ _ ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200 (s_1 : Set G),     MeasurableSet s_1 \u2192       \u2191\u2191\u03bc s_1 < \u22a4 \u2192 \u2191\u2191(Measure.inv \u03bd) s_1 < \u22a4 \u2192 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc s_1, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4 ** intro A hA _ h3A ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 A : Set G hA : MeasurableSet A a\u271d : \u2191\u2191\u03bc A < \u22a4 h3A : \u2191\u2191(Measure.inv \u03bd) A < \u22a4 \u22a2 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4 ** simp only [\u03bd.inv_apply] at h3A ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 A : Set G hA : MeasurableSet A a\u271d : \u2191\u2191\u03bc A < \u22a4 h3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4 \u22a2 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4 ** apply ae_lt_top (measurable_measure_mul_right \u03bd sm) ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 A : Set G hA : MeasurableSet A a\u271d : \u2191\u2191\u03bc A < \u22a4 h3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4 \u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4 ** have h1 := measure_mul_lintegral_eq \u03bc \u03bd sm (A\u207b\u00b9.indicator 1) (measurable_one.indicator hA.inv) ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 A : Set G hA : MeasurableSet A a\u271d : \u2191\u2191\u03bc A < \u22a4 h3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4 h1 : \u2191\u2191\u03bc s * \u222b\u207b (y : G), indicator A\u207b\u00b9 1 y \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * indicator A\u207b\u00b9 1 x\u207b\u00b9 \u2202\u03bc \u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4 ** rw [lintegral_indicator _ hA.inv] at h1 ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 A : Set G hA : MeasurableSet A a\u271d : \u2191\u2191\u03bc A < \u22a4 h3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4 h1 : \u2191\u2191\u03bc s * \u222b\u207b (a : G) in A\u207b\u00b9, OfNat.ofNat 1 a \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * indicator A\u207b\u00b9 1 x\u207b\u00b9 \u2202\u03bc \u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4 ** simp_rw [Pi.one_apply, set_lintegral_one, \u2190 image_inv, indicator_image inv_injective, image_inv, \u2190\n  indicator_mul_right _ fun x => \u03bd ((fun y => y * x) \u207b\u00b9' s), Function.comp, Pi.one_apply,\n  mul_one] at h1 ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 A : Set G hA : MeasurableSet A a\u271d : \u2191\u2191\u03bc A < \u22a4 h3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4 h1 : \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 = \u222b\u207b (x : G), indicator A (fun a => \u2191\u2191\u03bd ((fun y => y * a) \u207b\u00b9' s)) x \u2202\u03bc \u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4 ** rw [\u2190 lintegral_indicator _ hA, \u2190 h1] ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 A : Set G hA : MeasurableSet A a\u271d : \u2191\u2191\u03bc A < \u22a4 h3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4 h1 : \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 = \u222b\u207b (x : G), indicator A (fun a => \u2191\u2191\u03bd ((fun y => y * a) \u207b\u00b9' s)) x \u2202\u03bc \u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 \u2260 \u22a4 ** exact ENNReal.mul_ne_top h\u03bcs h3A.ne ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_preimage_linearEquiv ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2243\u2097[\u211d] E s : Set E \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' s) = ENNReal.ofReal |\u2191LinearMap.det \u2191(LinearEquiv.symm f)| * \u2191\u2191\u03bc s ** have A : LinearMap.det (f : E \u2192\u2097[\u211d] E) \u2260 0 := (LinearEquiv.isUnit_det' f).ne_zero ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2243\u2097[\u211d] E s : Set E A : \u2191LinearMap.det \u2191f \u2260 0 \u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' s) = ENNReal.ofReal |\u2191LinearMap.det \u2191(LinearEquiv.symm f)| * \u2191\u2191\u03bc s ** convert addHaar_preimage_linearMap \u03bc A s ** case h.e'_3.h.e'_5.h.e'_1.h.e'_3 E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F f : E \u2243\u2097[\u211d] E s : Set E A : \u2191LinearMap.det \u2191f \u2260 0 \u22a2 \u2191LinearMap.det \u2191(LinearEquiv.symm f) = (\u2191LinearMap.det \u2191f)\u207b\u00b9 ** simp only [LinearEquiv.det_coe_symm] ** Qed",
        "informal": ""
    },
    {
        "formal": "StieltjesFunction.measure_Ioo ** f : StieltjesFunction a b : \u211d \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a) ** rcases le_or_lt b a with (hab | hab) ** case inl f : StieltjesFunction a b : \u211d hab : b \u2264 a \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a) ** simp only [hab, measure_empty, Ioo_eq_empty, not_lt] ** case inl f : StieltjesFunction a b : \u211d hab : b \u2264 a \u22a2 0 = ofReal (leftLim (\u2191f) b - \u2191f a) ** symm ** case inl f : StieltjesFunction a b : \u211d hab : b \u2264 a \u22a2 ofReal (leftLim (\u2191f) b - \u2191f a) = 0 ** simp [ENNReal.ofReal_eq_zero, f.mono.leftLim_le hab] ** case inr f : StieltjesFunction a b : \u211d hab : a < b \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a) ** have A : Disjoint (Ioo a b) {b} := by simp ** case inr f : StieltjesFunction a b : \u211d hab : a < b A : Disjoint (Ioo a b) {b} \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a) ** have D : f b - f a = f b - leftLim f b + (leftLim f b - f a) := by abel ** case inr f : StieltjesFunction a b : \u211d hab : a < b A : Disjoint (Ioo a b) {b} D : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a) \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a) ** have := f.measure_Ioc a b ** case inr f : StieltjesFunction a b : \u211d hab : a < b A : Disjoint (Ioo a b) {b} D : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a) this : \u2191\u2191(StieltjesFunction.measure f) (Ioc a b) = ofReal (\u2191f b - \u2191f a) \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a) ** simp only [\u2190 Ioo_union_Icc_eq_Ioc hab le_rfl, measure_singleton,\n  measure_union A (measurableSet_singleton b), Icc_self] at this ** case inr f : StieltjesFunction a b : \u211d hab : a < b A : Disjoint (Ioo a b) {b} D : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a) this : \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) + ofReal (\u2191f b - leftLim (\u2191f) b) = ofReal (\u2191f b - \u2191f a) \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a) ** rw [D, ENNReal.ofReal_add, add_comm] at this ** f : StieltjesFunction a b : \u211d hab : a < b \u22a2 Disjoint (Ioo a b) {b} ** simp ** f : StieltjesFunction a b : \u211d hab : a < b A : Disjoint (Ioo a b) {b} \u22a2 \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a) ** abel ** case inr f : StieltjesFunction a b : \u211d hab : a < b A : Disjoint (Ioo a b) {b} D : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a) this :   ofReal (\u2191f b - leftLim (\u2191f) b) + \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) =     ofReal (\u2191f b - leftLim (\u2191f) b) + ofReal (leftLim (\u2191f) b - \u2191f a) \u22a2 \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) = ofReal (leftLim (\u2191f) b - \u2191f a) ** simpa only [ENNReal.add_right_inj ENNReal.ofReal_ne_top] ** case inr.hp f : StieltjesFunction a b : \u211d hab : a < b A : Disjoint (Ioo a b) {b} D : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a) this :   \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) + ofReal (\u2191f b - leftLim (\u2191f) b) =     ofReal (\u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)) \u22a2 0 \u2264 \u2191f b - leftLim (\u2191f) b ** simp only [f.mono.leftLim_le le_rfl, sub_nonneg] ** case inr.hq f : StieltjesFunction a b : \u211d hab : a < b A : Disjoint (Ioo a b) {b} D : \u2191f b - \u2191f a = \u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a) this :   \u2191\u2191(StieltjesFunction.measure f) (Ioo a b) + ofReal (\u2191f b - leftLim (\u2191f) b) =     ofReal (\u2191f b - leftLim (\u2191f) b + (leftLim (\u2191f) b - \u2191f a)) \u22a2 0 \u2264 leftLim (\u2191f) b - \u2191f a ** simp only [f.mono.le_leftLim hab, sub_nonneg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_measure_preimage_mul_right_lt_top_of_ne_zero ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 \u22a2 \u2200\u1d50 (x : G) \u2202\u03bc, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4 ** refine' (ae_measure_preimage_mul_right_lt_top \u03bd \u03bd sm h3s).filter_mono _ ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 \u22a2 ae \u03bc \u2264 ae \u03bd ** refine' (absolutelyContinuous_of_isMulLeftInvariant \u03bc \u03bd _).ae_le ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 \u22a2 \u03bd \u2260 0 ** refine' mt _ h2s ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 \u22a2 \u03bd = 0 \u2192 \u2191\u2191\u03bd s = 0 ** intro h\u03bd ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 h\u03bd : \u03bd = 0 \u22a2 \u2191\u2191\u03bd s = 0 ** rw [h\u03bd, Measure.coe_zero, Pi.zero_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "RegularExpression.add_rmatch_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P Q : RegularExpression \u03b1 x : List \u03b1 \u22a2 rmatch (P + Q) x = true \u2194 rmatch P x = true \u2228 rmatch Q x = true ** induction' x with _ _ ih generalizing P Q ** case nil \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P\u271d Q\u271d P Q : RegularExpression \u03b1 \u22a2 rmatch (P + Q) [] = true \u2194 rmatch P [] = true \u2228 rmatch Q [] = true ** simp only [rmatch, matchEpsilon, Bool.or_coe_iff] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P\u271d Q\u271d : RegularExpression \u03b1 head\u271d : \u03b1 tail\u271d : List \u03b1 ih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true P Q : RegularExpression \u03b1 \u22a2 rmatch (P + Q) (head\u271d :: tail\u271d) = true \u2194 rmatch P (head\u271d :: tail\u271d) = true \u2228 rmatch Q (head\u271d :: tail\u271d) = true ** repeat' rw [rmatch] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P\u271d Q\u271d : RegularExpression \u03b1 head\u271d : \u03b1 tail\u271d : List \u03b1 ih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true P Q : RegularExpression \u03b1 \u22a2 rmatch (deriv (P + Q) head\u271d) tail\u271d = true \u2194 rmatch (deriv P head\u271d) tail\u271d = true \u2228 rmatch (deriv Q head\u271d) tail\u271d = true ** rw [deriv_add] ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P\u271d Q\u271d : RegularExpression \u03b1 head\u271d : \u03b1 tail\u271d : List \u03b1 ih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true P Q : RegularExpression \u03b1 \u22a2 rmatch (deriv P head\u271d + deriv Q head\u271d) tail\u271d = true \u2194     rmatch (deriv P head\u271d) tail\u271d = true \u2228 rmatch (deriv Q head\u271d) tail\u271d = true ** exact ih _ _ ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 dec : DecidableEq \u03b1 a b : \u03b1 P\u271d Q\u271d : RegularExpression \u03b1 head\u271d : \u03b1 tail\u271d : List \u03b1 ih : \u2200 (P Q : RegularExpression \u03b1), rmatch (P + Q) tail\u271d = true \u2194 rmatch P tail\u271d = true \u2228 rmatch Q tail\u271d = true P Q : RegularExpression \u03b1 \u22a2 rmatch (deriv (P + Q) head\u271d) tail\u271d = true \u2194 rmatch (deriv P head\u271d) tail\u271d = true \u2228 rmatch Q (head\u271d :: tail\u271d) = true ** rw [rmatch] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.degree_finSuccEquiv ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R h : f \u2260 0 \u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f) ** have h\u2080 : \u2200 {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f := fun f => rfl ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R h : f \u2260 0 h\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f \u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f) ** have h\u2081 : \u2200 {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 \u03b2), f \u2218 (fun x => x) = f := fun f => rfl ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R h : f \u2260 0 h\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f h\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f \u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f) ** have h\u2082 : WithBot.some = Nat.cast := rfl ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R h : f \u2260 0 h\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f h\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f h\u2082 : WithBot.some = Nat.cast \u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f) ** have h' : ((finSuccEquiv R n f).support.sup fun x => x) = degreeOf 0 f := by\n  rw [degreeOf_eq_sup, finSuccEquiv_support f, Finset.sup_image, h\u2080] ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R h : f \u2260 0 h\u2080 : \u2200 {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f h\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f h\u2082 : WithBot.some = Nat.cast h' : (Finset.sup (Polynomial.support (\u2191(finSuccEquiv R n) f)) fun x => x) = degreeOf 0 f \u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f) ** rw [Polynomial.degree, \u2190 h', \u2190 h\u2082, Finset.coe_sup_of_nonempty (support_finSuccEquiv_nonempty h),\n  Finset.max_eq_sup_coe, h\u2081] ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 f : MvPolynomial (Fin (n + 1)) R h : f \u2260 0 h\u2080 : \u2200 {\u03b1 : Type ?u.1271073} {\u03b2 : Type ?u.1271076} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f h\u2081 : \u2200 {\u03b1 : Type ?u.1271138} {\u03b2 : Type ?u.1271141} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f h\u2082 : WithBot.some = Nat.cast \u22a2 (Finset.sup (Polynomial.support (\u2191(finSuccEquiv R n) f)) fun x => x) = degreeOf 0 f ** rw [degreeOf_eq_sup, finSuccEquiv_support f, Finset.sup_image, h\u2080] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.univ_pi_update ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2\u271d : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i\u271d : \u03b9 inst\u271d : DecidableEq \u03b9 \u03b2 : \u03b9 \u2192 Type u_4 i : \u03b9 f : (j : \u03b9) \u2192 \u03b1 j a : \u03b1 i t : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j) \u22a2 (pi univ fun j => t j (update f i a j)) = {x | x i \u2208 t i a} \u2229 pi {i}\u1d9c fun j => t j (f j) ** rw [compl_eq_univ_diff, \u2190 pi_update_of_mem (mem_univ _)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.dropWhile ** l m r : List Char p : Char \u2192 Bool \u22a2 ValidFor (l ++ List.takeWhile p m) (List.dropWhile p m) r     (Substring.dropWhile       { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },         stopPos := { byteIdx := utf8Len l + utf8Len m } }       p) ** simp only [Substring.dropWhile, takeWhileAux_of_valid] ** l m r : List Char p : Char \u2192 Bool \u22a2 ValidFor (l ++ List.takeWhile p m) (List.dropWhile p m) r     { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l + utf8Len (List.takeWhile p m) },       stopPos := { byteIdx := utf8Len l + utf8Len m } } ** refine' .of_eq .. <;> simp ** case refine'_3 l m r : List Char p : Char \u2192 Bool \u22a2 utf8Len l + utf8Len m = utf8Len l + utf8Len (List.takeWhile p m) + utf8Len (List.dropWhile p m) ** rw [Nat.add_assoc, \u2190 utf8Len_append (m.takeWhile p), List.takeWhile_append_dropWhile] ** Qed",
        "informal": ""
    },
    {
        "formal": "isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompactSpace \u03b1 \u22a2 IsFiniteMeasure \u03bc \u2194 IsFiniteMeasureOnCompacts \u03bc ** constructor <;> intros ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompactSpace \u03b1 a\u271d : IsFiniteMeasure \u03bc \u22a2 IsFiniteMeasureOnCompacts \u03bc ** infer_instance ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompactSpace \u03b1 a\u271d : IsFiniteMeasureOnCompacts \u03bc \u22a2 IsFiniteMeasure \u03bc ** exact CompactSpace.isFiniteMeasure ** Qed",
        "informal": ""
    },
    {
        "formal": "PosNum.natSize_to_nat ** \u03b1 : Type u_1 n : PosNum \u22a2 natSize n = Nat.size \u2191n ** rw [\u2190 size_eq_natSize, size_to_nat] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.iIndepFun.cgf_sum ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s : Finset \u03b9 h_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) \u22a2 cgf (\u2211 i in s, X i) \u03bc t = \u2211 i in s, cgf (X i) \u03bc t ** simp_rw [cgf] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s : Finset \u03b9 h_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) \u22a2 log (mgf (\u2211 i in s, X i) \u03bc t) = \u2211 x in s, log (mgf (X x) \u03bc t) ** rw [\u2190 log_prod _ _ fun j hj => ?_] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s : Finset \u03b9 h_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) \u22a2 log (mgf (\u2211 i in s, X i) \u03bc t) = log (\u220f i in s, mgf (X i) \u03bc t) ** rw [h_indep.mgf_sum h_meas] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s : Finset \u03b9 h_int : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable fun \u03c9 => rexp (t * X i \u03c9) j : \u03b9 hj : j \u2208 s \u22a2 mgf (X j) \u03bc t \u2260 0 ** exact (mgf_pos (h_int j hj)).ne' ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.limsup_lintegral_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : \u2200 (n : \u2115), Measurable (f n) h_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g h_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4 \u22a2 \u2a05 n, \u222b\u207b (a : \u03b1), \u2a06 i, \u2a06 (_ : i \u2265 n), f i a \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2a05 n, \u2a06 i, \u2a06 (_ : i \u2265 n), f i a \u2202\u03bc ** refine' (lintegral_iInf _ _ _).symm ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : \u2200 (n : \u2115), Measurable (f n) h_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g h_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4 \u22a2 \u2200 (n : \u2115), Measurable fun a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a ** intro n ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : \u2200 (n : \u2115), Measurable (f n) h_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g h_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4 n : \u2115 \u22a2 Measurable fun a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a ** exact measurable_biSup _ (to_countable _) (fun i _ \u21a6 hf_meas i) ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : \u2200 (n : \u2115), Measurable (f n) h_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g h_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4 \u22a2 Antitone fun n a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a ** intro n m hnm a ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : \u2200 (n : \u2115), Measurable (f n) h_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g h_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4 n m : \u2115 hnm : n \u2264 m a : \u03b1 \u22a2 (fun n a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a) m a \u2264 (fun n a => \u2a06 i, \u2a06 (_ : i \u2265 n), f i a) n a ** exact iSup_le_iSup_of_subset fun i hi => le_trans hnm hi ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : \u2200 (n : \u2115), Measurable (f n) h_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g h_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4 \u22a2 \u222b\u207b (a : \u03b1), \u2a06 i, \u2a06 (_ : i \u2265 0), f i a \u2202\u03bc \u2260 \u22a4 ** refine' ne_top_of_le_ne_top h_fin (lintegral_mono_ae _) ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : \u2200 (n : \u2115), Measurable (f n) h_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g h_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2a06 i, \u2a06 (_ : i \u2265 0), f i a \u2264 g a ** refine' (ae_all_iff.2 h_bound).mono fun n hn => _ ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : \u2200 (n : \u2115), Measurable (f n) h_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g h_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4 n : \u03b1 hn : \u2200 (i : \u2115), f i n \u2264 g n \u22a2 \u2a06 i, \u2a06 (_ : i \u2265 0), f i n \u2264 g n ** exact iSup_le fun i => iSup_le fun _ => hn i ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : \u2200 (n : \u2115), Measurable (f n) h_bound : \u2200 (n : \u2115), f n \u2264\u1d50[\u03bc] g h_fin : \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2260 \u22a4 \u22a2 \u222b\u207b (a : \u03b1), \u2a05 n, \u2a06 i, \u2a06 (_ : i \u2265 n), f i a \u2202\u03bc = \u222b\u207b (a : \u03b1), limsup (fun n => f n a) atTop \u2202\u03bc ** simp only [limsup_eq_iInf_iSup_of_nat] ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.ne_neg_self ** n : \u2115 hn : Odd n a : ZMod n ha : a \u2260 0 \u22a2 a \u2260 -a ** rwa [Ne, eq_neg_iff_add_eq_zero, add_self_eq_zero_iff_eq_zero hn] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_withDensity_iff ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u03b1 \u2192 Prop f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 (\u2200\u1d50 (x : \u03b1) \u2202withDensity \u03bc f, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x \u2260 0 \u2192 p x ** rw [ae_iff, ae_iff, withDensity_apply_eq_zero hf, iff_iff_eq] ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u03b1 \u2192 Prop f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 (\u2191\u2191\u03bc ({x | f x \u2260 0} \u2229 {a | \u00acp a}) = 0) = (\u2191\u2191\u03bc {a | \u00ac(f a \u2260 0 \u2192 p a)} = 0) ** congr ** case e_a.e_a \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u03b1 \u2192 Prop f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 {x | f x \u2260 0} \u2229 {a | \u00acp a} = {a | \u00ac(f a \u2260 0 \u2192 p a)} ** ext x ** case e_a.e_a.h \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u03b1 \u2192 Prop f : \u03b1 \u2192 \u211d\u22650\u221e hf : Measurable f x : \u03b1 \u22a2 x \u2208 {x | f x \u2260 0} \u2229 {a | \u00acp a} \u2194 x \u2208 {a | \u00ac(f a \u2260 0 \u2192 p a)} ** simp only [exists_prop, mem_inter_iff, iff_self_iff, mem_setOf_eq, not_forall] ** Qed",
        "informal": ""
    },
    {
        "formal": "Real.borel_eq_generateFrom_Ici_rat ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 \u22a2 borel \u211d = MeasurableSpace.generateFrom (\u22c3 a, {Ici \u2191a}) ** rw [borel_eq_generateFrom_Iio_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s t u : Set \u03b1 \u22a2 MeasurableSpace.generateFrom (range fun a => Iio \u2191a) = MeasurableSpace.generateFrom (range fun a => Ici \u2191a) ** refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>\nrintro _ \u27e8q, rfl\u27e9 <;>\ndsimp only <;>\n[rw [\u2190 compl_Ici]; rw [\u2190 compl_Iio]] <;>\nexact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.ofDual_min' ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1\u1d52\u1d48 hs : Finset.Nonempty s \u22a2 \u2191ofDual (min' s hs) = max' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s)) ** rw [\u2190 WithBot.coe_eq_coe] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1\u1d52\u1d48 hs : Finset.Nonempty s \u22a2 \u2191(\u2191ofDual (min' s hs)) = \u2191(max' (image (\u2191ofDual) s) (_ : Finset.Nonempty (image (\u2191ofDual) s))) ** simp only [min'_eq_inf', id_eq, ofDual_inf', Function.comp_apply, coe_sup', max'_eq_sup',\n  sup_image] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 s\u271d : Finset \u03b1 H : Finset.Nonempty s\u271d x : \u03b1 s : Finset \u03b1\u1d52\u1d48 hs : Finset.Nonempty s \u22a2 sup s (WithBot.some \u2218 fun x => \u2191ofDual x) = sup s ((WithBot.some \u2218 fun x => x) \u2218 \u2191ofDual) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.integral_eq_integral_meas_lt ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f \u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) ** have key := lintegral_eq_lintegral_meas_lt \u03bc f_nn f_intble.aemeasurable ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} \u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) ** have lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u221e := Integrable.lintegral_lt_top f_intble ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 \u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) ** have rhs_finite : \u222b\u207b (t : \u211d) in Set.Ioi 0, \u03bc {a | t < f a} < \u221e := by simp only [\u2190 key, lhs_finite] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 \u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) ** have rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u03bc {a | t < f a} < \u221e :=\n  fun t ht \u21a6 measure_gt_lt_top f_intble ht ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 \u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) ** convert (ENNReal.toReal_eq_toReal lhs_finite.ne rhs_finite.ne).mpr key ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 \u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 ** simp only [\u2190 key, lhs_finite] ** case h.e'_2 \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 \u22a2 \u222b (\u03c9 : \u03b1), f \u03c9 \u2202\u03bc = ENNReal.toReal (\u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc) ** exact integral_eq_lintegral_of_nonneg_ae f_nn f_intble.aestronglyMeasurable ** case h.e'_3 \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 \u22a2 \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) = ENNReal.toReal (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}) ** have aux := @integral_eq_lintegral_of_nonneg_ae _ _ ((volume : Measure \u211d).restrict (Set.Ioi 0))\n  (fun t \u21a6 ENNReal.toReal (\u03bc {a : \u03b1 | t < f a})) ?_ ?_ ** case h.e'_3.refine_3 \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 aux :   \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =     ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}))) \u22a2 \u222b (t : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) = ENNReal.toReal (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}) ** rw [aux] ** case h.e'_3.refine_3 \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 aux :   \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =     ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}))) \u22a2 ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}))) =     ENNReal.toReal (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}) ** congr 1 ** case h.e'_3.refine_3.e_a \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 aux :   \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =     ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}))) \u22a2 \u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1})) = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} ** apply set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall _) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 aux :   \u222b (a : \u211d) in Ioi 0, ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}) =     ENNReal.toReal (\u222b\u207b (a : \u211d) in Ioi 0, ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a_1 | a < f a_1}))) \u22a2 \u2200 (x : \u211d), x \u2208 Ioi 0 \u2192 ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bc {a | x < f a})) = \u2191\u2191\u03bc {a | x < f a} ** exact fun t t_pos \u21a6 ENNReal.ofReal_toReal (rhs_integrand_finite t t_pos).ne ** case h.e'_3.refine_1 \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 \u22a2 0 \u2264\u1da0[ae (Measure.restrict volume (Ioi 0))] fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) ** exact eventually_of_forall (fun x \u21a6 by simp only [Pi.zero_apply, ENNReal.toReal_nonneg]) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 x : \u211d \u22a2 OfNat.ofNat 0 x \u2264 (fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})) x ** simp only [Pi.zero_apply, ENNReal.toReal_nonneg] ** case h.e'_3.refine_2 \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 \u22a2 AEStronglyMeasurable (fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a})) (Measure.restrict volume (Ioi 0)) ** apply Measurable.aestronglyMeasurable ** case h.e'_3.refine_2.hf \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 \u22a2 Measurable fun t => ENNReal.toReal (\u2191\u2191\u03bc {a | t < f a}) ** refine Measurable.ennreal_toReal ?_ ** case h.e'_3.refine_2.hf \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d f_intble : Integrable f f_nn : 0 \u2264\u1da0[ae \u03bc] f key : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} lhs_finite : \u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc < \u22a4 rhs_finite : \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a} < \u22a4 rhs_integrand_finite : \u2200 (t : \u211d), t > 0 \u2192 \u2191\u2191\u03bc {a | t < f a} < \u22a4 \u22a2 Measurable fun t => \u2191\u2191\u03bc {a | t < f a} ** exact Antitone.measurable (fun _ _ hst \u21a6 measure_mono (fun _ h \u21a6 lt_of_le_of_lt hst h)) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.add' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f g : \u03b1 \u2192 \u03b2 hf : Integrable f hg : Integrable g a : \u03b1 \u22a2 \u2191\u2016f a + g a\u2016\u208a \u2264 \u2191\u2016f a\u2016\u208a + \u2191\u2016g a\u2016\u208a ** exact_mod_cast nnnorm_add_le _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.variance_mul ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 variance (fun \u03c9 => c * X \u03c9) \u03bc = c ^ 2 * variance X \u03bc ** rw [variance, evariance_mul, ENNReal.toReal_mul, ENNReal.toReal_ofReal (sq_nonneg _)] ** \u03a9 : Type u_1 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d \u03bc\u271d : Measure \u03a9 c : \u211d X : \u03a9 \u2192 \u211d \u03bc : Measure \u03a9 \u22a2 c ^ 2 * ENNReal.toReal (evariance (fun \u03c9 => X \u03c9) \u03bc) = c ^ 2 * variance X \u03bc ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.addHaar_image_le_mul_of_det_lt ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m \u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0, \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s ** apply nhdsWithin_le_nhds ** case a E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0 ** let d := ENNReal.ofReal |A.det| ** case a E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0 ** obtain \u27e8\u03b5, h\u03b5, \u03b5pos\u27e9 :\n  \u2203 \u03b5 : \u211d, \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5 := by\n  have HC : IsCompact (A '' closedBall 0 1) :=\n    (ProperSpace.isCompact_closedBall _ _).image A.continuous\n  have L0 :\n    Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (A '' closedBall 0 1))) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n    apply Tendsto.mono_left _ nhdsWithin_le_nhds\n    exact tendsto_measure_cthickening_of_isCompact HC\n  have L1 :\n    Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n    apply L0.congr' _\n    filter_upwards [self_mem_nhdsWithin] with r hr\n    rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]\n  have L2 :\n    Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (d * \u03bc (closedBall 0 1))) := by\n    convert L1\n    exact (addHaar_image_continuousLinearMap _ _ _).symm\n  have I : d * \u03bc (closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n    (ENNReal.mul_lt_mul_right (measure_closedBall_pos \u03bc _ zero_lt_one).ne'\n          measure_closedBall_lt_top.ne).2\n      hm\n  have H :\n    \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n    (tendsto_order.1 L2).2 _ I\n  exact (H.and self_mem_nhdsWithin).exists ** case a.intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0 ** have : Iio (\u27e8\u03b5, \u03b5pos.le\u27e9 : \u211d\u22650) \u2208 \ud835\udcdd (0 : \u211d\u22650) := by apply Iio_mem_nhds; exact \u03b5pos ** case a.intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0 ** filter_upwards [this] ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u22a2 \u2200 (a : \u211d\u22650),     a \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2192       \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s ** intro \u03b4 h\u03b4 s f hf ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s ** have J : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d\u22650\u221e), \u03bc (f '' s) \u2264 m * (\u03bc s + a) := by\n  filter_upwards [self_mem_nhdsWithin] with a ha\n  change 0 < a at ha\n  obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n    \u2203 (t : Set E) (r : E \u2192 \u211d),\n      t.Countable \u2227\n        t \u2286 s \u2227\n          (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n            (s \u2286 \u22c3 x \u2208 t, closedBall x (r x)) \u2227\n              (\u2211' x : \u21a5t, \u03bc (closedBall (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n    Besicovitch.exists_closedBall_covering_tsum_measure_le \u03bc ha.ne' (fun _ => Ioi 0) s\n      fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [half_pos \u03b4pos, \u03b4pos]\u27e9\n  haveI : Encodable t := t_count.toEncodable\n  calc\n    \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 closedBall x (r x))) := by\n      rw [biUnion_eq_iUnion] at st\n      apply measure_mono\n      rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n      exact image_subset _ (subset_inter (Subset.refl _) st)\n    _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 closedBall x (r x))) := (measure_iUnion_le _)\n    _ \u2264 \u2211' x : t, m * \u03bc (closedBall x (r x)) :=\n      (ENNReal.tsum_le_tsum fun x => I x (r x) (ts x.2) (rpos x x.2).le)\n    _ \u2264 m * (\u03bc s + a) := by rw [ENNReal.tsum_mul_left]; exact mul_le_mul_left' \u03bct _ ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) J : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s ** have L : Tendsto (fun a => (m : \u211d\u22650\u221e) * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (m * (\u03bc s + 0))) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  apply ENNReal.Tendsto.const_mul (tendsto_const_nhds.add tendsto_id)\n  simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) J : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) L : Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0))) \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s ** rw [add_zero] at L ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) J : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) L : Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * \u2191\u2191\u03bc s)) \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s ** exact ge_of_tendsto L J ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5 ** have HC : IsCompact (A '' closedBall 0 1) :=\n  (ProperSpace.isCompact_closedBall _ _).image A.continuous ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) \u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5 ** have L0 :\n  Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (A '' closedBall 0 1))) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  exact tendsto_measure_cthickening_of_isCompact HC ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) \u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5 ** have L1 :\n  Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n  apply L0.congr' _\n  filter_upwards [self_mem_nhdsWithin] with r hr\n  rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) L1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) \u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5 ** have L2 :\n  Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (d * \u03bc (closedBall 0 1))) := by\n  convert L1\n  exact (addHaar_image_continuousLinearMap _ _ _).symm ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) L1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) L2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1))) \u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5 ** have I : d * \u03bc (closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n  (ENNReal.mul_lt_mul_right (measure_closedBall_pos \u03bc _ zero_lt_one).ne'\n        measure_closedBall_lt_top.ne).2\n    hm ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) L1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) L2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1))) I : d * \u2191\u2191\u03bc (closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5 ** have H :\n  \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n  (tendsto_order.1 L2).2 _ I ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) L1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) L2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1))) I : d * \u2191\u2191\u03bc (closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) H : \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (closedBall 0 b + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5 ** exact (H.and self_mem_nhdsWithin).exists ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) \u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) ** apply Tendsto.mono_left _ nhdsWithin_le_nhds ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) \u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) ** exact tendsto_measure_cthickening_of_isCompact HC ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) \u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) ** apply L0.congr' _ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) \u22a2 (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun \u03b5 =>     \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) ** filter_upwards [self_mem_nhdsWithin] with r hr ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) r : \u211d hr : r \u2208 Ioi 0 \u22a2 \u2191\u2191\u03bc (cthickening r (\u2191A '' closedBall 0 1)) = \u2191\u2191\u03bc (closedBall 0 r + \u2191A '' closedBall 0 1) ** rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) L1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) \u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1))) ** convert L1 ** case h.e'_5.h.e'_3 E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| HC : IsCompact (\u2191A '' closedBall 0 1) L0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) L1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1))) \u22a2 d * \u2191\u2191\u03bc (closedBall 0 1) = \u2191\u2191\u03bc (\u2191A '' closedBall 0 1) ** exact (addHaar_image_continuousLinearMap _ _ _).symm ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 \u22a2 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 ** apply Iio_mem_nhds ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 \u22a2 0 < { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } ** exact \u03b5pos ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 \u22a2 \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) ** intro x r xs r0 ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r K : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) ** have :\n  A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) =\n    {f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n  rw [smul_add, \u2190 add_assoc, add_comm {f x}, add_assoc, smul_closedBall _ _ \u03b5pos.le, smul_zero,\n    singleton_add_closedBall_zero, \u2190 image_smul_set \u211d E E A, smul_closedBall _ _ zero_le_one,\n    smul_zero, Real.norm_eq_abs, abs_of_nonneg r0, mul_one, mul_comm] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r K : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) this : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) ** rw [this] at K ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r K : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) this : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) ** calc\n  \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u03bc ({f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5)) :=\n    measure_mono K\n  _ = ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n    simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add,\n      measure_preimage_add]\n  _ \u2264 ENNReal.ofReal (r ^ finrank \u211d E) * (m * \u03bc (closedBall 0 1)) := by\n    rw [add_comm]; exact mul_le_mul_left' h\u03b5.le _\n  _ = m * \u03bc (closedBall x r) := by simp only [addHaar_closedBall' \u03bc _ r0]; ring ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r \u22a2 f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) ** rintro y \u27e8z, \u27e8zs, zr\u27e9, rfl\u27e9 ** case intro.intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r z : E zs : z \u2208 s zr : z \u2208 closedBall x r \u22a2 f z \u2208 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) ** apply Set.mem_add.2 \u27e8A (z - x), f z - f x - A (z - x) + f x, _, _, _\u27e9 ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r z : E zs : z \u2208 s zr : z \u2208 closedBall x r \u22a2 \u2191A (z - x) \u2208 \u2191A '' closedBall 0 r ** apply mem_image_of_mem ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r z : E zs : z \u2208 s zr : z \u2208 closedBall x r \u22a2 z - x \u2208 closedBall 0 r ** simpa only [dist_eq_norm, mem_closedBall, mem_closedBall_zero_iff, sub_zero] using zr ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r z : E zs : z \u2208 s zr : z \u2208 closedBall x r \u22a2 f z - f x - \u2191A (z - x) + f x \u2208 closedBall (f x) (\u03b5 * r) ** rw [mem_closedBall_iff_norm, add_sub_cancel] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r z : E zs : z \u2208 s zr : z \u2208 closedBall x r \u22a2 \u2016f z - f x - \u2191A (z - x)\u2016 \u2264 \u03b5 * r ** calc\n  \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b4 * \u2016z - x\u2016 := hf _ zs _ xs\n  _ \u2264 \u03b5 * r :=\n    mul_le_mul (le_of_lt h\u03b4) (mem_closedBall_iff_norm.1 zr) (norm_nonneg _) \u03b5pos.le ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r z : E zs : z \u2208 s zr : z \u2208 closedBall x r \u22a2 \u2191A (z - x) + (f z - f x - \u2191A (z - x) + f x) = f z ** simp only [map_sub, Pi.sub_apply] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r z : E zs : z \u2208 s zr : z \u2208 closedBall x r \u22a2 \u2191A z - \u2191A x + (f z - f x - (\u2191A z - \u2191A x) + f x) = f z ** abel ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r K : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) \u22a2 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) ** rw [smul_add, \u2190 add_assoc, add_comm {f x}, add_assoc, smul_closedBall _ _ \u03b5pos.le, smul_zero,\n  singleton_add_closedBall_zero, \u2190 image_smul_set \u211d E E A, smul_closedBall _ _ zero_le_one,\n  smul_zero, Real.norm_eq_abs, abs_of_nonneg r0, mul_one, mul_comm] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r K : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) this : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u22a2 \u2191\u2191\u03bc ({f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)) =     ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) ** simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add,\n  measure_preimage_add] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r K : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) this : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u2264     ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1)) ** rw [add_comm] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r K : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) this : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) \u2264     ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1)) ** exact mul_le_mul_left' h\u03b5.le _ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r K : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) this : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1)) = \u2191m * \u2191\u2191\u03bc (closedBall x r) ** simp only [addHaar_closedBall' \u03bc _ r0] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 x : E r : \u211d xs : x \u2208 s r0 : 0 \u2264 r K : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) this : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1)) =     \u2191m * (ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)) ** ring ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) \u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) ** filter_upwards [self_mem_nhdsWithin] with a ha ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : a \u2208 Ioi 0 \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) ** change 0 < a at ha ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) ** obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n  \u2203 (t : Set E) (r : E \u2192 \u211d),\n    t.Countable \u2227\n      t \u2286 s \u2227\n        (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n          (s \u2286 \u22c3 x \u2208 t, closedBall x (r x)) \u2227\n            (\u2211' x : \u21a5t, \u03bc (closedBall (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n  Besicovitch.exists_closedBall_covering_tsum_measure_le \u03bc ha.ne' (fun _ => Ioi 0) s\n    fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [half_pos \u03b4pos, \u03b4pos]\u27e9 ** case h.intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a t : Set E r : E \u2192 \u211d t_count : Set.Countable t ts : t \u2286 s rpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x st : s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) ** haveI : Encodable t := t_count.toEncodable ** case h.intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a t : Set E r : E \u2192 \u211d t_count : Set.Countable t ts : t \u2286 s rpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x st : s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a this : Encodable \u2191t \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) ** calc\n  \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 closedBall x (r x))) := by\n    rw [biUnion_eq_iUnion] at st\n    apply measure_mono\n    rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n    exact image_subset _ (subset_inter (Subset.refl _) st)\n  _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 closedBall x (r x))) := (measure_iUnion_le _)\n  _ \u2264 \u2211' x : t, m * \u03bc (closedBall x (r x)) :=\n    (ENNReal.tsum_le_tsum fun x => I x (r x) (ts x.2) (rpos x x.2).le)\n  _ \u2264 m * (\u03bc s + a) := by rw [ENNReal.tsum_mul_left]; exact mul_le_mul_left' \u03bct _ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4\u271d : \u211d\u22650 h\u03b4 : \u03b4\u271d \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4\u271d I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a x : E x\u271d : x \u2208 s \u03b4 : \u211d \u03b4pos : \u03b4 > 0 \u22a2 \u03b4 / 2 \u2208 (fun x => Ioi 0) x \u2229 Ioo 0 \u03b4 ** simp [half_pos \u03b4pos, \u03b4pos] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a t : Set E r : E \u2192 \u211d t_count : Set.Countable t ts : t \u2286 s rpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x st : s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a this : Encodable \u2191t \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x))) ** rw [biUnion_eq_iUnion] at st ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a t : Set E r : E \u2192 \u211d t_count : Set.Countable t ts : t \u2286 s rpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x st : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x) \u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a this : Encodable \u2191t \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x))) ** apply measure_mono ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a t : Set E r : E \u2192 \u211d t_count : Set.Countable t ts : t \u2286 s rpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x st : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x) \u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a this : Encodable \u2191t \u22a2 f '' s \u2286 \u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x)) ** rw [\u2190 image_iUnion, \u2190 inter_iUnion] ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a t : Set E r : E \u2192 \u211d t_count : Set.Countable t ts : t \u2286 s rpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x st : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x) \u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a this : Encodable \u2191t \u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 i, closedBall (\u2191i) (r \u2191i)) ** exact image_subset _ (subset_inter (Subset.refl _) st) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a t : Set E r : E \u2192 \u211d t_count : Set.Countable t ts : t \u2286 s rpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x st : s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a this : Encodable \u2191t \u22a2 \u2211' (x : \u2191t), \u2191m * \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) ** rw [ENNReal.tsum_mul_left] ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) a : \u211d\u22650\u221e ha : 0 < a t : Set E r : E \u2192 \u211d t_count : Set.Countable t ts : t \u2286 s rpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x st : s \u2286 \u22c3 x \u2208 t, closedBall x (r x) \u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a this : Encodable \u2191t \u22a2 \u2191m * \u2211' (i : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191i) (r \u2191i)) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) ** exact mul_le_mul_left' \u03bct _ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) J : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) \u22a2 Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0))) ** apply Tendsto.mono_left _ nhdsWithin_le_nhds ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) J : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) \u22a2 Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0))) ** apply ENNReal.Tendsto.const_mul (tendsto_const_nhds.add tendsto_id) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s\u271d : Set E f\u271d : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc A : E \u2192L[\u211d] E m : \u211d\u22650 hm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m d : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A| \u03b5 : \u211d h\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u03b5pos : 0 < \u03b5 this : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0 \u03b4 : \u211d\u22650 h\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } s : Set E f : E \u2192 E hf : ApproximatesLinearOn f A s \u03b4 I : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r) J : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a) \u22a2 \u2191\u2191\u03bc s + 0 \u2260 0 \u2228 \u2191m \u2260 \u22a4 ** simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.mkMetric_nnreal_smul ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e c : \u211d\u22650 hc : c \u2260 0 \u22a2 mkMetric (c \u2022 m) = c \u2022 mkMetric m ** rw [ENNReal.smul_def, ENNReal.smul_def,\n  mkMetric_smul m ENNReal.coe_ne_top (ENNReal.coe_ne_zero.mpr hc)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Substring.ValidFor.takeWhile ** l m r : List Char p : Char \u2192 Bool \u22a2 ValidFor l (List.takeWhile p m) (List.dropWhile p m ++ r)     (Substring.takeWhile       { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },         stopPos := { byteIdx := utf8Len l + utf8Len m } }       p) ** simp only [Substring.takeWhile, takeWhileAux_of_valid] ** l m r : List Char p : Char \u2192 Bool \u22a2 ValidFor l (List.takeWhile p m) (List.dropWhile p m ++ r)     { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },       stopPos := { byteIdx := utf8Len l + utf8Len (List.takeWhile p m) } } ** refine' .of_eq .. <;> simp ** case refine'_1 l m r : List Char p : Char \u2192 Bool \u22a2 m ++ r = List.takeWhile p m ++ (List.dropWhile p m ++ r) ** rw [\u2190 List.append_assoc, List.takeWhile_append_dropWhile] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_integral_abs_condexp_le ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc ** by_cases hnm : m \u2264 m0 ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc  case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : \u00acm \u2264 m0 \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc ** by_cases hfint : Integrable f \u03bc ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc  case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : \u00acIntegrable f \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f this : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc ** rw [this, \u2190 integral_indicator] ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f this : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u22a2 \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), Set.indicator s (fun x => |f x|) x \u2202\u03bc  case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f this : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u22a2 MeasurableSet s ** swap ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f this : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u22a2 \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), Set.indicator s (fun x => |f x|) x \u2202\u03bc ** refine' (integral_abs_condexp_le _).trans\n  (le_of_eq <| integral_congr_ae <| eventually_of_forall fun x => _) ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f this : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc x : \u03b1 \u22a2 (fun x => |Set.indicator s f x|) x = (fun x => Set.indicator s (fun x => |f x|) x) x ** simp_rw [\u2190 Real.norm_eq_abs, norm_indicator_eq_indicator_norm] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : \u00acm \u2264 m0 \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc ** simp_rw [condexp_of_not_le hnm, Pi.zero_apply, abs_zero, integral_zero] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : \u00acm \u2264 m0 \u22a2 0 \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc ** exact integral_nonneg fun x => abs_nonneg _ ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : \u00acIntegrable f \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc ** simp only [condexp_undef hfint, Pi.zero_apply, abs_zero, integral_const, Algebra.id.smul_eq_mul,\n  mul_zero] ** case neg \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : \u00acIntegrable f \u22a2 0 \u2264 \u222b (x : \u03b1) in s, |f x| \u2202\u03bc ** exact integral_nonneg fun x => abs_nonneg _ ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f \u22a2 \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc ** rw [\u2190 integral_indicator] ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f \u22a2 \u222b (x : \u03b1), Set.indicator s (fun x => |(\u03bc[f|m]) x|) x \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc  \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f \u22a2 MeasurableSet s ** swap ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f \u22a2 \u222b (x : \u03b1), Set.indicator s (fun x => |(\u03bc[f|m]) x|) x \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc ** refine' integral_congr_ae _ ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f \u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) =\u1d50[\u03bc] fun x => |(\u03bc[Set.indicator s f|m]) x| ** have : (fun x => |(\u03bc[s.indicator f|m]) x|) =\u1d50[\u03bc] fun x => |s.indicator (\u03bc[f|m]) x| :=\n  EventuallyEq.fun_comp (condexp_indicator hfint hs) _ ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f this : (fun x => |(\u03bc[Set.indicator s f|m]) x|) =\u1d50[\u03bc] fun x => |Set.indicator s (\u03bc[f|m]) x| \u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) =\u1d50[\u03bc] fun x => |(\u03bc[Set.indicator s f|m]) x| ** refine' EventuallyEq.trans (eventually_of_forall fun x => _) this.symm ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f this : (fun x => |(\u03bc[Set.indicator s f|m]) x|) =\u1d50[\u03bc] fun x => |Set.indicator s (\u03bc[f|m]) x| x : \u03b1 \u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) x = |Set.indicator s (\u03bc[f|m]) x| ** rw [\u2190 Real.norm_eq_abs, norm_indicator_eq_indicator_norm] ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f this : (fun x => |(\u03bc[Set.indicator s f|m]) x|) =\u1d50[\u03bc] fun x => |Set.indicator s (\u03bc[f|m]) x| x : \u03b1 \u22a2 (fun x => Set.indicator s (fun x => |(\u03bc[f|m]) x|) x) x = Set.indicator s (fun a => \u2016(\u03bc[f|m]) a\u2016) x ** rfl ** \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f \u22a2 MeasurableSet s ** exact hnm _ hs ** case pos \u03b1 : Type u_1 m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 hs : MeasurableSet s f : \u03b1 \u2192 \u211d hnm : m \u2264 m0 hfint : Integrable f this : \u222b (x : \u03b1) in s, |(\u03bc[f|m]) x| \u2202\u03bc = \u222b (x : \u03b1), |(\u03bc[Set.indicator s f|m]) x| \u2202\u03bc \u22a2 MeasurableSet s ** exact hnm _ hs ** Qed",
        "informal": ""
    },
    {
        "formal": "ae_restrict_of_ae_restrict_inter_Ioo ** \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x ** let T : s \u00d7 s \u2192 Set \u211d := fun p => Ioo p.1 p.2 ** \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x ** let u := \u22c3 i : \u21a5s \u00d7 \u21a5s, T i ** \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x ** have hfinite : (s \\ u).Finite := s.finite_diff_iUnion_Ioo' ** \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x ** obtain \u27e8A, A_count, hA\u27e9 :\n  \u2203 A : Set (\u21a5s \u00d7 \u21a5s), A.Countable \u2227 \u22c3 i \u2208 A, T i = \u22c3 i : \u21a5s \u00d7 \u21a5s, T i :=\n  isOpen_iUnion_countable _ fun p => isOpen_Ioo ** case intro.intro \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x ** apply ae_restrict_of_ae_restrict_of_subset this ** case intro.intro \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p), p x ** rw [ae_restrict_union_iff, ae_restrict_biUnion_iff _ A_count] ** case intro.intro \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p \u22a2 (\u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x) \u2227 \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T i), p x ** constructor ** \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i \u22a2 s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p ** intro x hx ** \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i x : \u211d hx : x \u2208 s \u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p ** by_cases h'x : x \u2208 \u22c3 i : \u21a5s \u00d7 \u21a5s, T i ** case pos \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i x : \u211d hx : x \u2208 s h'x : x \u2208 \u22c3 i, T i \u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p ** rw [\u2190 hA] at h'x ** case pos \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i x : \u211d hx : x \u2208 s h'x : x \u2208 \u22c3 i \u2208 A, T i \u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p ** obtain \u27e8p, pA, xp\u27e9 : \u2203 p : \u21a5s \u00d7 \u21a5s, p \u2208 A \u2227 x \u2208 T p := by\n  simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using h'x ** case pos.intro.intro \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p\u271d : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p\u271d x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i x : \u211d hx : x \u2208 s h'x : x \u2208 \u22c3 i \u2208 A, T i p : \u2191s \u00d7 \u2191s pA : p \u2208 A xp : x \u2208 T p \u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p ** right ** case pos.intro.intro.h \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p\u271d : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p\u271d x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i x : \u211d hx : x \u2208 s h'x : x \u2208 \u22c3 i \u2208 A, T i p : \u2191s \u00d7 \u2191s pA : p \u2208 A xp : x \u2208 T p \u22a2 x \u2208 \u22c3 p \u2208 A, s \u2229 T p ** exact mem_biUnion pA \u27e8hx, xp\u27e9 ** \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i x : \u211d hx : x \u2208 s h'x : x \u2208 \u22c3 i \u2208 A, T i \u22a2 \u2203 p, p \u2208 A \u2227 x \u2208 T p ** simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using h'x ** case neg \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i x : \u211d hx : x \u2208 s h'x : \u00acx \u2208 \u22c3 i, T i \u22a2 x \u2208 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p ** exact Or.inl \u27e8hx, h'x\u27e9 ** case intro.intro.left \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x ** have : \u03bc.restrict (s \\ u) = 0 := by simp only [restrict_eq_zero, hfinite.measure_zero] ** case intro.intro.left \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this\u271d : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p this : Measure.restrict \u03bc (s \\ u) = 0 \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x ** simp only [this, ae_zero, eventually_bot] ** \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p \u22a2 Measure.restrict \u03bc (s \\ u) = 0 ** simp only [restrict_eq_zero, hfinite.measure_zero] ** case intro.intro.right \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p \u22a2 \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T i), p x ** rintro \u27e8\u27e8a, as\u27e9, \u27e8b, bs\u27e9\u27e9 - ** case intro.intro.right.mk.mk.mk \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p a : \u211d as : a \u2208 s b : \u211d bs : b \u2208 s \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T ({ val := a, property := as }, { val := b, property := bs })), p x ** dsimp ** case intro.intro.right.mk.mk.mk \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p a : \u211d as : a \u2208 s b : \u211d bs : b \u2208 s \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x ** rcases le_or_lt b a with (hba | hab) ** case intro.intro.right.mk.mk.mk.inl \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p a : \u211d as : a \u2208 s b : \u211d bs : b \u2208 s hba : b \u2264 a \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x ** simp only [Ioo_eq_empty_of_le hba, inter_empty, restrict_empty, ae_zero, eventually_bot] ** case intro.intro.right.mk.mk.mk.inr \u03bc : Measure \u211d inst\u271d : NoAtoms \u03bc s : Set \u211d p : \u211d \u2192 Prop h : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x T : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.1 \u2191p.2 u : Set \u211d := \u22c3 i, T i hfinite : Set.Finite (s \\ u) A : Set (\u2191s \u00d7 \u2191s) A_count : Set.Countable A hA : \u22c3 i \u2208 A, T i = \u22c3 i, T i this : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p a : \u211d as : a \u2208 s b : \u211d bs : b \u2208 s hab : a < b \u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x ** exact h a b as bs hab ** Qed",
        "informal": ""
    },
    {
        "formal": "Finite.Set.subset ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s t : Set \u03b1 inst\u271d : Finite \u2191s h : t \u2286 s \u22a2 Finite \u2191t ** rw [\u2190 sep_eq_of_subset h] ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x s t : Set \u03b1 inst\u271d : Finite \u2191s h : t \u2286 s \u22a2 Finite \u2191{x | x \u2208 s \u2227 x \u2208 t} ** infer_instance ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.induction ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 P : (\u03b1 \u2192 E) \u2192 Prop h_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (indicator s fun x => c) h_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Integrable f \u2192 Integrable g \u2192 P f \u2192 P g \u2192 P (f + g) h_closed : IsClosed {f | P \u2191\u2191f} h_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Integrable f \u2192 P f \u2192 P g \u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Integrable f \u2192 P f ** simp only [\u2190 mem\u2112p_one_iff_integrable] at * ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f : \u03b1 \u2192 E p : \u211d\u22650\u221e \u03bc : Measure \u03b1 P : (\u03b1 \u2192 E) \u2192 Prop h_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (indicator s fun x => c) h_closed : IsClosed {f | P \u2191\u2191f} h_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f 1 \u2192 Mem\u2112p g 1 \u2192 P f \u2192 P g \u2192 P (f + g) h_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f 1 \u2192 P f \u2192 P g \u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Mem\u2112p f 1 \u2192 P f ** exact Mem\u2112p.induction one_ne_top (P := P) h_ind h_add h_closed h_ae ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.le_ncard_diff ** \u03b1 : Type u_1 s\u271d t\u271d s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard t \u2264 ncard s + ncard (t \\ s) ** rw [add_comm] ** \u03b1 : Type u_1 s\u271d t\u271d s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d \u22a2 ncard t \u2264 ncard (t \\ s) + ncard s ** apply ncard_le_ncard_diff_add_ncard _ _ hs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_eq_zero_iff ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc h0 : p \u2260 0 \u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0 ** by_cases h_top : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc h0 : p \u2260 0 h_top : \u00acp = \u22a4 \u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0 ** rw [snorm_eq_snorm' h0 h_top] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc h0 : p \u2260 0 h_top : \u00acp = \u22a4 \u22a2 snorm' f (ENNReal.toReal p) \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0 ** exact snorm'_eq_zero_iff (ENNReal.toReal_pos h0 h_top) hf ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 E hf : AEStronglyMeasurable f \u03bc h0 : p \u2260 0 h_top : p = \u22a4 \u22a2 snorm f p \u03bc = 0 \u2194 f =\u1d50[\u03bc] 0 ** rw [h_top, snorm_exponent_top, snormEssSup_eq_zero_iff] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_eq_iff_surjOn_mapsTo ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 p : Set \u03b3 f f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2 g g\u2081 g\u2082 : \u03b2 \u2192 \u03b3 f' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1 g' : \u03b3 \u2192 \u03b2 a : \u03b1 b : \u03b2 \u22a2 f '' s = t \u2194 SurjOn f s t \u2227 MapsTo f s t ** refine' \u27e8_, fun h => h.1.image_eq_of_mapsTo h.2\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 p : Set \u03b3 f f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2 g g\u2081 g\u2082 : \u03b2 \u2192 \u03b3 f' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1 g' : \u03b3 \u2192 \u03b2 a : \u03b1 b : \u03b2 \u22a2 f '' s = t \u2192 SurjOn f s t \u2227 MapsTo f s t ** rintro rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 s s\u2081 s\u2082 : Set \u03b1 t\u2081 t\u2082 : Set \u03b2 p : Set \u03b3 f f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2 g g\u2081 g\u2082 : \u03b2 \u2192 \u03b3 f' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1 g' : \u03b3 \u2192 \u03b2 a : \u03b1 b : \u03b2 \u22a2 SurjOn f s (f '' s) \u2227 MapsTo f s (f '' s) ** exact \u27e8s.surjOn_image f, s.mapsTo_image f\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBSet.find?_insert_of_ne ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp h : cmp v' v \u2260 Ordering.eq \u22a2 find? (insert t v) v' = find? t v' ** refine Option.ext fun u =>\n  find?_some.trans <| .trans (and_congr_left fun h' => ?_) find?_some.symm ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp h : cmp v' v \u2260 Ordering.eq u : \u03b1 h' : cmp v' u = Ordering.eq \u22a2 u \u2208 toList (insert t v) \u2194 u \u2208 toList t ** rw [mem_toList_insert, or_iff_left, and_iff_left] ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp h : cmp v' v \u2260 Ordering.eq u : \u03b1 h' : cmp v' u = Ordering.eq \u22a2 find? t v \u2260 some u ** exact mt (fun h => by rwa [TransCmp.cmp_congr_right (find?_some_eq_eq h)]) h ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp h\u271d : cmp v' v \u2260 Ordering.eq u : \u03b1 h' : cmp v' u = Ordering.eq h : find? t v = some u \u22a2 cmp v' v = Ordering.eq ** rwa [TransCmp.cmp_congr_right (find?_some_eq_eq h)] ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' v : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp h : cmp v' v \u2260 Ordering.eq u : \u03b1 h' : cmp v' u = Ordering.eq \u22a2 \u00acu = v ** rintro rfl ** \u03b1 : Type u_1 cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v' : \u03b1 inst\u271d : TransCmp cmp t : RBSet \u03b1 cmp u : \u03b1 h' : cmp v' u = Ordering.eq h : cmp v' u \u2260 Ordering.eq \u22a2 False ** contradiction ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.mem\u2112p_congr_ae ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hfg : f =\u1d50[\u03bc] g \u22a2 Mem\u2112p f p \u2194 Mem\u2112p g p ** simp only [Mem\u2112p, snorm_congr_ae hfg, aestronglyMeasurable_congr hfg] ** Qed",
        "informal": ""
    },
    {
        "formal": "subset_piiUnionInter ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03c0 : \u03b9 \u2192 Set (Set \u03b1) S : Set \u03b9 i : \u03b9 his : i \u2208 S \u22a2 \u03c0 i \u2286 piiUnionInter \u03c0 S ** have h_ss : {i} \u2286 S := by\n  intro j hj\n  rw [mem_singleton_iff] at hj\n  rwa [hj] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03c0 : \u03b9 \u2192 Set (Set \u03b1) S : Set \u03b9 i : \u03b9 his : i \u2208 S h_ss : {i} \u2286 S \u22a2 \u03c0 i \u2286 piiUnionInter \u03c0 S ** refine' Subset.trans _ (piiUnionInter_mono_right h_ss) ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03c0 : \u03b9 \u2192 Set (Set \u03b1) S : Set \u03b9 i : \u03b9 his : i \u2208 S h_ss : {i} \u2286 S \u22a2 \u03c0 i \u2286 piiUnionInter \u03c0 {i} ** rw [piiUnionInter_singleton] ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03c0 : \u03b9 \u2192 Set (Set \u03b1) S : Set \u03b9 i : \u03b9 his : i \u2208 S h_ss : {i} \u2286 S \u22a2 \u03c0 i \u2286 \u03c0 i \u222a {univ} ** exact subset_union_left _ _ ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03c0 : \u03b9 \u2192 Set (Set \u03b1) S : Set \u03b9 i : \u03b9 his : i \u2208 S \u22a2 {i} \u2286 S ** intro j hj ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03c0 : \u03b9 \u2192 Set (Set \u03b1) S : Set \u03b9 i : \u03b9 his : i \u2208 S j : \u03b9 hj : j \u2208 {i} \u22a2 j \u2208 S ** rw [mem_singleton_iff] at hj ** \u03b1 : Type u_1 \u03b9 : Type u_2 \u03c0 : \u03b9 \u2192 Set (Set \u03b1) S : Set \u03b9 i : \u03b9 his : i \u2208 S j : \u03b9 hj : j = i \u22a2 j \u2208 S ** rwa [hj] ** Qed",
        "informal": ""
    },
    {
        "formal": "StieltjesFunction.outer_Ioc ** f : StieltjesFunction a b : \u211d \u22a2 \u2191(StieltjesFunction.outer f) (Ioc a b) = ofReal (\u2191f b - \u2191f a) ** refine'\n  le_antisymm\n    (by\n      rw [\u2190 f.length_Ioc]\n      apply outer_le_length)\n    (le_iInf\u2082 fun s hs => ENNReal.le_of_forall_pos_le_add fun \u03b5 \u03b5pos h => _) ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5 ** let \u03b4 := \u03b5 / 2 ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5 ** have \u03b4pos : 0 < (\u03b4 : \u211d\u22650\u221e) := by simpa using \u03b5pos.ne' ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5 ** rcases ENNReal.exists_pos_sum_of_countable \u03b4pos.ne' \u2115 with \u27e8\u03b5', \u03b5'0, h\u03b5\u27e9 ** case intro.intro f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 \u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5 ** obtain \u27e8a', ha', aa'\u27e9 : \u2203 a', f a' - f a < \u03b4 \u2227 a < a' := by\n  have A : ContinuousWithinAt (fun r => f r - f a) (Ioi a) a := by\n    refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n    exact (f.right_continuous a).mono Ioi_subset_Ici_self\n  have B : f a - f a < \u03b4 := by rwa [sub_self, NNReal.coe_pos, \u2190 ENNReal.coe_pos]\n  exact (((tendsto_order.1 A).2 _ B).and self_mem_nhdsWithin).exists ** case intro.intro.intro.intro f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' \u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5 ** have : \u2200 i, \u2203 p : \u211d \u00d7 \u211d, s i \u2286 Ioo p.1 p.2 \u2227\n    (ofReal (f p.2 - f p.1) : \u211d\u22650\u221e) < f.length (s i) + \u03b5' i := by\n  intro i\n  have hl :=\n    ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_ne_zero.2 (\u03b5'0 i).ne')\n  conv at hl =>\n    lhs\n    rw [length]\n  simp only [iInf_lt_iff, exists_prop] at hl\n  rcases hl with \u27e8p, q', spq, hq'\u27e9\n  have : ContinuousWithinAt (fun r => ofReal (f r - f p)) (Ioi q') q' := by\n    apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt\n    refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n    exact (f.right_continuous q').mono Ioi_subset_Ici_self\n  rcases (((tendsto_order.1 this).2 _ hq').and self_mem_nhdsWithin).exists with \u27e8q, hq, q'q\u27e9\n  exact \u27e8\u27e8p, q\u27e9, spq.trans (Ioc_subset_Ioo_right q'q), hq\u27e9 ** case intro.intro.intro.intro f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' this : \u2200 (i : \u2115), \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i) \u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5 ** choose g hg using this ** case intro.intro.intro.intro f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' g : \u2115 \u2192 \u211d \u00d7 \u211d hg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i) \u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5 ** have I_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2 :=\n  calc\n    Icc a' b \u2286 Ioc a b := fun x hx => \u27e8aa'.trans_le hx.1, hx.2\u27e9\n    _ \u2286 \u22c3 i, s i := hs\n    _ \u2286 \u22c3 i, Ioo (g i).1 (g i).2 := iUnion_mono fun i => (hg i).1 ** case intro.intro.intro.intro f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' g : \u2115 \u2192 \u211d \u00d7 \u211d hg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i) I_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2 \u22a2 ofReal (\u2191f b - \u2191f a) \u2264 \u2211' (i : \u2115), length f (s i) + \u2191\u03b5 ** calc\n  ofReal (f b - f a) = ofReal (f b - f a' + (f a' - f a)) := by rw [sub_add_sub_cancel]\n  _ \u2264 ofReal (f b - f a') + ofReal (f a' - f a) := ENNReal.ofReal_add_le\n  _ \u2264 \u2211' i, ofReal (f (g i).2 - f (g i).1) + ofReal \u03b4 :=\n    (add_le_add (f.length_subadditive_Icc_Ioo I_subset) (ENNReal.ofReal_le_ofReal ha'.le))\n  _ \u2264 \u2211' i, (f.length (s i) + \u03b5' i) + \u03b4 :=\n    (add_le_add (ENNReal.tsum_le_tsum fun i => (hg i).2.le)\n      (by simp only [ENNReal.ofReal_coe_nnreal, le_rfl]))\n  _ = \u2211' i, f.length (s i) + \u2211' i, (\u03b5' i : \u211d\u22650\u221e) + \u03b4 := by rw [ENNReal.tsum_add]\n  _ \u2264 \u2211' i, f.length (s i) + \u03b4 + \u03b4 := (add_le_add (add_le_add le_rfl h\u03b5.le) le_rfl)\n  _ = \u2211' i : \u2115, f.length (s i) + \u03b5 := by simp [add_assoc, ENNReal.add_halves] ** f : StieltjesFunction a b : \u211d \u22a2 \u2191(StieltjesFunction.outer f) (Ioc a b) \u2264 ofReal (\u2191f b - \u2191f a) ** rw [\u2190 f.length_Ioc] ** f : StieltjesFunction a b : \u211d \u22a2 \u2191(StieltjesFunction.outer f) (Ioc a b) \u2264 length f (Ioc a b) ** apply outer_le_length ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u22a2 0 < \u2191\u03b4 ** simpa using \u03b5pos.ne' ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 \u22a2 \u2203 a', \u2191f a' - \u2191f a < \u2191\u03b4 \u2227 a < a' ** have A : ContinuousWithinAt (fun r => f r - f a) (Ioi a) a := by\n  refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n  exact (f.right_continuous a).mono Ioi_subset_Ici_self ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 A : ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a \u22a2 \u2203 a', \u2191f a' - \u2191f a < \u2191\u03b4 \u2227 a < a' ** have B : f a - f a < \u03b4 := by rwa [sub_self, NNReal.coe_pos, \u2190 ENNReal.coe_pos] ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 A : ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a B : \u2191f a - \u2191f a < \u2191\u03b4 \u22a2 \u2203 a', \u2191f a' - \u2191f a < \u2191\u03b4 \u2227 a < a' ** exact (((tendsto_order.1 A).2 _ B).and self_mem_nhdsWithin).exists ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 \u22a2 ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a ** refine' ContinuousWithinAt.sub _ continuousWithinAt_const ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 \u22a2 ContinuousWithinAt (fun r => \u2191f r) (Ioi a) a ** exact (f.right_continuous a).mono Ioi_subset_Ici_self ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 A : ContinuousWithinAt (fun r => \u2191f r - \u2191f a) (Ioi a) a \u22a2 \u2191f a - \u2191f a < \u2191\u03b4 ** rwa [sub_self, NNReal.coe_pos, \u2190 ENNReal.coe_pos] ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' \u22a2 \u2200 (i : \u2115), \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i) ** intro i ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 \u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i) ** have hl :=\n  ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_ne_zero.2 (\u03b5'0 i).ne') ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 hl : length f (s i) < length f (s i) + \u2191(\u03b5' i) \u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i) ** conv at hl =>\n  lhs\n  rw [length] ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 hl : \u2a05 a, \u2a05 b, \u2a05 (_ : s i \u2286 Ioc a b), ofReal (\u2191f b - \u2191f a) < length f (s i) + \u2191(\u03b5' i) \u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i) ** simp only [iInf_lt_iff, exists_prop] at hl ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 hl : \u2203 i_1 i_2, s i \u2286 Ioc i_1 i_2 \u2227 ofReal (\u2191f i_2 - \u2191f i_1) < length f (s i) + \u2191(\u03b5' i) \u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i) ** rcases hl with \u27e8p, q', spq, hq'\u27e9 ** case intro.intro.intro f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 p q' : \u211d spq : s i \u2286 Ioc p q' hq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i) \u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i) ** have : ContinuousWithinAt (fun r => ofReal (f r - f p)) (Ioi q') q' := by\n  apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt\n  refine' ContinuousWithinAt.sub _ continuousWithinAt_const\n  exact (f.right_continuous q').mono Ioi_subset_Ici_self ** case intro.intro.intro f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 p q' : \u211d spq : s i \u2286 Ioc p q' hq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i) this : ContinuousWithinAt (fun r => ofReal (\u2191f r - \u2191f p)) (Ioi q') q' \u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i) ** rcases (((tendsto_order.1 this).2 _ hq').and self_mem_nhdsWithin).exists with \u27e8q, hq, q'q\u27e9 ** case intro.intro.intro.intro.intro f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 p q' : \u211d spq : s i \u2286 Ioc p q' hq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i) this : ContinuousWithinAt (fun r => ofReal (\u2191f r - \u2191f p)) (Ioi q') q' q : \u211d hq : ofReal (\u2191f q - \u2191f p) < length f (s i) + \u2191(\u03b5' i) q'q : q' < q \u22a2 \u2203 p, s i \u2286 Ioo p.1 p.2 \u2227 ofReal (\u2191f p.2 - \u2191f p.1) < length f (s i) + \u2191(\u03b5' i) ** exact \u27e8\u27e8p, q\u27e9, spq.trans (Ioc_subset_Ioo_right q'q), hq\u27e9 ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 p q' : \u211d spq : s i \u2286 Ioc p q' hq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i) \u22a2 ContinuousWithinAt (fun r => ofReal (\u2191f r - \u2191f p)) (Ioi q') q' ** apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 p q' : \u211d spq : s i \u2286 Ioc p q' hq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i) \u22a2 ContinuousWithinAt (fun r => \u2191f r - \u2191f p) (Ioi q') q' ** refine' ContinuousWithinAt.sub _ continuousWithinAt_const ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' i : \u2115 p q' : \u211d spq : s i \u2286 Ioc p q' hq' : ofReal (\u2191f q' - \u2191f p) < length f (s i) + \u2191(\u03b5' i) \u22a2 ContinuousWithinAt (fun r => \u2191f r) (Ioi q') q' ** exact (f.right_continuous q').mono Ioi_subset_Ici_self ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' g : \u2115 \u2192 \u211d \u00d7 \u211d hg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i) I_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2 \u22a2 ofReal (\u2191f b - \u2191f a) = ofReal (\u2191f b - \u2191f a' + (\u2191f a' - \u2191f a)) ** rw [sub_add_sub_cancel] ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' g : \u2115 \u2192 \u211d \u00d7 \u211d hg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i) I_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2 \u22a2 ofReal \u2191\u03b4 \u2264 \u2191\u03b4 ** simp only [ENNReal.ofReal_coe_nnreal, le_rfl] ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' g : \u2115 \u2192 \u211d \u00d7 \u211d hg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i) I_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2 \u22a2 \u2211' (i : \u2115), (length f (s i) + \u2191(\u03b5' i)) + \u2191\u03b4 = \u2211' (i : \u2115), length f (s i) + \u2211' (i : \u2115), \u2191(\u03b5' i) + \u2191\u03b4 ** rw [ENNReal.tsum_add] ** f : StieltjesFunction a b : \u211d s : \u2115 \u2192 Set \u211d hs : Ioc a b \u2286 \u22c3 i, s i \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 h : \u2211' (i : \u2115), length f (s i) < \u22a4 \u03b4 : \u211d\u22650 := \u03b5 / 2 \u03b4pos : 0 < \u2191\u03b4 \u03b5' : \u2115 \u2192 \u211d\u22650 \u03b5'0 : \u2200 (i : \u2115), 0 < \u03b5' i h\u03b5 : \u2211' (i : \u2115), \u2191(\u03b5' i) < \u2191\u03b4 a' : \u211d ha' : \u2191f a' - \u2191f a < \u2191\u03b4 aa' : a < a' g : \u2115 \u2192 \u211d \u00d7 \u211d hg : \u2200 (i : \u2115), s i \u2286 Ioo (g i).1 (g i).2 \u2227 ofReal (\u2191f (g i).2 - \u2191f (g i).1) < length f (s i) + \u2191(\u03b5' i) I_subset : Icc a' b \u2286 \u22c3 i, Ioo (g i).1 (g i).2 \u22a2 \u2211' (i : \u2115), length f (s i) + \u2191\u03b4 + \u2191\u03b4 = \u2211' (i : \u2115), length f (s i) + \u2191\u03b5 ** simp [add_assoc, ENNReal.add_halves] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.PairingHeapImp.Heap.size_deleteMin ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool a : \u03b1 s' s : Heap \u03b1 h : NoSibling s eq : deleteMin le s = some (a, s') \u22a2 size s = size s' + 1 ** cases h with cases eq | node a c => rw [size_combine, size, size] ** case node.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool a : \u03b1 c : Heap \u03b1 \u22a2 size (node a c nil) = size (combine le c) + 1 ** rw [size_combine, size, size] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.aestronglyMeasurable'_condexpL1Clm ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } \u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc ** refine' @Lp.induction _ _ _ _ _ _ _ ENNReal.one_ne_top\n  (fun f : \u03b1 \u2192\u2081[\u03bc] F' => AEStronglyMeasurable' m (condexpL1Clm F' hm \u03bc f) \u03bc) _ _ _ f ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } \u22a2 \u2200 (c : F') {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4),     (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc)       \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c) ** intro c s hs h\u03bcs ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s\u271d : Set \u03b1 f : { x // x \u2208 Lp F' 1 } c : F' s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) \u2191(simpleFunc.indicatorConst 1 hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c))) \u03bc ** rw [condexpL1Clm_indicatorConst hs h\u03bcs.ne c] ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s\u271d : Set \u03b1 f : { x // x \u2208 Lp F' 1 } c : F' s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpInd F' hm \u03bc s) c)) \u03bc ** exact aestronglyMeasurable'_condexpInd hs h\u03bcs.ne c ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } \u22a2 \u2200 \u2983f g : \u03b1 \u2192 F'\u2984 (hf : Mem\u2112p f 1) (hg : Mem\u2112p g 1),     Disjoint (Function.support f) (Function.support g) \u2192       (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) (Mem\u2112p.toLp f hf) \u2192         (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) (Mem\u2112p.toLp g hg) \u2192           (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg) ** intro f g hf hg _ hfm hgm ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d\u00b9 g\u271d : \u03b1 \u2192 F' s : Set \u03b1 f\u271d : { x // x \u2208 Lp F' 1 } f g : \u03b1 \u2192 F' hf : Mem\u2112p f 1 hg : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hfm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf))) \u03bc hgm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc \u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg))) \u03bc ** rw [(condexpL1Clm F' hm \u03bc).map_add] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d\u00b9 g\u271d : \u03b1 \u2192 F' s : Set \u03b1 f\u271d : { x // x \u2208 Lp F' 1 } f g : \u03b1 \u2192 F' hf : Mem\u2112p f 1 hg : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hfm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf))) \u03bc hgm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc \u22a2 AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf) + \u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc ** refine' AEStronglyMeasurable'.congr _ (coeFn_add _ _).symm ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d\u00b9 g\u271d : \u03b1 \u2192 F' s : Set \u03b1 f\u271d : { x // x \u2208 Lp F' 1 } f g : \u03b1 \u2192 F' hf : Mem\u2112p f 1 hg : Mem\u2112p g 1 a\u271d : Disjoint (Function.support f) (Function.support g) hfm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf))) \u03bc hgm : AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc \u22a2 AEStronglyMeasurable' m     (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp f hf)) + \u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) (Mem\u2112p.toLp g hg))) \u03bc ** exact AEStronglyMeasurable'.add hfm hgm ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } \u22a2 IsClosed {f | (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) f} ** have : {f : Lp F' 1 \u03bc | AEStronglyMeasurable' m (condexpL1Clm F' hm \u03bc f) \u03bc} =\n    condexpL1Clm F' hm \u03bc \u207b\u00b9' {f | AEStronglyMeasurable' m f \u03bc} := rfl ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } this :   {f | AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc} =     \u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc} \u22a2 IsClosed {f | (fun f => AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc) f} ** rw [this] ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } this :   {f | AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc} =     \u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc} \u22a2 IsClosed (\u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc}) ** refine' IsClosed.preimage (condexpL1Clm F' hm \u03bc).continuous _ ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 f : { x // x \u2208 Lp F' 1 } this :   {f | AEStronglyMeasurable' m (\u2191\u2191(\u2191(condexpL1Clm F' hm \u03bc) f)) \u03bc} =     \u2191(condexpL1Clm F' hm \u03bc) \u207b\u00b9' {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc} \u22a2 IsClosed {f | AEStronglyMeasurable' m (\u2191\u2191f) \u03bc} ** exact isClosed_aeStronglyMeasurable' hm ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.exists_subset_or_subset_of_two_mul_lt_card ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n\u271d : \u2115 inst\u271d : DecidableEq \u03b1 X Y : Finset \u03b1 n : \u2115 hXY : 2 * n < card (X \u222a Y) \u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y) ** have h\u2081 : (X \u2229 (Y \\ X)).card = 0 := Finset.card_eq_zero.mpr (Finset.inter_sdiff_self X Y) ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n\u271d : \u2115 inst\u271d : DecidableEq \u03b1 X Y : Finset \u03b1 n : \u2115 hXY : 2 * n < card (X \u222a Y) h\u2081 : card (X \u2229 (Y \\ X)) = 0 \u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y) ** have h\u2082 : (X \u222a Y).card = X.card + (Y \\ X).card := by\n  rw [\u2190 card_union_add_card_inter X (Y \\ X), Finset.union_sdiff_self_eq_union, h\u2081, add_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n\u271d : \u2115 inst\u271d : DecidableEq \u03b1 X Y : Finset \u03b1 n : \u2115 hXY : 2 * n < card (X \u222a Y) h\u2081 : card (X \u2229 (Y \\ X)) = 0 h\u2082 : card (X \u222a Y) = card X + card (Y \\ X) \u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y) ** rw [h\u2082, two_mul] at hXY ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n\u271d : \u2115 inst\u271d : DecidableEq \u03b1 X Y : Finset \u03b1 n : \u2115 hXY : n + n < card X + card (Y \\ X) h\u2081 : card (X \u2229 (Y \\ X)) = 0 h\u2082 : card (X \u222a Y) = card X + card (Y \\ X) \u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y) ** rcases lt_or_lt_of_add_lt_add hXY with (h | h) ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n\u271d : \u2115 inst\u271d : DecidableEq \u03b1 X Y : Finset \u03b1 n : \u2115 hXY : 2 * n < card (X \u222a Y) h\u2081 : card (X \u2229 (Y \\ X)) = 0 \u22a2 card (X \u222a Y) = card X + card (Y \\ X) ** rw [\u2190 card_union_add_card_inter X (Y \\ X), Finset.union_sdiff_self_eq_union, h\u2081, add_zero] ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n\u271d : \u2115 inst\u271d : DecidableEq \u03b1 X Y : Finset \u03b1 n : \u2115 hXY : n + n < card X + card (Y \\ X) h\u2081 : card (X \u2229 (Y \\ X)) = 0 h\u2082 : card (X \u222a Y) = card X + card (Y \\ X) h : n < card X \u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y) ** exact \u27e8X, h, Or.inl (Finset.Subset.refl X)\u27e9 ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n\u271d : \u2115 inst\u271d : DecidableEq \u03b1 X Y : Finset \u03b1 n : \u2115 hXY : n + n < card X + card (Y \\ X) h\u2081 : card (X \u2229 (Y \\ X)) = 0 h\u2082 : card (X \u222a Y) = card X + card (Y \\ X) h : n < card (Y \\ X) \u22a2 \u2203 C, n < card C \u2227 (C \u2286 X \u2228 C \u2286 Y) ** exact \u27e8Y \\ X, h, Or.inr (Finset.sdiff_subset Y X)\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.not_frequently_of_upcrossings_lt_top ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : upcrossings a b f \u03c9 \u2260 \u22a4 \u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9) ** rw [\u2190 lt_top_iff_ne_top, upcrossings_lt_top_iff] at h\u03c9 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 \u2264 k \u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9) ** replace h\u03c9 : \u2203 k, \u2200 N, upcrossingsBefore a b f N \u03c9 < k ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k \u22a2 \u00ac((\u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a) \u2227 \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9) ** rintro \u27e8h\u2081, h\u2082\u27e9 ** case intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2203\u1da0 (n : \u2115) in atTop, f n \u03c9 < a h\u2082 : \u2203\u1da0 (n : \u2115) in atTop, b < f n \u03c9 \u22a2 False ** rw [frequently_atTop] at h\u2081 h\u2082 ** case intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a h\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9 \u22a2 False ** refine' Classical.not_not.2 h\u03c9 _ ** case intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a h\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9 \u22a2 \u00ac\u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k ** push_neg ** case intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a h\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9 \u22a2 \u2200 (k : \u2115), \u2203 N, k \u2264 upcrossingsBefore a b f N \u03c9 ** intro k ** case intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a h\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9 k : \u2115 \u22a2 \u2203 N, k \u2264 upcrossingsBefore a b f N \u03c9 ** induction' k with k ih ** case h\u03c9 \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 \u2264 k \u22a2 \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k ** obtain \u27e8k, hk\u27e9 := h\u03c9 ** case h\u03c9.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b k : \u2115 hk : \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 \u2264 k \u22a2 \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k ** exact \u27e8k + 1, fun N => lt_of_le_of_lt (hk N) k.lt_succ_self\u27e9 ** case intro.zero \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a h\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9 \u22a2 \u2203 N, Nat.zero \u2264 upcrossingsBefore a b f N \u03c9 ** simp only [Nat.zero_eq, zero_le, exists_const] ** case intro.succ \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a h\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9 k : \u2115 ih : \u2203 N, k \u2264 upcrossingsBefore a b f N \u03c9 \u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9 ** obtain \u27e8N, hN\u27e9 := ih ** case intro.succ.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a h\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9 k N : \u2115 hN : k \u2264 upcrossingsBefore a b f N \u03c9 \u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9 ** obtain \u27e8N\u2081, hN\u2081, hN\u2081'\u27e9 := h\u2081 N ** case intro.succ.intro.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a h\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9 k N : \u2115 hN : k \u2264 upcrossingsBefore a b f N \u03c9 N\u2081 : \u2115 hN\u2081 : N\u2081 \u2265 N hN\u2081' : f N\u2081 \u03c9 < a \u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9 ** obtain \u27e8N\u2082, hN\u2082, hN\u2082'\u27e9 := h\u2082 N\u2081 ** case intro.succ.intro.intro.intro.intro.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \u2131 : Filtration \u2115 m0 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c9 : \u03a9 R : \u211d\u22650 hab : a < b h\u03c9 : \u2203 k, \u2200 (N : \u2115), upcrossingsBefore a b f N \u03c9 < k h\u2081 : \u2200 (a_1 : \u2115), \u2203 b, b \u2265 a_1 \u2227 f b \u03c9 < a h\u2082 : \u2200 (a : \u2115), \u2203 b_1, b_1 \u2265 a \u2227 b < f b_1 \u03c9 k N : \u2115 hN : k \u2264 upcrossingsBefore a b f N \u03c9 N\u2081 : \u2115 hN\u2081 : N\u2081 \u2265 N hN\u2081' : f N\u2081 \u03c9 < a N\u2082 : \u2115 hN\u2082 : N\u2082 \u2265 N\u2081 hN\u2082' : b < f N\u2082 \u03c9 \u22a2 \u2203 N, Nat.succ k \u2264 upcrossingsBefore a b f N \u03c9 ** exact \u27e8N\u2082 + 1, Nat.succ_le_of_lt <|\n  lt_of_le_of_lt hN (upcrossingsBefore_lt_of_exists_upcrossing hab hN\u2081 hN\u2081' hN\u2082 hN\u2082')\u27e9 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.edist_toL1_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192 \u03b2 hf : Integrable f \u22a2 edist (toL1 f hf) 0 = \u222b\u207b (a : \u03b1), edist (f a) 0 \u2202\u03bc ** simp only [toL1, Lp.edist_toLp_zero, snorm, one_ne_zero, snorm', one_toReal, ENNReal.rpow_one,\n  ne_eq, not_false_eq_true, div_self, ite_false] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192 \u03b2 hf : Integrable f \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), edist (f a) 0 \u2202\u03bc ** simp [edist_eq_coe_nnnorm] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.set_integral_condexp ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c F inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hf : Integrable f hs : MeasurableSet s \u22a2 \u222b (x : \u03b1) in s, (\u03bc[f|m]) x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** rw [set_integral_congr_ae (hm s hs) ((condexp_ae_eq_condexpL1 hm f).mono fun x hx _ => hx)] ** \u03b1 : Type u_1 F : Type u_2 F' : Type u_3 \ud835\udd5c : Type u_4 p : \u211d\u22650\u221e inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \ud835\udd5c F inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \ud835\udd5c F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : CompleteSpace F' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f g : \u03b1 \u2192 F' s : Set \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hf : Integrable f hs : MeasurableSet s \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191(condexpL1 hm \u03bc f) x \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2202\u03bc ** exact set_integral_condexpL1 hf hs ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.set_integral_deterministic' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } E : Type u_4 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f : \u03b2 \u2192 E g : \u03b1 \u2192 \u03b2 a : \u03b1 hg : Measurable g hf : StronglyMeasurable f s : Set \u03b2 hs : MeasurableSet s inst\u271d : Decidable (g a \u2208 s) \u22a2 \u222b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0 ** rw [kernel.deterministic_apply, set_integral_dirac' hf _ hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.image\u2082_inter_left ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 \u03b6 : Type u_11 \u03b6' : Type u_12 \u03bd : Type u_13 inst\u271d\u2078 : DecidableEq \u03b1' inst\u271d\u2077 : DecidableEq \u03b2' inst\u271d\u2076 : DecidableEq \u03b3 inst\u271d\u2075 : DecidableEq \u03b3' inst\u271d\u2074 : DecidableEq \u03b4 inst\u271d\u00b3 : DecidableEq \u03b4' inst\u271d\u00b2 : DecidableEq \u03b5 inst\u271d\u00b9 : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 inst\u271d : DecidableEq \u03b1 hf : Injective2 f \u22a2 \u2191(image\u2082 f (s \u2229 s') t) = \u2191(image\u2082 f s t \u2229 image\u2082 f s' t) ** push_cast ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 \u03b6 : Type u_11 \u03b6' : Type u_12 \u03bd : Type u_13 inst\u271d\u2078 : DecidableEq \u03b1' inst\u271d\u2077 : DecidableEq \u03b2' inst\u271d\u2076 : DecidableEq \u03b3 inst\u271d\u2075 : DecidableEq \u03b3' inst\u271d\u2074 : DecidableEq \u03b4 inst\u271d\u00b3 : DecidableEq \u03b4' inst\u271d\u00b2 : DecidableEq \u03b5 inst\u271d\u00b9 : DecidableEq \u03b5' f f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 g g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 s s' : Finset \u03b1 t t' : Finset \u03b2 u u' : Finset \u03b3 a a' : \u03b1 b b' : \u03b2 c : \u03b3 inst\u271d : DecidableEq \u03b1 hf : Injective2 f \u22a2 image2 f (\u2191s \u2229 \u2191s') \u2191t = image2 f \u2191s \u2191t \u2229 image2 f \u2191s' \u2191t ** exact image2_inter_left hf ** Qed",
        "informal": ""
    },
    {
        "formal": "rieszContentAux_sup_le ** X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 ** apply NNReal.le_of_forall_pos_le_add ** case h X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5 ** intro \u03b5 \u03b5pos ** case h X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5 ** obtain \u27e8f1, f_test_function_K1\u27e9 := exists_lt_rieszContentAux_add_pos \u039b K1 (half_pos \u03b5pos) ** case h.intro X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 f1 : X \u2192\u1d47 \u211d\u22650 f_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2 \u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5 ** obtain \u27e8f2, f_test_function_K2\u27e9 := exists_lt_rieszContentAux_add_pos \u039b K2 (half_pos \u03b5pos) ** case h.intro.intro X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 f1 : X \u2192\u1d47 \u211d\u22650 f_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2 f2 : X \u2192\u1d47 \u211d\u22650 f_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2 f_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x \u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5 ** apply (rieszContentAux_le \u039b f_test_function_union).trans (le_of_lt _) ** X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 f1 : X \u2192\u1d47 \u211d\u22650 f_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2 f2 : X \u2192\u1d47 \u211d\u22650 f_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2 f_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x \u22a2 \u2191\u039b (f1 + f2) < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5 ** rw [map_add] ** X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 f1 : X \u2192\u1d47 \u211d\u22650 f_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2 f2 : X \u2192\u1d47 \u211d\u22650 f_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2 f_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x \u22a2 \u2191\u039b f1 + \u2191\u039b f2 < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5 ** apply lt_of_lt_of_le (_root_.add_lt_add f_test_function_K1.right f_test_function_K2.right)\n  (le_of_eq _) ** X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 f1 : X \u2192\u1d47 \u211d\u22650 f_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2 f2 : X \u2192\u1d47 \u211d\u22650 f_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2 f_test_function_union : \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x \u22a2 rieszContentAux \u039b K1 + \u03b5 / 2 + (rieszContentAux \u039b K2 + \u03b5 / 2) = rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5 ** rw [add_assoc, add_comm (\u03b5 / 2), add_assoc, add_halves \u03b5, add_assoc] ** X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 f1 : X \u2192\u1d47 \u211d\u22650 f_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2 f2 : X \u2192\u1d47 \u211d\u22650 f_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2 \u22a2 \u2200 (x : X), x \u2208 K1 \u2294 K2 \u2192 1 \u2264 \u2191(f1 + f2) x ** rintro x (x_in_K1 | x_in_K2) ** case inl X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 f1 : X \u2192\u1d47 \u211d\u22650 f_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2 f2 : X \u2192\u1d47 \u211d\u22650 f_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2 x : X x_in_K1 : x \u2208 \u2191K1 \u22a2 1 \u2264 \u2191(f1 + f2) x ** exact le_add_right (f_test_function_K1.left x x_in_K1) ** case inr X : Type u_1 inst\u271d : TopologicalSpace X \u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650 K1 K2 : Compacts X \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 f1 : X \u2192\u1d47 \u211d\u22650 f_test_function_K1 : (\u2200 (x : X), x \u2208 K1 \u2192 1 \u2264 \u2191f1 x) \u2227 \u2191\u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2 f2 : X \u2192\u1d47 \u211d\u22650 f_test_function_K2 : (\u2200 (x : X), x \u2208 K2 \u2192 1 \u2264 \u2191f2 x) \u2227 \u2191\u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2 x : X x_in_K2 : x \u2208 \u2191K2 \u22a2 1 \u2264 \u2191(f1 + f2) x ** exact le_add_left (f_test_function_K2.left x x_in_K2) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.LebesgueDecomposition.sup_mem_measurableLE ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : f \u2208 measurableLE \u03bc \u03bd hg : g \u2208 measurableLE \u03bc \u03bd \u22a2 (fun a => f a \u2294 g a) \u2208 measurableLE \u03bc \u03bd ** simp_rw [ENNReal.sup_eq_max] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : f \u2208 measurableLE \u03bc \u03bd hg : g \u2208 measurableLE \u03bc \u03bd \u22a2 (fun a => max (f a) (g a)) \u2208 measurableLE \u03bc \u03bd ** refine' \u27e8Measurable.max hf.1 hg.1, fun A hA => _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : f \u2208 measurableLE \u03bc \u03bd hg : g \u2208 measurableLE \u03bc \u03bd A : Set \u03b1 hA : MeasurableSet A \u22a2 \u222b\u207b (x : \u03b1) in A, (fun a => max (f a) (g a)) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A ** have h\u2081 := hA.inter (measurableSet_le hf.1 hg.1) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : f \u2208 measurableLE \u03bc \u03bd hg : g \u2208 measurableLE \u03bc \u03bd A : Set \u03b1 hA : MeasurableSet A h\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a}) \u22a2 \u222b\u207b (x : \u03b1) in A, (fun a => max (f a) (g a)) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A ** have h\u2082 := hA.inter (measurableSet_lt hg.1 hf.1) ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : f \u2208 measurableLE \u03bc \u03bd hg : g \u2208 measurableLE \u03bc \u03bd A : Set \u03b1 hA : MeasurableSet A h\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a}) h\u2082 : MeasurableSet (A \u2229 {a | g a < f a}) \u22a2 \u222b\u207b (x : \u03b1) in A, (fun a => max (f a) (g a)) x \u2202\u03bc \u2264 \u2191\u2191\u03bd A ** rw [set_lintegral_max hf.1 hg.1] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : f \u2208 measurableLE \u03bc \u03bd hg : g \u2208 measurableLE \u03bc \u03bd A : Set \u03b1 hA : MeasurableSet A h\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a}) h\u2082 : MeasurableSet (A \u2229 {a | g a < f a}) \u22a2 \u222b\u207b (x : \u03b1) in A \u2229 {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in A \u2229 {x | g x < f x}, f x \u2202\u03bc \u2264 \u2191\u2191\u03bd A ** refine' (add_le_add (hg.2 _ h\u2081) (hf.2 _ h\u2082)).trans_eq _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : f \u2208 measurableLE \u03bc \u03bd hg : g \u2208 measurableLE \u03bc \u03bd A : Set \u03b1 hA : MeasurableSet A h\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a}) h\u2082 : MeasurableSet (A \u2229 {a | g a < f a}) \u22a2 \u2191\u2191\u03bd (A \u2229 {a | f a \u2264 g a}) + \u2191\u2191\u03bd (A \u2229 {a | g a < f a}) = \u2191\u2191\u03bd A ** simp only [\u2190 not_le, \u2190 compl_setOf, \u2190 diff_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hf : f \u2208 measurableLE \u03bc \u03bd hg : g \u2208 measurableLE \u03bc \u03bd A : Set \u03b1 hA : MeasurableSet A h\u2081 : MeasurableSet (A \u2229 {a | f a \u2264 g a}) h\u2082 : MeasurableSet (A \u2229 {a | g a < f a}) \u22a2 \u2191\u2191\u03bd (A \u2229 {a | f a \u2264 g a}) + \u2191\u2191\u03bd (A \\ {a | f a \u2264 g a}) = \u2191\u2191\u03bd A ** exact measure_inter_add_diff _ (measurableSet_le hf.1 hg.1) ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.Cofix.bisim_rel ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y \u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y ** let r' (x y) := x = y \u2228 r x y ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y \u22a2 \u2200 (x y : Cofix F \u03b1), r x y \u2192 x = y ** intro x y rxy ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F \u03b1 rxy : r x y \u22a2 x = y ** apply Cofix.bisim_aux r' ** case a n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F \u03b1 rxy : r x y \u22a2 r' x y ** right ** case a.h n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F \u03b1 rxy : r x y \u22a2 r x y ** exact rxy ** case h' n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F \u03b1 rxy : r x y \u22a2 \u2200 (x : Cofix F \u03b1), r' x x ** intro x ** case h' n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y : Cofix F \u03b1 rxy : r x\u271d y x : Cofix F \u03b1 \u22a2 r' x x ** left ** case h'.h n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y : Cofix F \u03b1 rxy : r x\u271d y x : Cofix F \u03b1 \u22a2 x = x ** rfl ** case h n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x y : Cofix F \u03b1 rxy : r x y \u22a2 \u2200 (x y : Cofix F \u03b1), r' x y \u2192 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y ** intro x y r'xy ** case h n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 r'xy : r' x y \u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y ** cases r'xy ** case h.inl n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 h\u271d : x = y \u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y  case h.inr n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 h\u271d : r x y \u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y ** case inl h =>\n  rw [h] ** case h.inr n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 h\u271d : r x y \u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y ** case inr r'xy =>\n  have : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h\n  rw [\u2190 Quot.factor_mk_eq _ _ this]\n  dsimp\n  rw [appendFun_comp_id]\n  rw [@comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r)),\n    @comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r))]\n  rw [h _ _ r'xy] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h\u271d : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 h : x = y \u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y ** rw [h] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 r'xy : r x y \u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y ** have : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 r'xy : r x y this : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y \u22a2 (TypeVec.id ::: Quot.mk r') <$$> dest x = (TypeVec.id ::: Quot.mk r') <$$> dest y ** rw [\u2190 Quot.factor_mk_eq _ _ this] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 r'xy : r x y this : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y \u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$$> dest x =     (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$$> dest y ** dsimp ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 r'xy : r x y this : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y \u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$$>       dest x =     (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$$>       dest y ** rw [appendFun_comp_id] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 r'xy : r x y this : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y \u22a2 ((TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) \u229a         (TypeVec.id ::: Quot.mk fun x y => r x y)) <$$>       dest x =     ((TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) \u229a         (TypeVec.id ::: Quot.mk fun x y => r x y)) <$$>       dest y ** rw [@comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r)),\n  @comp_map _ _ _ q _ _ _ (appendFun id (Quot.mk r))] ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F \u03b1 : TypeVec.{u} n r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop h : \u2200 (x y : Cofix F \u03b1), r x y \u2192 (TypeVec.id ::: Quot.mk r) <$$> dest x = (TypeVec.id ::: Quot.mk r) <$$> dest y r' : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop := fun x y => x = y \u2228 r x y x\u271d y\u271d : Cofix F \u03b1 rxy : r x\u271d y\u271d x y : Cofix F \u03b1 r'xy : r x y this : \u2200 (x y : Cofix F \u03b1), r x y \u2192 r' x y \u22a2 (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) <$$>       (TypeVec.id ::: Quot.mk r) <$$> dest x =     (TypeVec.id ::: Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this) <$$>       (TypeVec.id ::: Quot.mk r) <$$> dest y ** rw [h _ _ r'xy] ** Qed",
        "informal": ""
    },
    {
        "formal": "Rat.divInt_mul_right ** n d a : Int a0 : a \u2260 0 \u22a2 n * a /. (d * a) = n /. d ** simp [\u2190 divInt_mul_left (d := d) a0, Int.mul_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "tendsto_measure_cthickening_of_isClosed ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b3 : PseudoEMetricSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 x : \u03b1 \u03b5 : \u211d\u22650\u221e \u03bc : Measure \u03b1 s : Set \u03b1 hs : \u2203 R, R > 0 \u2227 \u2191\u2191\u03bc (cthickening R s) \u2260 \u22a4 h's : IsClosed s \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc s)) ** convert tendsto_measure_cthickening hs ** case h.e'_5.h.e'_3.h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9 : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b3 : PseudoEMetricSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : OpensMeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 x : \u03b1 \u03b5 : \u211d\u22650\u221e \u03bc : Measure \u03b1 s : Set \u03b1 hs : \u2203 R, R > 0 \u2227 \u2191\u2191\u03bc (cthickening R s) \u2260 \u22a4 h's : IsClosed s \u22a2 s = closure s ** exact h's.closure_eq.symm ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Lp.norm_const_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : IsFiniteMeasure \u03bc c : E \u22a2 \u2016\u2191(Lp.const p \u03bc) c\u2016 \u2264 \u2016c\u2016 * ENNReal.toReal (\u2191\u2191\u03bc Set.univ) ^ (1 / ENNReal.toReal p) ** rw [\u2190 indicatorConstLp_univ] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : IsFiniteMeasure \u03bc c : E \u22a2 \u2016indicatorConstLp p (_ : MeasurableSet Set.univ) (_ : \u2191\u2191\u03bc Set.univ \u2260 \u22a4) c\u2016 \u2264     \u2016c\u2016 * ENNReal.toReal (\u2191\u2191\u03bc Set.univ) ^ (1 / ENNReal.toReal p) ** exact norm_indicatorConstLp_le ** Qed",
        "informal": ""
    },
    {
        "formal": "Group.card_pow_eq_card_pow_card_univ ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k \u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G) ** have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr \u27e81\u27e9 ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G \u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G) ** by_cases hS : S = \u2205 ** case neg F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 \u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G) ** obtain \u27e8a, ha\u27e9 := Set.nonempty_iff_ne_empty.2 hS ** case neg.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S \u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G) ** have key : \u2200 (a) (s t : Set G) [Fintype s] [Fintype t],\n    (\u2200 b : G, b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card s \u2264 Fintype.card t := by\n  refine' fun a s t _ _ h \u21a6 Fintype.card_le_of_injective (fun \u27e8b, hb\u27e9 \u21a6 \u27e8a * b, h b hb\u27e9) _\n  rintro \u27e8b, hb\u27e9 \u27e8c, hc\u27e9 hbc\n  exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc)) ** case neg.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t \u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G) ** have mono : Monotone (fun n \u21a6 Fintype.card (\u21a5(S ^ n)) : \u2115 \u2192 \u2115) :=\n  monotone_nat_of_le_succ fun n \u21a6 key a _ _ fun b hb \u21a6 Set.mul_mem_mul ha hb ** case neg.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) \u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G) ** refine' card_pow_eq_card_pow_card_univ_aux mono (fun n \u21a6 set_fintype_card_le_univ (S ^ n))\n  fun n h \u21a6 le_antisymm (mono (n + 1).le_succ) (key a\u207b\u00b9 (S ^ (n + 2)) (S ^ (n + 1)) _) ** case neg.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) \u22a2 \u2200 (b : G), b \u2208 S ^ (n + 2) \u2192 a\u207b\u00b9 * b \u2208 S ^ (n + 1) ** replace h\u2082 : {a} * S ^ n = S ^ (n + 1) ** case neg.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) h\u2082 : {a} * S ^ n = S ^ (n + 1) \u22a2 \u2200 (b : G), b \u2208 S ^ (n + 2) \u2192 a\u207b\u00b9 * b \u2208 S ^ (n + 1) ** rw [pow_succ', \u2190 h\u2082, mul_assoc, \u2190 pow_succ', h\u2082] ** case neg.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) h\u2082 : {a} * S ^ n = S ^ (n + 1) \u22a2 \u2200 (b : G), b \u2208 {a} * S ^ (n + 1) \u2192 a\u207b\u00b9 * b \u2208 S ^ (n + 1) ** rintro _ \u27e8b, c, hb, hc, rfl\u27e9 ** case neg.intro.intro.intro.intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) h\u2082 : {a} * S ^ n = S ^ (n + 1) b c : G hb : b \u2208 {a} hc : c \u2208 S ^ (n + 1) \u22a2 a\u207b\u00b9 * (fun x x_1 => x * x_1) b c \u2208 S ^ (n + 1) ** rwa [Set.mem_singleton_iff.mp hb, inv_mul_cancel_left] ** case pos F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : S = \u2205 \u22a2 \u2200 (k : \u2115), Fintype.card G \u2264 k \u2192 Fintype.card \u2191(S ^ k) = Fintype.card \u2191(S ^ Fintype.card G) ** refine' fun k hk \u21a6 Fintype.card_congr _ ** case pos F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : S = \u2205 k : \u2115 hk : Fintype.card G \u2264 k \u22a2 \u2191(S ^ k) \u2243 \u2191(S ^ Fintype.card G) ** rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S \u22a2 \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t ** refine' fun a s t _ _ h \u21a6 Fintype.card_le_of_injective (fun \u27e8b, hb\u27e9 \u21a6 \u27e8a * b, h b hb\u27e9) _ ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a\u271d : G ha : a\u271d \u2208 S a : G s t : Set G x\u271d\u00b9 : Fintype \u2191s x\u271d : Fintype \u2191t h : \u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t \u22a2 Function.Injective fun x =>     match x with     | { val := b, property := hb } => { val := a * b, property := (_ : a * b \u2208 t) } ** rintro \u27e8b, hb\u27e9 \u27e8c, hc\u27e9 hbc ** case mk.mk F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a\u271d : G ha : a\u271d \u2208 S a : G s t : Set G x\u271d\u00b9 : Fintype \u2191s x\u271d : Fintype \u2191t h : \u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t b : G hb : b \u2208 s c : G hc : c \u2208 s hbc :   (fun x =>         match x with         | { val := b, property := hb } => { val := a * b, property := (_ : a * b \u2208 t) })       { val := b, property := hb } =     (fun x =>         match x with         | { val := b, property := hb } => { val := a * b, property := (_ : a * b \u2208 t) })       { val := c, property := hc } \u22a2 { val := b, property := hb } = { val := c, property := hc } ** exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc)) ** case h\u2082 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) \u22a2 {a} * S ^ n = S ^ (n + 1) ** have : Fintype (Set.singleton a * S ^ n) := by\n  classical!\n  apply fintypeMul ** case h\u2082 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) this : Fintype \u2191(Set.singleton a * S ^ n) \u22a2 {a} * S ^ n = S ^ (n + 1) ** refine' Set.eq_of_subset_of_card_le _ (le_trans (ge_of_eq h) _) ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) \u22a2 Fintype \u2191(Set.singleton a * S ^ n) ** classical! ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) em\u271d : (a : Prop) \u2192 Decidable a \u22a2 Fintype \u2191(Set.singleton a * S ^ n) ** apply fintypeMul ** case h\u2082.refine'_1 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) this : Fintype \u2191(Set.singleton a * S ^ n) \u22a2 {a} * S ^ n \u2286 S ^ (n + 1) ** exact mul_subset_mul (Set.singleton_subset_iff.mpr ha) Set.Subset.rfl ** case h\u2082.refine'_2 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 G : Type u_5 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : Fintype G S : Set G inst\u271d : (k : \u2115) \u2192 DecidablePred fun x => x \u2208 S ^ k hG : 0 < Fintype.card G hS : \u00acS = \u2205 a : G ha : a \u2208 S key :   \u2200 (a : G) (s t : Set G) [inst : Fintype \u2191s] [inst_1 : Fintype \u2191t],     (\u2200 (b : G), b \u2208 s \u2192 a * b \u2208 t) \u2192 Fintype.card \u2191s \u2264 Fintype.card \u2191t mono : Monotone fun n => Fintype.card \u2191(S ^ n) n : \u2115 h : Fintype.card \u2191(S ^ n) = Fintype.card \u2191(S ^ (n + 1)) this : Fintype \u2191(Set.singleton a * S ^ n) \u22a2 Fintype.card \u2191(S ^ n) \u2264 Fintype.card \u2191({a} * S ^ n) ** convert key a (S ^ n) ({a} * S ^ n) fun b hb \u21a6 Set.mul_mem_mul (Set.mem_singleton a) hb ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.tendsto_integral_truncation ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f \u22a2 Tendsto (fun A => \u222b (x : \u03b1), truncation f A x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : \u03b1), f x \u2202\u03bc)) ** refine' tendsto_integral_filter_of_dominated_convergence (fun x => abs (f x)) _ _ _ _ ** case refine'_1 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f \u22a2 \u2200\u1da0 (n : \u211d) in atTop, AEStronglyMeasurable (fun x => truncation f n x) \u03bc ** exact eventually_of_forall fun A => hf.aestronglyMeasurable.truncation ** case refine'_2 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f \u22a2 \u2200\u1da0 (n : \u211d) in atTop, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016truncation f n a\u2016 \u2264 (fun x => |f x|) a ** apply eventually_of_forall fun A => ?_ ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f A : \u211d \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016truncation f A a\u2016 \u2264 (fun x => |f x|) a ** apply eventually_of_forall fun x => ?_ ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f A : \u211d x : \u03b1 \u22a2 \u2016truncation f A x\u2016 \u2264 (fun x => |f x|) x ** rw [Real.norm_eq_abs] ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f A : \u211d x : \u03b1 \u22a2 |truncation f A x| \u2264 (fun x => |f x|) x ** exact abs_truncation_le_abs_self _ _ _ ** case refine'_3 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f \u22a2 Integrable fun x => |f x| ** apply hf.abs ** case refine'_4 \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => truncation f n a) atTop (\ud835\udcdd (f a)) ** apply eventually_of_forall fun x => ?_ ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f x : \u03b1 \u22a2 Tendsto (fun n => truncation f n x) atTop (\ud835\udcdd (f x)) ** apply tendsto_const_nhds.congr' _ ** \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f x : \u03b1 \u22a2 (fun x_1 => f x) =\u1da0[atTop] fun n => truncation f n x ** filter_upwards [Ioi_mem_atTop (abs (f x))] with A hA ** case h \u03b1 : Type u_1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f\u271d f : \u03b1 \u2192 \u211d hf : Integrable f x : \u03b1 A : \u211d hA : A \u2208 Set.Ioi |f x| \u22a2 f x = truncation f A x ** exact (truncation_eq_self hA).symm ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.gcd_gcd_self_left_right ** m n : Nat \u22a2 gcd (gcd n m) m = gcd n m ** rw [gcd_comm, gcd_gcd_self_right_right] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm_map_measure ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b2 : Type u_5 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 E hg : AEStronglyMeasurable g (Measure.map f \u03bc) hf : AEMeasurable f \u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc ** by_cases hp_zero : p = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b2 : Type u_5 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 E hg : AEStronglyMeasurable g (Measure.map f \u03bc) hf : AEMeasurable f hp_zero : \u00acp = 0 \u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc ** by_cases hp_top : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b2 : Type u_5 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 E hg : AEStronglyMeasurable g (Measure.map f \u03bc) hf : AEMeasurable f hp_zero : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc ** simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b2 : Type u_5 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 E hg : AEStronglyMeasurable g (Measure.map f \u03bc) hf : AEMeasurable f hp_zero : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 (\u222b\u207b (x : \u03b2), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202Measure.map f \u03bc) ^ (1 / ENNReal.toReal p) =     (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) ** rw [lintegral_map' (hg.ennnorm.pow_const p.toReal) hf] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b2 : Type u_5 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 E hg : AEStronglyMeasurable g (Measure.map f \u03bc) hf : AEMeasurable f hp_zero : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g (f a)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =     (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) ** rfl ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b2 : Type u_5 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 E hg : AEStronglyMeasurable g (Measure.map f \u03bc) hf : AEMeasurable f hp_zero : p = 0 \u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc ** simp only [hp_zero, snorm_exponent_zero] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b2 : Type u_5 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 E hg : AEStronglyMeasurable g (Measure.map f \u03bc) hf : AEMeasurable f hp_zero : \u00acp = 0 hp_top : p = \u22a4 \u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc ** simp_rw [hp_top, snorm_exponent_top] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G \u03b2 : Type u_5 m\u03b2 : MeasurableSpace \u03b2 f : \u03b1 \u2192 \u03b2 g : \u03b2 \u2192 E hg : AEStronglyMeasurable g (Measure.map f \u03bc) hf : AEMeasurable f hp_zero : \u00acp = 0 hp_top : p = \u22a4 \u22a2 snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc ** exact snormEssSup_map_measure hg hf ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.suppPreservation_iff_liftpPreservation ** n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F \u22a2 SuppPreservation \u2194 LiftPPreservation ** constructor <;> intro h ** case mp n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h : SuppPreservation \u22a2 LiftPPreservation ** rintro \u03b1 p \u27e8a, f\u27e9 ** case mp.mk n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h : SuppPreservation \u03b1 : TypeVec.{u} n p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 \u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f } ** have h' := h ** case mp.mk n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h : SuppPreservation \u03b1 : TypeVec.{u} n p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 h' : SuppPreservation \u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f } ** rw [suppPreservation_iff_isUniform] at h' ** case mp.mk n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h : SuppPreservation \u03b1 : TypeVec.{u} n p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 h' : IsUniform \u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f } ** dsimp only [SuppPreservation, supp] at h ** case mp.mk n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h :   \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (x : \u2191(P F) \u03b1),     (fun i => {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P (abs x) \u2192 P i y}) = fun i =>       {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P x \u2192 P i y} \u03b1 : TypeVec.{u} n p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 h' : IsUniform \u22a2 LiftP p (abs { fst := a, snd := f }) \u2194 LiftP p { fst := a, snd := f } ** simp only [liftP_iff_of_isUniform, supp_eq_of_isUniform, MvPFunctor.liftP_iff', h',\n  image_univ, mem_range, exists_imp] ** case mp.mk n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h :   \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (x : \u2191(P F) \u03b1),     (fun i => {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P (abs x) \u2192 P i y}) = fun i =>       {y | \u2200 \u2983P : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\u2984, LiftP P x \u2192 P i y} \u03b1 : TypeVec.{u} n p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 h' : IsUniform \u22a2 (\u2200 (i : Fin2 n) (u : \u03b1 i) (x : MvPFunctor.B (P F) a i), f i x = u \u2192 p u) \u2194     \u2200 (i : Fin2 n) (x : MvPFunctor.B (P F) a i), p (f i x) ** constructor <;> intros <;> subst_vars <;> solve_by_elim ** case mpr n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h : LiftPPreservation \u22a2 SuppPreservation ** rintro \u03b1 \u27e8a, f\u27e9 ** case mpr.mk n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h : LiftPPreservation \u03b1 : TypeVec.{u} n a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 \u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f } ** simp only [LiftPPreservation] at h ** case mpr.mk n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h : \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop) (x : \u2191(P F) \u03b1), LiftP p (abs x) \u2194 LiftP p x \u03b1 : TypeVec.{u} n a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 \u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f } ** ext ** case mpr.mk.h.h n : \u2115 F : TypeVec.{u} n \u2192 Type u_1 inst\u271d : MvFunctor F q : MvQPF F h : \u2200 \u2983\u03b1 : TypeVec.{u} n\u2984 (p : \u2983i : Fin2 n\u2984 \u2192 \u03b1 i \u2192 Prop) (x : \u2191(P F) \u03b1), LiftP p (abs x) \u2194 LiftP p x \u03b1 : TypeVec.{u} n a : (P F).A f : MvPFunctor.B (P F) a \u27f9 \u03b1 x\u271d\u00b9 : Fin2 n x\u271d : \u03b1 x\u271d\u00b9 \u22a2 x\u271d \u2208 supp (abs { fst := a, snd := f }) x\u271d\u00b9 \u2194 x\u271d \u2208 supp { fst := a, snd := f } x\u271d\u00b9 ** simp only [supp, h, mem_setOf_eq] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.filterMap_eq_filterMapTR ** \u22a2 @filterMap = @filterMapTR ** funext \u03b1 \u03b2 f l ** case h.h.h.h \u03b1 : Type u_2 \u03b2 : Type u_1 f : \u03b1 \u2192 Option \u03b2 l : List \u03b1 \u22a2 filterMap f l = filterMapTR f l ** exact (go l #[]).symm ** \u03b1 : Type u_2 \u03b2 : Type u_1 f : \u03b1 \u2192 Option \u03b2 l : List \u03b1 acc : Array \u03b2 \u22a2 filterMapTR.go f [] acc = acc.data ++ filterMap f [] ** simp [filterMapTR.go, filterMap] ** \u03b1 : Type u_2 \u03b2 : Type u_1 f : \u03b1 \u2192 Option \u03b2 l : List \u03b1 a : \u03b1 as : List \u03b1 acc : Array \u03b2 \u22a2 filterMapTR.go f (a :: as) acc = acc.data ++ filterMap f (a :: as) ** simp [filterMapTR.go, filterMap, go as] ** \u03b1 : Type u_2 \u03b2 : Type u_1 f : \u03b1 \u2192 Option \u03b2 l : List \u03b1 a : \u03b1 as : List \u03b1 acc : Array \u03b2 \u22a2 (match f a with     | none => acc.data ++ filterMap f as     | some b => acc.data ++ b :: filterMap f as) =     acc.data ++       match f a with       | none => filterMap f as       | some b => b :: filterMap f as ** split <;> simp [*] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.IndepFun.ae_eq ** \u03a9 : Type u_1 \u03b9 : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d : \u03a9 \u2192 \u03b2 g\u271d : \u03a9 \u2192 \u03b2' m\u03b2 : MeasurableSpace \u03b2 f g f' g' : \u03a9 \u2192 \u03b2 hfg : IndepFun f g hf : f =\u1d50[\u03bc] f' hg : g =\u1d50[\u03bc] g' \u22a2 IndepFun f' g' ** refine kernel.IndepFun.ae_eq hfg ?_ ?_ <;>\n  simp only [ae_dirac_eq, Filter.eventually_pure, kernel.const_apply] ** case refine_1 \u03a9 : Type u_1 \u03b9 : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d : \u03a9 \u2192 \u03b2 g\u271d : \u03a9 \u2192 \u03b2' m\u03b2 : MeasurableSpace \u03b2 f g f' g' : \u03a9 \u2192 \u03b2 hfg : IndepFun f g hf : f =\u1d50[\u03bc] f' hg : g =\u1d50[\u03bc] g' \u22a2 f =\u1d50[\u03bc] f'  case refine_2 \u03a9 : Type u_1 \u03b9 : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 m\u03a9 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 f\u271d : \u03a9 \u2192 \u03b2 g\u271d : \u03a9 \u2192 \u03b2' m\u03b2 : MeasurableSpace \u03b2 f g f' g' : \u03a9 \u2192 \u03b2 hfg : IndepFun f g hf : f =\u1d50[\u03bc] f' hg : g =\u1d50[\u03bc] g' \u22a2 g =\u1d50[\u03bc] g' ** exacts [hf, hg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.addHaar_smul ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F r : \u211d s : Set E \u22a2 \u2191\u2191\u03bc (r \u2022 s) = ENNReal.ofReal |r ^ finrank \u211d E| * \u2191\u2191\u03bc s ** rcases ne_or_eq r 0 with (h | rfl) ** case inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E \u22a2 \u2191\u2191\u03bc (0 \u2022 s) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s ** rcases eq_empty_or_nonempty s with (rfl | hs) ** case inr.inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : Set.Nonempty s \u22a2 \u2191\u2191\u03bc (0 \u2022 s) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s ** rw [zero_smul_set hs, \u2190 singleton_zero] ** case inr.inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : Set.Nonempty s \u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s ** by_cases h : finrank \u211d E = 0 ** case inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F r : \u211d s : Set E h : r \u2260 0 \u22a2 \u2191\u2191\u03bc (r \u2022 s) = ENNReal.ofReal |r ^ finrank \u211d E| * \u2191\u2191\u03bc s ** rw [\u2190 preimage_smul_inv\u2080 h, addHaar_preimage_smul \u03bc (inv_ne_zero h), inv_pow, inv_inv] ** case inr.inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u22a2 \u2191\u2191\u03bc (0 \u2022 \u2205) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc \u2205 ** simp only [measure_empty, mul_zero, smul_set_empty] ** case pos E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : Set.Nonempty s h : finrank \u211d E = 0 \u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s ** haveI : Subsingleton E := finrank_zero_iff.1 h ** case pos E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : Set.Nonempty s h : finrank \u211d E = 0 this : Subsingleton E \u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s ** simp only [h, one_mul, ENNReal.ofReal_one, abs_one, Subsingleton.eq_univ_of_nonempty hs,\n  pow_zero, Subsingleton.eq_univ_of_nonempty (singleton_nonempty (0 : E))] ** case neg E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : Set.Nonempty s h : \u00acfinrank \u211d E = 0 \u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s ** haveI : Nontrivial E := nontrivial_of_finrank_pos (bot_lt_iff_ne_bot.2 h) ** case neg E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s : Set E hs : Set.Nonempty s h : \u00acfinrank \u211d E = 0 this : Nontrivial E \u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s ** simp only [h, zero_mul, ENNReal.ofReal_zero, abs_zero, Ne.def, not_false_iff,\n  zero_pow', measure_singleton] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.martingale_of_condexp_sub_eq_zero_nat ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 \u22a2 Martingale f \ud835\udca2 \u03bc ** refine' martingale_iff.2 \u27e8supermartingale_of_condexp_sub_nonneg_nat hadp hint fun i => _,\n  submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => (hf i).symm.le\u27e9 ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 i : \u2115 \u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i] ** rw [\u2190 neg_sub] ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 i : \u2115 \u22a2 0 \u2264\u1d50[\u03bc] \u03bc[-(f (i + 1) - f i)|\u2191\ud835\udca2 i] ** refine' (EventuallyEq.trans _ (condexp_neg _).symm).le ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 i : \u2115 \u22a2 0 =\u1d50[\u03bc] -\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] ** filter_upwards [hf i] with x hx ** case h \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2074 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 \ud835\udca2 : Filtration \u2115 m0 inst\u271d : IsFiniteMeasure \u03bc f : \u2115 \u2192 \u03a9 \u2192 \u211d hadp : Adapted \ud835\udca2 f hint : \u2200 (i : \u2115), Integrable (f i) hf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0 i : \u2115 x : \u03a9 hx : (\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x = OfNat.ofNat 0 x \u22a2 OfNat.ofNat 0 x = (-\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x ** simpa only [Pi.zero_apply, Pi.neg_apply, zero_eq_neg] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.maximal_ineq ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) hadd :   ENNReal.ofReal (\u222b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) =     ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}, f n \u03c9 \u2202\u03bc) +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}, f n \u03c9 \u2202\u03bc) \u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2264     ENNReal.ofReal       (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}, f n \u03c9 \u2202\u03bc) ** rwa [hadd, ENNReal.add_le_add_iff_right ENNReal.ofReal_ne_top] at this ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 ENNReal.ofReal (\u222b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) =     ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}, f n \u03c9 \u2202\u03bc) +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}, f n \u03c9 \u2202\u03bc) ** rw [\u2190 ENNReal.ofReal_add, \u2190 integral_union] ** case hfs \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 IntegrableOn (fun \u03c9 => f n \u03c9) {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}  case hft \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 IntegrableOn (fun \u03c9 => f n \u03c9) {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}  case hp \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 0 \u2264 \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}, f n \u03c9 \u2202\u03bc  case hq \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 0 \u2264 \u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5}, f n \u03c9 \u2202\u03bc ** exacts [(hsub.integrable _).integrableOn, (hsub.integrable _).integrableOn,\n  integral_nonneg (hnonneg _), integral_nonneg (hnonneg _)] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 ENNReal.ofReal (\u222b (\u03c9 : \u03a9), f n \u03c9 \u2202\u03bc) =     ENNReal.ofReal       (\u222b (x : \u03a9) in         {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a           {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},         f n x \u2202\u03bc) ** rw [\u2190 integral_univ] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 ENNReal.ofReal (\u222b (x : \u03a9) in Set.univ, f n x \u2202\u03bc) =     ENNReal.ofReal       (\u222b (x : \u03a9) in         {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a           {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},         f n x \u2202\u03bc) ** convert rfl ** case h.e'_3.h.e'_1.h.e'_6.h.e'_4 \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a       {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} =     Set.univ ** ext \u03c9 ** case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u03c9 : \u03a9 \u22a2 \u03c9 \u2208       {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a         {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2194     \u03c9 \u2208 Set.univ ** change (\u03b5 : \u211d) \u2264 _ \u2228 _ < (\u03b5 : \u211d) \u2194 _ ** case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u03c9 : \u03a9 \u22a2 (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2228       (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 \u2194     \u03c9 \u2208 Set.univ ** simp only [le_or_lt, Set.mem_univ] ** case hst \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 Disjoint {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}     {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} ** rw [disjoint_iff_inf_le] ** case hst \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2293       {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2264     \u22a5 ** rintro \u03c9 \u27e8h\u03c9\u2081, h\u03c9\u2082\u27e9 ** case hst.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u03c9 : \u03a9 h\u03c9\u2081 : \u03c9 \u2208 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} h\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u22a2 \u03c9 \u2208 \u22a5 ** change (\u03b5 : \u211d) \u2264 _ at h\u03c9\u2081 ** case hst.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u03c9 : \u03a9 h\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} h\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9 \u22a2 \u03c9 \u2208 \u22a5 ** change _ < (\u03b5 : \u211d) at h\u03c9\u2082 ** case hst.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u03c9 : \u03a9 h\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9 h\u03c9\u2082 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 \u22a2 \u03c9 \u2208 \u22a5 ** exact (not_le.2 h\u03c9\u2082) h\u03c9\u2081 ** case ht \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 this :   \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) \u22a2 MeasurableSet {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} ** exact measurableSet_lt (Finset.measurable_range_sup'' fun n _ =>\n  (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)) measurable_const ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 \u03b5 \u2022 \u2191\u2191\u03bc {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           f n \u03c9 \u2202\u03bc) \u2264     ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},           stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) ** refine' add_le_add (smul_le_stoppedValue_hitting hsub _)\n  (ENNReal.ofReal_le_ofReal (set_integral_mono_on (hsub.integrable n).integrableOn\n    (Integrable.integrableOn (hsub.integrable_stoppedValue\n      (hitting_isStoppingTime hsub.adapted measurableSet_Ici) hitting_le))\n        (measurableSet_lt (Finset.measurable_range_sup'' fun n _ =>\n          (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)) measurable_const) _)) ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 \u2200 (x : \u03a9),     x \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2192       f n x \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x ** intro \u03c9 h\u03c9 ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u22a2 f n \u03c9 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 ** rw [Set.mem_setOf_eq] at h\u03c9 ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 \u22a2 f n \u03c9 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 ** have : hitting f {y : \u211d | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n := by\n  classical simp only [hitting, Set.mem_setOf_eq, exists_prop, Pi.coe_nat, Nat.cast_id,\n    ite_eq_right_iff, forall_exists_index, and_imp]\n  intro m hm h\u03b5m\n  exact False.elim\n    ((not_le.2 h\u03c9) ((le_sup'_iff _).2 \u27e8m, mem_range.2 (Nat.lt_succ_of_le hm.2), h\u03b5m\u27e9)) ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 this : hitting f {y | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n \u22a2 f n \u03c9 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 ** simp_rw [stoppedValue, this, le_rfl] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 \u22a2 hitting f {y | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n ** classical simp only [hitting, Set.mem_setOf_eq, exists_prop, Pi.coe_nat, Nat.cast_id,\n  ite_eq_right_iff, forall_exists_index, and_imp] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 \u22a2 \u2200 (x : \u2115), x \u2208 Set.Icc 0 n \u2192 \u2191\u03b5 \u2264 f x \u03c9 \u2192 sInf (Set.Icc 0 n \u2229 {i | \u2191\u03b5 \u2264 f i \u03c9}) = n ** intro m hm h\u03b5m ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 m : \u2115 hm : m \u2208 Set.Icc 0 n h\u03b5m : \u2191\u03b5 \u2264 f m \u03c9 \u22a2 sInf (Set.Icc 0 n \u2229 {i | \u2191\u03b5 \u2264 f i \u03c9}) = n ** exact False.elim\n  ((not_le.2 h\u03c9) ((le_sup'_iff _).2 \u27e8m, mem_range.2 (Nat.lt_succ_of_le hm.2), h\u03b5m\u27e9)) ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 \u22a2 hitting f {y | \u2191\u03b5 \u2264 y} 0 n \u03c9 = n ** simp only [hitting, Set.mem_setOf_eq, exists_prop, Pi.coe_nat, Nat.cast_id,\nite_eq_right_iff, forall_exists_index, and_imp] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},           stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) +       ENNReal.ofReal         (\u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},           stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) =     ENNReal.ofReal (\u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) ** rw [\u2190 ENNReal.ofReal_add, \u2190 integral_union] ** case hp \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 0 \u2264     \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9},       stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc  case hq \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 0 \u2264     \u222b (\u03c9 : \u03a9) in {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},       stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc ** exacts [integral_nonneg fun x => hnonneg _ _, integral_nonneg fun x => hnonneg _ _] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 ENNReal.ofReal       (\u222b (x : \u03a9) in         {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a           {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},         stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc) =     ENNReal.ofReal (\u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) ** rw [\u2190 integral_univ (\u03bc := \u03bc)] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 ENNReal.ofReal       (\u222b (x : \u03a9) in         {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a           {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5},         stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc) =     ENNReal.ofReal (\u222b (x : \u03a9) in Set.univ, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc) ** convert rfl ** case h.e'_3.h.e'_1.h.e'_6.h.e'_4 \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 Set.univ =     {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a       {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} ** ext \u03c9 ** case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 \u22a2 \u03c9 \u2208 Set.univ \u2194     \u03c9 \u2208       {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u222a         {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} ** change _ \u2194 (\u03b5 : \u211d) \u2264 _ \u2228 _ < (\u03b5 : \u211d) ** case h.e'_3.h.e'_1.h.e'_6.h.e'_4.h \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 \u22a2 \u03c9 \u2208 Set.univ \u2194     (\u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) \u2228       (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 ** simp only [le_or_lt, Set.mem_univ] ** case hst \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 Disjoint {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9}     {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} ** rw [disjoint_iff_inf_le] ** case hst \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} \u2293       {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u2264     \u22a5 ** rintro \u03c9 \u27e8h\u03c9\u2081, h\u03c9\u2082\u27e9 ** case hst.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9\u2081 : \u03c9 \u2208 {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} h\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} \u22a2 \u03c9 \u2208 \u22a5 ** change (\u03b5 : \u211d) \u2264 _ at h\u03c9\u2081 ** case hst.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9\u2082 : \u03c9 \u2208 {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} h\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9 \u22a2 \u03c9 \u2208 \u22a5 ** change _ < (\u03b5 : \u211d) at h\u03c9\u2082 ** case hst.intro \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u03c9 : \u03a9 h\u03c9\u2081 : \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9 h\u03c9\u2082 : (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5 \u22a2 \u03c9 \u2208 \u22a5 ** exact (not_le.2 h\u03c9\u2082) h\u03c9\u2081 ** case ht \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 MeasurableSet {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} ** exact measurableSet_lt (Finset.measurable_range_sup'' fun n _ =>\n  (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)) measurable_const ** case hfs \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 IntegrableOn (fun \u03c9 => stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9)     {\u03c9 | \u2191\u03b5 \u2264 sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9} ** exact Integrable.integrableOn (hsub.integrable_stoppedValue\n  (hitting_isStoppingTime hsub.adapted measurableSet_Ici) hitting_le) ** case hft \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 IntegrableOn (fun \u03c9 => stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9)     {\u03c9 | (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k \u03c9) < \u2191\u03b5} ** exact Integrable.integrableOn (hsub.integrable_stoppedValue\n  (hitting_isStoppingTime hsub.adapted measurableSet_Ici) hitting_le) ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 ENNReal.ofReal (\u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc) \u2264 ENNReal.ofReal (\u222b (x : \u03a9), f n x \u2202\u03bc) ** refine' ENNReal.ofReal_le_ofReal _ ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 \u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc \u2264 \u222b (x : \u03a9), f n x \u2202\u03bc ** rw [\u2190 stoppedValue_const f n] ** \u03a9 : Type u_1 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 \ud835\udca2 : Filtration \u2115 m0 f : \u2115 \u2192 \u03a9 \u2192 \u211d \u03c4 \u03c0 : \u03a9 \u2192 \u2115 inst\u271d : IsFiniteMeasure \u03bc hsub : Submartingale f \ud835\udca2 \u03bc hnonneg : 0 \u2264 f \u03b5 : \u211d\u22650 n : \u2115 \u22a2 \u222b (\u03c9 : \u03a9), stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc \u2264 \u222b (x : \u03a9), stoppedValue f (fun x => n) x \u2202\u03bc ** exact hsub.expected_stoppedValue_mono (hitting_isStoppingTime hsub.adapted measurableSet_Ici)\n  (isStoppingTime_const _ _) (fun \u03c9 => hitting_le \u03c9) (fun _ => le_rfl : \u2200 _, n \u2264 n) ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.disjoint_pi ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 \u03b2 : \u03b9 \u2192 Type u_3 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) i : \u03b9 inst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i) \u22a2 Disjoint (pi s t\u2081) (pi s t\u2082) \u2194 \u2203 i, i \u2208 s \u2227 Disjoint (t\u2081 i) (t\u2082 i) ** simp only [disjoint_iff_inter_eq_empty, \u2190 pi_inter_distrib, pi_eq_empty_iff'] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_mul_left_Icc ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a b c : \u03b1 ha : 0 \u2264 a hbc : b \u2264 c \u22a2 (fun x x_1 => x * x_1) a '' Icc b c = Icc (a * b) (a * c) ** convert image_mul_right_Icc hbc ha using 1 <;> simp only [mul_comm _ a] ** Qed",
        "informal": ""
    },
    {
        "formal": "exists_signed_sum_aux ** \u03b1\u271d \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 \u22a2 \u2203 \u03b2 t sgn g,     (\u2200 (b : \u03b2), g b \u2208 s) \u2227       card t = \u2211 a in s, Int.natAbs (f a) \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 b in t, if g b = a then \u2191(sgn b) else 0) = f a ** refine'\n  \u27e8(\u03a3 _ : { x // x \u2208 s }, \u2115), Finset.univ.sigma fun a => range (f a).natAbs,\n    fun a => sign (f a.1), fun a => a.1, fun a => a.1.2, _, _\u27e9 ** case refine'_1 \u03b1\u271d \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 \u22a2 card (Finset.sigma univ fun a => range (Int.natAbs (f \u2191a))) = \u2211 a in s, Int.natAbs (f a) ** simp [sum_attach (f := fun a => (f a).natAbs)] ** case refine'_2 \u03b1\u271d \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 \u22a2 \u2200 (a : \u03b1),     a \u2208 s \u2192       (\u2211 b in Finset.sigma univ fun a => range (Int.natAbs (f \u2191a)),           if (fun a => \u2191a.fst) b = a then \u2191((fun a => \u2191sign (f \u2191a.fst)) b) else 0) =         f a ** intro x hx ** case refine'_2 \u03b1\u271d \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : \u03b1 \u2192 \u2124 x : \u03b1 hx : x \u2208 s \u22a2 (\u2211 b in Finset.sigma univ fun a => range (Int.natAbs (f \u2191a)),       if (fun a => \u2191a.fst) b = x then \u2191((fun a => \u2191sign (f \u2191a.fst)) b) else 0) =     f x ** simp [sum_sigma, hx, \u2190 Int.sign_eq_sign, Int.sign_mul_abs, mul_comm |f _|,\n  sum_attach (s := s) (f := fun y => if y = x then f y else 0)] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.RBNode.size_lt_depth ** \u03b1 : Type u_1 \u22a2 0 < 1 ** decide ** \u03b1 : Type u_1 c\u271d : RBColor a : RBNode \u03b1 v\u271d : \u03b1 b : RBNode \u03b1 \u22a2 size (node c\u271d a v\u271d b) < 2 ^ depth (node c\u271d a v\u271d b) ** rw [size, depth, Nat.add_right_comm, Nat.pow_succ, Nat.mul_two] ** \u03b1 : Type u_1 c\u271d : RBColor a : RBNode \u03b1 v\u271d : \u03b1 b : RBNode \u03b1 \u22a2 size a + 1 + size b < 2 ^ max (depth a) (depth b) + 2 ^ max (depth a) (depth b) ** refine Nat.add_le_add\n  (Nat.lt_of_lt_of_le a.size_lt_depth ?_) (Nat.lt_of_lt_of_le b.size_lt_depth ?_) ** case refine_1 \u03b1 : Type u_1 c\u271d : RBColor a : RBNode \u03b1 v\u271d : \u03b1 b : RBNode \u03b1 \u22a2 2 ^ depth a \u2264 2 ^ max (depth a) (depth b) ** exact Nat.pow_le_pow_of_le_right (by decide) (Nat.le_max_left ..) ** \u03b1 : Type u_1 c\u271d : RBColor a : RBNode \u03b1 v\u271d : \u03b1 b : RBNode \u03b1 \u22a2 2 > 0 ** decide ** case refine_2 \u03b1 : Type u_1 c\u271d : RBColor a : RBNode \u03b1 v\u271d : \u03b1 b : RBNode \u03b1 \u22a2 2 ^ depth b \u2264 2 ^ max (depth a) (depth b) ** exact Nat.pow_le_pow_of_le_right (by decide) (Nat.le_max_right ..) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_liminf_eq_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc (liminf s atTop) = 0 ** rw [\u2190 le_zero_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc (liminf s atTop) \u2264 0 ** have : liminf s atTop \u2264 limsup s atTop := liminf_le_limsup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4 this : liminf s atTop \u2264 limsup s atTop \u22a2 \u2191\u2191\u03bc (liminf s atTop) \u2264 0 ** exact (\u03bc.mono this).trans (by simp [measure_limsup_eq_zero h]) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 s : \u2115 \u2192 Set \u03b1 h : \u2211' (i : \u2115), \u2191\u2191\u03bc (s i) \u2260 \u22a4 this : liminf s atTop \u2264 limsup s atTop \u22a2 \u2191\u2191\u03bc (limsup s atTop) \u2264 0 ** simp [measure_limsup_eq_zero h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux2 ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** have :\n  Tendsto (fun \u03b5 : \u211d\u22650 => (\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) + 2 * \u03b5 * \u03bc s) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd ((\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) + 2 * (0 : \u211d\u22650) * \u03bc s)) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  refine' tendsto_const_nhds.add _\n  refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)\n  exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x this :   Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s)) \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** simp only [add_zero, zero_mul, mul_zero, ENNReal.coe_zero] at this ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x this :   Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc)) \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc ** apply ge_of_tendsto this ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x this :   Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc)) \u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,     \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191c * \u2191\u2191\u03bc s ** filter_upwards [self_mem_nhdsWithin] ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x this :   Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc)) \u22a2 \u2200 (a : \u211d\u22650),     a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191a * \u2191\u2191\u03bc s ** rintro \u03b5 (\u03b5pos : 0 < \u03b5) ** case h E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x this :   Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc)) \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 \u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s ** exact addHaar_image_le_lintegral_abs_det_fderiv_aux1 \u03bc hs hf' \u03b5pos ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u22a2 Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)     (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s)) ** apply Tendsto.mono_left _ nhdsWithin_le_nhds ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u22a2 Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0)     (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s)) ** refine' tendsto_const_nhds.add _ ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910 * \u2191\u2191\u03bc s)) ** refine' ENNReal.Tendsto.mul_const _ (Or.inr h's) ** E : Type u_1 F : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : FiniteDimensional \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F s : Set E f : E \u2192 E f' : E \u2192 E \u2192L[\u211d] E inst\u271d\u00b2 : MeasurableSpace E inst\u271d\u00b9 : BorelSpace E \u03bc : Measure E inst\u271d : IsAddHaarMeasure \u03bc hs : MeasurableSet s h's : \u2191\u2191\u03bc s \u2260 \u22a4 hf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x \u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910)) ** exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_congr_left ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f : \u03b1 \u2192 E \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f ** by_cases hf : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f : \u03b1 \u2192 E hf : Integrable f \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f ** simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left T T' hT hT' h] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' h : T = T' f : \u03b1 \u2192 E hf : \u00acIntegrable f \u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f ** simp_rw [setToFun_undef _ hf] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_const_add_Ici ** M : Type u_1 inst\u271d\u00b9 : OrderedCancelAddCommMonoid M inst\u271d : ExistsAddOfLE M a b c d : M \u22a2 (fun x => a + x) '' Ici b = Ici (a + b) ** simp only [add_comm a, image_add_const_Ici] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_eq_div_smul ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd hs : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 \u22a2 \u03bc = (\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd ** ext1 t ht ** case h G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd hs : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 t : Set G ht : MeasurableSet t \u22a2 \u2191\u2191\u03bc t = \u2191\u2191((\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd) t ** rw [smul_apply, smul_eq_mul, mul_comm, \u2190 mul_div_assoc, mul_comm,\n  measure_mul_measure_eq \u03bc \u03bd hs ht h2s h3s, mul_div_assoc, ENNReal.mul_div_cancel' h2s h3s] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.map_add_left_Icc ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1 inst\u271d\u00b9 : ExistsAddOfLE \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 \u22a2 map (addLeftEmbedding c) (Icc a b) = Icc (c + a) (c + b) ** rw [\u2190 coe_inj, coe_map, coe_Icc, coe_Icc] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1 inst\u271d\u00b9 : ExistsAddOfLE \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 \u22a2 \u2191(addLeftEmbedding c) '' Set.Icc a b = Set.Icc (c + a) (c + b) ** exact Set.image_const_add_Icc _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_map ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 hf : Measurable f hg : Measurable g \u22a2 \u222b\u207b (a : \u03b2), f a \u2202Measure.map g \u03bc = \u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc ** erw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral (hf.comp hg)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 hf : Measurable f hg : Measurable g \u22a2 \u2a06 n, SimpleFunc.lintegral (eapprox f n) (Measure.map g \u03bc) = \u2a06 n, SimpleFunc.lintegral (eapprox (f \u2218 g) n) \u03bc ** congr with n : 1 ** case e_s.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 hf : Measurable f hg : Measurable g n : \u2115 \u22a2 SimpleFunc.lintegral (eapprox f n) (Measure.map g \u03bc) = SimpleFunc.lintegral (eapprox (f \u2218 g) n) \u03bc ** convert SimpleFunc.lintegral_map _ hg ** case h.e'_3.h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 hf : Measurable f hg : Measurable g n : \u2115 \u22a2 eapprox (f \u2218 g) n = comp (eapprox f n) g hg ** ext1 x ** case h.e'_3.h.e'_3.H \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 hf : Measurable f hg : Measurable g n : \u2115 x : \u03b1 \u22a2 \u2191(eapprox (f \u2218 g) n) x = \u2191(comp (eapprox f n) g hg) x ** simp only [eapprox_comp hf hg, coe_comp] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_union_add_inter' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 hs : MeasurableSet s t : Set \u03b1 \u22a2 \u2191\u2191\u03bc (s \u222a t) + \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t ** rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.strong_law_aux1 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -             \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <         \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** have c_pos : 0 < c := zero_lt_one.trans c_one ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -             \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <         \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** have hX : \u2200 i, AEStronglyMeasurable (X i) \u2119 := fun i =>\n  (hident i).symm.aestronglyMeasurable_snd hint.1 ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -             \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <         \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** have A : \u2200 i, StronglyMeasurable (indicator (Set.Ioc (-i : \u211d) i) id) := fun i =>\n  stronglyMeasurable_id.indicator measurableSet_Ioc ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 -             \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) (fun i => truncation (X i) \u2191i) a| <         \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** set Y := fun n : \u2115 => truncation (X n) n ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** set S := fun n => \u2211 i in range n, Y i with hS ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** let u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** have u_mono : Monotone u := fun i j hij => Nat.floor_mono (pow_le_pow c_one.le hij) ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** have I1 : \u2200 K, \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264 2 * \ud835\udd3c[X 0] := by\n  intro K\n  calc\n    \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264\n        \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[truncation (X 0) j ^ 2] := by\n      apply sum_le_sum fun j _ => ?_\n      refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _))\n      rw [(hident j).truncation.variance_eq]\n      exact variance_le_expectation_sq (hX 0).truncation\n    _ \u2264 2 * \ud835\udd3c[X 0] := sum_variance_truncation_le hint (hnonneg 0) K ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** let C := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \ud835\udd3c[X 0]) ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C I3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** have I4 : (\u2211' i, \u2119 {\u03c9 | (u i * \u03b5 : \u211d) \u2264 |S (u i) \u03c9 - \ud835\udd3c[S (u i)]|}) < \u221e :=\n  (le_of_tendsto_of_tendsto' (ENNReal.tendsto_nat_tsum _) tendsto_const_nhds I3).trans_lt\n    ENNReal.ofReal_lt_top ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C I3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) I4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4 \u22a2 \u2200\u1d50 (\u03c9 : \u03a9),     \u2200\u1da0 (n : \u2115) in atTop,       |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** filter_upwards [ae_eventually_not_mem I4.ne] with \u03c9 h\u03c9 ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C I3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) I4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4 \u03c9 : \u03a9 h\u03c9 : \u2200\u1da0 (n : \u2115) in atTop, \u00ac\u2191(u n) * \u03b5 \u2264 |S (u n) \u03c9 - \u222b (a : \u03a9), S (u n) a| \u22a2 \u2200\u1da0 (n : \u2115) in atTop,     |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** simp_rw [not_le, mul_comm, sum_apply] at h\u03c9 ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C I3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) I4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4 \u03c9 : \u03a9 h\u03c9 :   \u2200\u1da0 (n : \u2115) in atTop,     |\u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) \u03c9 - \u222b (a : \u03a9), \u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) a| <       \u03b5 * \u2191\u230ac ^ n\u230b\u208a \u22a2 \u2200\u1da0 (n : \u2115) in atTop,     |\u2211 i in range \u230ac ^ n\u230b\u208a, truncation (X i) (\u2191i) \u03c9 - \u222b (a : \u03a9), Finset.sum (range \u230ac ^ n\u230b\u208a) Y a| < \u03b5 * \u2191\u230ac ^ n\u230b\u208a ** convert h\u03c9 ** case h.e'_2.h.h.e'_3.h.e'_3.h.e'_6.h.e'_7.h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C I3 : \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) I4 : \u2211' (i : \u2115), \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} < \u22a4 \u03c9 : \u03a9 h\u03c9 :   \u2200\u1da0 (n : \u2115) in atTop,     |\u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) \u03c9 - \u222b (a : \u03a9), \u2211 c in range \u230ac ^ n\u230b\u208a, truncation (X c) (\u2191c) a| <       \u03b5 * \u2191\u230ac ^ n\u230b\u208a x\u271d\u00b9 : \u2115 x\u271d : \u03a9 \u22a2 Finset.sum (range \u230ac ^ x\u271d\u00b9\u230b\u208a) Y x\u271d = \u2211 c in range \u230ac ^ x\u271d\u00b9\u230b\u208a, truncation (X c) (\u2191c) x\u271d ** simp only [sum_apply] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u \u22a2 \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a ** intro K ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u K : \u2115 \u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a ** calc\n  \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * Var[Y j] \u2264\n      \u2211 j in range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[truncation (X 0) j ^ 2] := by\n    apply sum_le_sum fun j _ => ?_\n    refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _))\n    rw [(hident j).truncation.variance_eq]\n    exact variance_le_expectation_sq (hX 0).truncation\n  _ \u2264 2 * \ud835\udd3c[X 0] := sum_variance_truncation_le hint (hnonneg 0) K ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u K : \u2115 \u22a2 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a ** apply sum_le_sum fun j _ => ?_ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u K j : \u2115 x\u271d : j \u2208 range K \u22a2 (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 (\u2191j ^ 2)\u207b\u00b9 * \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a ** refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _)) ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u K j : \u2115 x\u271d : j \u2208 range K \u22a2 variance (Y j) \u2119 \u2264 \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a ** rw [(hident j).truncation.variance_eq] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u K j : \u2115 x\u271d : j \u2208 range K \u22a2 variance (truncation (X 0) \u2191j) \u2119 \u2264 \u222b (a : \u03a9), (truncation (X 0) \u2191j ^ 2) a ** exact variance_le_expectation_sq (hX 0).truncation ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) \u22a2 \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C ** intro N ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 =     \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119 ** congr 1 with i ** case e_f.h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i : \u2115 \u22a2 (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 = (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119 ** congr 1 ** case e_f.h.e_a \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i : \u2115 \u22a2 variance (S (u i)) \u2119 = \u2211 j in range (u i), variance (Y j) \u2119 ** rw [hS, IndepFun.variance_sum] ** case e_f.h.e_a.hs \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i : \u2115 \u22a2 \u2200 (i_1 : \u2115), i_1 \u2208 range (u i) \u2192 Mem\u2112p (Y i_1) 2 ** intro j _ ** case e_f.h.e_a.hs \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i j : \u2115 a\u271d : j \u2208 range (u i) \u22a2 Mem\u2112p (Y j) 2 ** exact (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation ** case e_f.h.e_a.h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i : \u2115 \u22a2 Set.Pairwise \u2191(range (u i)) fun i j => IndepFun (Y i) (Y j) ** intro k _ l _ hkl ** case e_f.h.e_a.h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i k : \u2115 a\u271d\u00b9 : k \u2208 \u2191(range (u i)) l : \u2115 a\u271d : l \u2208 \u2191(range (u i)) hkl : k \u2260 l \u22a2 IndepFun (Y k) (Y l) ** exact (hindep hkl).comp (A k).measurable (A l).measurable ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * \u2211 j in range (u i), variance (Y j) \u2119 =     \u2211 j in range (u (N - 1)), (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 ** simp_rw [mul_sum, sum_mul, sum_sigma'] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 \u2211 x in Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a,       (\u2191\u230ac ^ x.fst\u230b\u208a ^ 2)\u207b\u00b9 * variance (truncation (X x.snd) \u2191x.snd) \u2119 =     \u2211 x in Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N),       (\u2191\u230ac ^ x.snd\u230b\u208a ^ 2)\u207b\u00b9 * variance (truncation (X x.fst) \u2191x.fst) \u2119 ** refine' sum_bij' (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun a _ => rfl)\n  (fun (p : \u03a3 _ : \u2115, \u2115) _ => (\u27e8p.2, p.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ _ _ ** case refine'_1 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a),     (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208       Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N) ** rintro \u27e8i, j\u27e9 hij ** case refine'_1.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208     Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N) ** simp only [mem_sigma, mem_range] at hij ** case refine'_1.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i j : \u2115 hij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a hij : i < N \u2227 j < \u230ac ^ i\u230b\u208a \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208     Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N) ** simp only [hij.1, hij.2, mem_sigma, mem_range, mem_filter, and_true_iff] ** case refine'_1.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i j : \u2115 hij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a hij : i < N \u2227 j < \u230ac ^ i\u230b\u208a \u22a2 j < \u230ac ^ (N - 1)\u230b\u208a ** exact hij.2.trans_le (u_mono (Nat.le_pred_of_lt hij.1)) ** case refine'_2 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)),     (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a ** rintro \u27e8i, j\u27e9 hij ** case refine'_2.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N) \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208     Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a ** simp only [mem_sigma, mem_range, mem_filter] at hij ** case refine'_2.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i j : \u2115 hij\u271d : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N) hij : i < \u230ac ^ (N - 1)\u230b\u208a \u2227 j < N \u2227 i < \u230ac ^ j\u230b\u208a \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij\u271d \u2208     Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a ** simp only [hij.2.1, hij.2.2, mem_sigma, mem_range, and_self_iff] ** case refine'_3 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a),     (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)         (_ :           (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208             Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)) =       a ** rintro \u27e8i, j\u27e9 hij ** case refine'_3.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)       (_ :         (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208           Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)) =     { fst := i, snd := j } ** rfl ** case refine'_4 \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 \u2200 (a : (_ : \u2115) \u00d7 \u2115) (ha : a \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N)),     (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) a ha)         (_ : (fun p x => { fst := p.snd, snd := p.fst }) a ha \u2208 Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a) =       a ** rintro \u27e8i, j\u27e9 hij ** case refine'_4.mk \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N i j : \u2115 hij : { fst := i, snd := j } \u2208 Finset.sigma (range \u230ac ^ (N - 1)\u230b\u208a) fun a => filter (fun i => a < \u230ac ^ i\u230b\u208a) (range N) \u22a2 (fun p x => { fst := p.snd, snd := p.fst }) ((fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij)       (_ :         (fun p x => { fst := p.snd, snd := p.fst }) { fst := i, snd := j } hij \u2208           Finset.sigma (range N) fun a => range \u230ac ^ a\u230b\u208a) =     { fst := i, snd := j } ** rfl ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 \u2211 j in range (u (N - 1)), (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264     \u2211 j in range (u (N - 1)), c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119 ** apply sum_le_sum fun j hj => ?_ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N j : \u2115 hj : j \u2208 range (u (N - 1)) \u22a2 (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264     c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119 ** rcases @eq_zero_or_pos _ _ j with (rfl | hj) ** case inr \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N j : \u2115 hj\u271d : j \u2208 range (u (N - 1)) hj : 0 < j \u22a2 (\u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y j) \u2119 \u2264     c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119 ** apply mul_le_mul_of_nonneg_right _ (variance_nonneg _ _) ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N j : \u2115 hj\u271d : j \u2208 range (u (N - 1)) hj : 0 < j \u22a2 \u2211 i in filter (fun i => j < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 ** convert sum_div_nat_floor_pow_sq_le_div_sq N (Nat.cast_pos.2 hj) c_one using 2 ** case inl \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 hj : 0 \u2208 range (u (N - 1)) \u22a2 (\u2211 i in filter (fun i => 0 < u i) (range N), (\u2191(u i) ^ 2)\u207b\u00b9) * variance (Y 0) \u2119 \u2264     c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u21910 ^ 2 * variance (Y 0) \u2119 ** simp only [Nat.cast_zero, zero_pow', Ne.def, bit0_eq_zero, Nat.one_ne_zero,\n  not_false_iff, div_zero, zero_mul] ** case inl \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 hj : 0 \u2208 range (u (N - 1)) \u22a2 (\u2211 x in filter (fun i => 0 < \u230ac ^ i\u230b\u208a) (range N), (\u2191\u230ac ^ x\u230b\u208a ^ 2)\u207b\u00b9) * variance (truncation (X 0) 0) \u2119 \u2264 0 ** simp only [Nat.cast_zero, truncation_zero, variance_zero, mul_zero, le_rfl] ** case h.e'_3.h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N j : \u2115 hj\u271d : j \u2208 range (u (N - 1)) hj : 0 < j \u22a2 filter (fun i => j < u i) (range N) = filter (fun x => \u2191j < \u2191\u230ac ^ x\u230b\u208a) (range N) ** simp only [Nat.cast_lt] ** case h.e'_3.a \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N j : \u2115 hj\u271d : j \u2208 range (u (N - 1)) hj : 0 < j x\u271d : \u2115 a\u271d : x\u271d \u2208 filter (fun x => \u2191j < \u2191\u230ac ^ x\u230b\u208a) (range N) \u22a2 (\u2191(u x\u271d) ^ 2)\u207b\u00b9 = 1 / \u2191\u230ac ^ x\u271d\u230b\u208a ^ 2 ** simp only [one_div] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 \u2211 j in range (u (N - 1)), c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / \u2191j ^ 2 * variance (Y j) \u2119 =     c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * \u2211 j in range (u (N - 1)), (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 ** simp_rw [mul_sum, div_eq_mul_inv, mul_assoc] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * \u2211 j in range (u (N - 1)), (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264     c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) ** apply mul_le_mul_of_nonneg_left (I1 _) ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 0 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 ** apply mul_nonneg (pow_nonneg c_pos.le _) ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) N : \u2115 \u22a2 0 \u2264 (c - 1)\u207b\u00b9 ^ 3 ** exact pow_nonneg (inv_nonneg.2 (sub_nonneg.2 c_one.le)) _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C \u22a2 \u2200 (N : \u2115), \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) ** intro N ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N : \u2115 \u22a2 \u2211 i in range N, \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264     \u2211 i in range N, ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2) ** refine' sum_le_sum fun i _ => _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N i : \u2115 x\u271d : i \u2208 range N \u22a2 \u2191\u2191\u2119 {\u03c9 | \u2191(u i) * \u03b5 \u2264 |S (u i) \u03c9 - \u222b (a : \u03a9), S (u i) a|} \u2264 ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2) ** apply meas_ge_le_variance_div_sq ** case hX \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N i : \u2115 x\u271d : i \u2208 range N \u22a2 Mem\u2112p (fun \u03c9 => S (u i) \u03c9) 2 ** exact mem\u2112p_finset_sum' _ fun j _ => (hident j).aestronglyMeasurable_fst.mem\u2112p_truncation ** case hc \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N i : \u2115 x\u271d : i \u2208 range N \u22a2 0 < \u2191(u i) * \u03b5 ** apply mul_pos (Nat.cast_pos.2 _) \u03b5pos ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N i : \u2115 x\u271d : i \u2208 range N \u22a2 0 < u i ** refine' zero_lt_one.trans_le _ ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N i : \u2115 x\u271d : i \u2208 range N \u22a2 1 \u2264 u i ** apply Nat.le_floor ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N i : \u2115 x\u271d : i \u2208 range N \u22a2 \u21911 \u2264 c ^ i ** rw [Nat.cast_one] ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N i : \u2115 x\u271d : i \u2208 range N \u22a2 1 \u2264 c ^ i ** apply one_le_pow_of_one_le c_one.le ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N : \u2115 \u22a2 \u2211 i in range N, ENNReal.ofReal (variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2) =     ENNReal.ofReal (\u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2) ** rw [ENNReal.ofReal_sum_of_nonneg fun i _ => ?_] ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N i : \u2115 x\u271d : i \u2208 range N \u22a2 0 \u2264 variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2 ** exact div_nonneg (variance_nonneg _ _) (sq_nonneg _) ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N : \u2115 \u22a2 ENNReal.ofReal (\u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2) \u2264 ENNReal.ofReal (\u03b5\u207b\u00b9 ^ 2 * C) ** apply ENNReal.ofReal_le_ofReal ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N : \u2115 \u22a2 \u2211 i in range N, variance (S (u i)) \u2119 / (\u2191(u i) * \u03b5) ^ 2 \u2264 \u03b5\u207b\u00b9 ^ 2 * C ** conv_lhs =>\n  enter [2, i]\n  rw [div_eq_inv_mul, \u2190 inv_pow, mul_inv, mul_comm _ \u03b5\u207b\u00b9, mul_pow, mul_assoc] ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N : \u2115 \u22a2 \u2211 i in range N, \u03b5\u207b\u00b9 ^ 2 * ((\u2191(u i))\u207b\u00b9 ^ 2 * variance (S (u i)) \u2119) \u2264 \u03b5\u207b\u00b9 ^ 2 * C ** rw [\u2190 mul_sum] ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N : \u2115 \u22a2 \u03b5\u207b\u00b9 ^ 2 * \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119 \u2264 \u03b5\u207b\u00b9 ^ 2 * C ** refine' mul_le_mul_of_nonneg_left _ (sq_nonneg _) ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N : \u2115 \u22a2 \u2211 x in range N, (\u2191(u x))\u207b\u00b9 ^ 2 * variance (S (u x)) \u2119 \u2264 C ** conv_lhs => enter [2, i]; rw [inv_pow] ** case h \u03a9 : Type u_1 inst\u271d\u00b9 : MeasureSpace \u03a9 inst\u271d : IsProbabilityMeasure \u2119 X : \u2115 \u2192 \u03a9 \u2192 \u211d hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hnonneg : \u2200 (i : \u2115) (\u03c9 : \u03a9), 0 \u2264 X i \u03c9 c : \u211d c_one : 1 < c \u03b5 : \u211d \u03b5pos : 0 < \u03b5 c_pos : 0 < c hX : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 A : \u2200 (i : \u211d), StronglyMeasurable (indicator (Set.Ioc (-i) i) id) Y : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => truncation (X n) \u2191n S : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n => \u2211 i in range n, Y i hS : S = fun n => \u2211 i in range n, Y i u : \u2115 \u2192 \u2115 := fun n => \u230ac ^ n\u230b\u208a u_mono : Monotone u I1 : \u2200 (K : \u2115), \u2211 j in range K, (\u2191j ^ 2)\u207b\u00b9 * variance (Y j) \u2119 \u2264 2 * \u222b (a : \u03a9), X 0 a C : \u211d := c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 * (2 * \u222b (a : \u03a9), X 0 a) I2 : \u2200 (N : \u2115), \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C N : \u2115 \u22a2 \u2211 i in range N, (\u2191(u i) ^ 2)\u207b\u00b9 * variance (S (u i)) \u2119 \u2264 C ** exact I2 N ** Qed",
        "informal": ""
    },
    {
        "formal": "IsCountablySpanning.prod ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 C : Set (Set \u03b1) D : Set (Set \u03b2) hC : IsCountablySpanning C hD : IsCountablySpanning D \u22a2 IsCountablySpanning (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D) ** rcases hC, hD with \u27e8\u27e8s, h1s, h2s\u27e9, t, h1t, h2t\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 C : Set (Set \u03b1) D : Set (Set \u03b2) s : \u2115 \u2192 Set \u03b1 h1s : \u2200 (n : \u2115), s n \u2208 C h2s : \u22c3 n, s n = univ t : \u2115 \u2192 Set \u03b2 h1t : \u2200 (n : \u2115), t n \u2208 D h2t : \u22c3 n, t n = univ \u22a2 IsCountablySpanning (image2 (fun x x_1 => x \u00d7\u02e2 x_1) C D) ** refine' \u27e8fun n => s n.unpair.1 \u00d7\u02e2 t n.unpair.2, fun n => mem_image2_of_mem (h1s _) (h1t _), _\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 C : Set (Set \u03b1) D : Set (Set \u03b2) s : \u2115 \u2192 Set \u03b1 h1s : \u2200 (n : \u2115), s n \u2208 C h2s : \u22c3 n, s n = univ t : \u2115 \u2192 Set \u03b2 h1t : \u2200 (n : \u2115), t n \u2208 D h2t : \u22c3 n, t n = univ \u22a2 \u22c3 n, (fun n => s (Nat.unpair n).1 \u00d7\u02e2 t (Nat.unpair n).2) n = univ ** rw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.get_set_ne ** \u03b1 : Type ?u.13187 a : Array \u03b1 i : Fin (size a) j : Nat v : \u03b1 hj : j < size a h : i.val \u2260 j \u22a2 j < size (set a i v) ** simp [*] ** \u03b1 : Type u_1 a : Array \u03b1 i : Fin (size a) j : Nat v : \u03b1 hj : j < size a h : i.val \u2260 j \u22a2 (set a i v)[j] = a[j] ** simp only [set, getElem_eq_data_get, List.get_set_ne h] ** Qed",
        "informal": ""
    },
    {
        "formal": "integral_withDensity_eq_integral_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** by_cases hE : CompleteSpace E ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : \u00acCompleteSpace E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** by_cases hg : Integrable g (\u03bc.withDensity fun x => f x) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc  case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : \u00acIntegrable g \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** refine' Integrable.induction\n  (P := fun g => \u222b a, g a \u2202\u03bc.withDensity (fun x => f x) = \u222b a, f a \u2022 g a \u2202\u03bc) _ _ _ _ hg ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : \u00acCompleteSpace E \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** simp [integral, hE] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : \u00acIntegrable g \u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc ** rw [integral_undef hg, integral_undef] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : \u00acIntegrable g \u22a2 \u00acIntegrable fun a => f a \u2022 g a ** rwa [\u2190 integrable_withDensity_iff_integrable_smul f_meas] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,     MeasurableSet s \u2192       \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u2192         (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc)           (indicator s fun x => c) ** intro c s s_meas hs ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 (\u222b (a : \u03b1), indicator s (fun x => c) a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) =     \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc ** rw [integral_indicator s_meas] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc ** simp_rw [\u2190 indicator_smul_apply, integral_indicator s_meas] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc ** simp only [s_meas, integral_const, Measure.restrict_apply', univ_inter, withDensity_apply] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1) in s, \u2191(f x) \u2202\u03bc) \u2022 c = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc ** rw [lintegral_coe_eq_integral, ENNReal.toReal_ofReal, \u2190 integral_smul_const] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2191(f x) \u2022 c \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc ** rfl ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 0 \u2264 \u222b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc ** exact integral_nonneg fun x => NNReal.coe_nonneg _ ** case pos.refine'_1.hfi \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 Integrable fun x => \u2191(f x) ** refine' \u27e8f_meas.coe_nnreal_real.aemeasurable.aestronglyMeasurable, _\u27e9 ** case pos.refine'_1.hfi \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u22a2 HasFiniteIntegral fun x => \u2191(f x) ** rw [withDensity_apply _ s_meas] at hs ** case pos.refine'_1.hfi \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4 \u22a2 HasFiniteIntegral fun x => \u2191(f x) ** rw [HasFiniteIntegral] ** case pos.refine'_1.hfi \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4 \u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016\u2191(f a)\u2016\u208a \u2202\u03bc < \u22a4 ** convert hs with x ** case h.e'_3.h.e'_4.h.h.e'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g c : E s : Set \u03b1 s_meas : MeasurableSet s hs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4 x : \u03b1 \u22a2 \u2016\u2191(f x)\u2016\u208a = f x ** simp only [NNReal.nnnorm_eq] ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,     Disjoint (support f_1) (support g) \u2192       Integrable f_1 \u2192         Integrable g \u2192           (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192             (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g \u2192               (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) (f_1 + g) ** intro u u' _ u_int u'_int h h' ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \u22a2 (\u222b (a : \u03b1), (u + u') a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u + u') a \u2202\u03bc ** change\n  (\u222b a : \u03b1, u a + u' a \u2202\u03bc.withDensity fun x : \u03b1 => \u2191(f x)) = \u222b a : \u03b1, f a \u2022 (u a + u' a) \u2202\u03bc ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u a + u' a) \u2202\u03bc ** simp_rw [smul_add] ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a + f a \u2022 u' a \u2202\u03bc ** rw [integral_add u_int u'_int, h, h', integral_add] ** case pos.refine'_2.hf \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \u22a2 Integrable fun a => f a \u2022 u a ** exact (integrable_withDensity_iff_integrable_smul f_meas).1 u_int ** case pos.refine'_2.hg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u u' : \u03b1 \u2192 E a\u271d : Disjoint (support u) (support u') u_int : Integrable u u'_int : Integrable u' h : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc h' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc \u22a2 Integrable fun a => f a \u2022 u' a ** exact (integrable_withDensity_iff_integrable_smul f_meas).1 u'_int ** case pos.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1} ** have C1 :\n  Continuous fun u : Lp E 1 (\u03bc.withDensity fun x => f x) =>\n    \u222b x, u x \u2202\u03bc.withDensity fun x => f x :=\n  continuous_integral ** case pos.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) \u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1} ** have C2 : Continuous fun u : Lp E 1 (\u03bc.withDensity fun x => f x) => \u222b x, f x \u2022 u x \u2202\u03bc := by\n  have : Continuous ((fun u : Lp E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 withDensitySMulLI (E := E) \u03bc f_meas) :=\n    continuous_integral.comp (withDensitySMulLI (E := E) \u03bc f_meas).continuous\n  convert this with u\n  simp only [Function.comp_apply, withDensitySMulLI_apply]\n  exact integral_congr_ae (mem\u21121_smul_of_L1_withDensity f_meas u).coeFn_toLp.symm ** case pos.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) C2 : Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc \u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1} ** exact isClosed_eq C1 C2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) \u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc ** have : Continuous ((fun u : Lp E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 withDensitySMulLI (E := E) \u03bc f_meas) :=\n  continuous_integral.comp (withDensitySMulLI (E := E) \u03bc f_meas).continuous ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) this : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) \u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc ** convert this with u ** case h.e'_5.h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) this : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) u : { x // x \u2208 Lp E 1 } \u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) u ** simp only [Function.comp_apply, withDensitySMulLI_apply] ** case h.e'_5.h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g C1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x) this : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) u : { x // x \u2208 Lp E 1 } \u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = \u222b (x : \u03b1), \u2191\u2191(Mem\u2112p.toLp (fun x => f x \u2022 \u2191\u2191u x) (_ : Mem\u2112p (fun x => f x \u2022 \u2191\u2191u x) 1)) x \u2202\u03bc ** exact integral_congr_ae (mem\u21121_smul_of_L1_withDensity f_meas u).coeFn_toLp.symm ** case pos.refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g \u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,     f_1 =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g \u2192       Integrable f_1 \u2192         (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192           (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g ** intro u v huv _ hu ** case pos.refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc \u22a2 (\u222b (a : \u03b1), v a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc ** rw [\u2190 integral_congr_ae huv, hu] ** case pos.refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc \u22a2 \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc ** apply integral_congr_ae ** case pos.refine'_4.h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc \u22a2 (fun a => f a \u2022 u a) =\u1d50[\u03bc] fun a => f a \u2022 v a ** filter_upwards [(ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 huv] with x hx ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc x : \u03b1 hx : \u2191(f x) \u2260 0 \u2192 u x = v x \u22a2 f x \u2022 u x = f x \u2022 v x ** rcases eq_or_ne (f x) 0 with (h'x | h'x) ** case h.inl \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc x : \u03b1 hx : \u2191(f x) \u2260 0 \u2192 u x = v x h'x : f x = 0 \u22a2 f x \u2022 u x = f x \u2022 v x ** simp only [h'x, zero_smul] ** case h.inr \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc x : \u03b1 hx : \u2191(f x) \u2260 0 \u2192 u x = v x h'x : f x \u2260 0 \u22a2 f x \u2022 u x = f x \u2022 v x ** rw [hx _] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2077 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2076 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : NormedSpace \u211d F f : \u03b1 \u2192 \u211d\u22650 f_meas : Measurable f g : \u03b1 \u2192 E hE : CompleteSpace E hg : Integrable g u v : \u03b1 \u2192 E huv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v a\u271d : Integrable u hu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc x : \u03b1 hx : \u2191(f x) \u2260 0 \u2192 u x = v x h'x : f x \u2260 0 \u22a2 \u2191(f x) \u2260 0 ** simpa only [Ne.def, ENNReal.coe_eq_zero] using h'x ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.insert_insert_of_ne ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a a' : \u03b1 b : \u03b2 a b' : \u03b2 a' s\u271d : Finmap \u03b2 h : a \u2260 a' s : AList \u03b2 \u22a2 insert a' b' (insert a b \u27e6s\u27e7) = insert a b (insert a' b' \u27e6s\u27e7) ** simp only [insert_toFinmap, AList.toFinmap_eq, AList.insert_insert_of_ne _ h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.haar.chaar_sup_le ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** let eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** have : Continuous eval := by\n  exact ((continuous_apply K\u2081).add (continuous_apply K\u2082)).sub (continuous_apply (K\u2081 \u2294 K\u2082)) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval \u22a2 chaar K\u2080 (K\u2081 \u2294 K\u2082) \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 ** rw [\u2190 sub_nonneg] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval \u22a2 0 \u2264 chaar K\u2080 K\u2081 + chaar K\u2080 K\u2082 - chaar K\u2080 (K\u2081 \u2294 K\u2082) ** show chaar K\u2080 \u2208 eval \u207b\u00b9' Ici (0 : \u211d) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval \u22a2 chaar K\u2080 \u2208 eval \u207b\u00b9' Ici 0 ** apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K\u2080 \u22a4) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval \u22a2 clPrehaar \u2191K\u2080 \u22a4 \u2286 eval \u207b\u00b9' Ici 0 ** unfold clPrehaar ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval \u22a2 closure (prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U}) \u2286 eval \u207b\u00b9' Ici 0 ** rw [IsClosed.closure_subset_iff] ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) \u22a2 Continuous eval ** exact ((continuous_apply K\u2081).add (continuous_apply K\u2082)).sub (continuous_apply (K\u2081 \u2294 K\u2082)) ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval \u22a2 prehaar \u2191K\u2080 '' {U | U \u2286 \u2191\u22a4.toOpens \u2227 IsOpen U \u2227 1 \u2208 U} \u2286 eval \u207b\u00b9' Ici 0 ** rintro _ \u27e8U, \u27e8_, h2U, h3U\u27e9, rfl\u27e9 ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U \u2208 eval \u207b\u00b9' Ici 0 ** simp only [mem_preimage, mem_Ici, sub_nonneg] ** case intro.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 prehaar (\u2191K\u2080) U (K\u2081 \u2294 K\u2082) \u2264 prehaar (\u2191K\u2080) U K\u2081 + prehaar (\u2191K\u2080) U K\u2082 ** apply prehaar_sup_le ** case intro.intro.intro.intro.hU G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Set.Nonempty (interior U) ** rw [h2U.interior_eq] ** case intro.intro.intro.intro.hU G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval U : Set G left\u271d : U \u2286 \u2191\u22a4.toOpens h2U : IsOpen U h3U : 1 \u2208 U \u22a2 Set.Nonempty U ** exact \u27e81, h3U\u27e9 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval \u22a2 IsClosed (eval \u207b\u00b9' Ici 0) ** apply continuous_iff_isClosed.mp this ** case a G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G inst\u271d : TopologicalGroup G K\u2080 : PositiveCompacts G K\u2081 K\u2082 : Compacts G eval : (Compacts G \u2192 \u211d) \u2192 \u211d := fun f => f K\u2081 + f K\u2082 - f (K\u2081 \u2294 K\u2082) this : Continuous eval \u22a2 IsClosed (Ici 0) ** exact isClosed_Ici ** Qed",
        "informal": ""
    },
    {
        "formal": "VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) ** let A := MeasureTheory.Measure.finiteSpanningSetsInOpen' \u03bc ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) ** rcases h'f.isSeparable_range with \u27e8t, t_count, ht\u27e9 ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) ** filter_upwards [main, v.ae_eventually_measure_pos] with x hx h'x ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a \u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) ** have M :\n  \u2200 c \u2208 t, Tendsto (fun a => (\u222b\u207b y in a, \u2016f y - c\u2016\u208a \u2202\u03bc) / \u03bc a)\n    (v.filterAt x) (\ud835\udcdd \u2016f x - c\u2016\u208a) := by\n  intro c hc\n  obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using mem_univ x\n  specialize hx n c hc\n  simp only [xn, indicator_of_mem] at hx\n  apply hx.congr' _\n  filter_upwards [v.eventually_filterAt_subset_of_nhds (IsOpen.mem_nhds (A.set_mem n) xn),\n    v.eventually_filterAt_measurableSet x] with a ha h'a\n  congr 1\n  apply set_lintegral_congr_fun h'a\n  apply eventually_of_forall fun y => ?_\n  intro hy\n  simp only [ha hy, indicator_of_mem] ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 0) ** apply ENNReal.tendsto_nhds_zero.2 fun \u03b5 \u03b5pos => ?_ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 \u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in filterAt v x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc x_1 \u2264 \u03b5 ** obtain \u27e8c, ct, xc\u27e9 : \u2203 c \u2208 t, (\u2016f x - c\u2016\u208a : \u211d\u22650\u221e) < \u03b5 / 2 := by\n  simp_rw [\u2190 edist_eq_coe_nnnorm_sub]\n  have : f x \u2208 closure t := ht (mem_range_self _)\n  exact EMetric.mem_closure_iff.1 this (\u03b5 / 2) (ENNReal.half_pos (ne_of_gt \u03b5pos)) ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 \u22a2 \u2200\u1da0 (x_1 : Set \u03b1) in filterAt v x, (\u222b\u207b (y : \u03b1) in x_1, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc x_1 \u2264 \u03b5 ** filter_upwards [(tendsto_order.1 (M c ct)).2 (\u03b5 / 2) xc, h'x, v.eventually_measure_lt_top x] with\n  a ha h'a h''a ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 a : Set \u03b1 ha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2 h'a : 0 < \u2191\u2191\u03bc a h''a : \u2191\u2191\u03bc a < \u22a4 \u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a \u2264 \u03b5 ** apply ENNReal.div_le_of_le_mul ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) ** simp_rw [ae_all_iff, ae_ball_iff t_count] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t \u22a2 \u2200 (i : \u2115) (i_1 : E),     i_1 \u2208 t \u2192       \u2200\u1d50 (x : \u03b1) \u2202\u03bc,         Tendsto           (fun a =>             (\u222b\u207b (y : \u03b1) in a,                 \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) i) (fun x => i_1) y\u2016\u208a \u2202\u03bc) /               \u2191\u2191\u03bc a)           (filterAt v x)           (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) i) (fun x => i_1) x\u2016\u208a) ** intro n c _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     Tendsto       (fun a =>         (\u222b\u207b (y : \u03b1) in a,             \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /           \u2191\u2191\u03bc a)       (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a) ** apply ae_tendsto_lintegral_div' ** case hf \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t \u22a2 Measurable fun y => \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a ** refine' (h'f.sub _).ennnorm ** case hf \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t \u22a2 StronglyMeasurable fun y => indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y ** exact stronglyMeasurable_const.indicator (IsOpen.measurableSet (A.set_mem n)) ** case h'f \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t \u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc \u2260 \u22a4 ** apply ne_of_lt ** case h'f.h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t \u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc < \u22a4 ** calc\n  (\u222b\u207b y, \u2191\u2016f y - (A.set n).indicator (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc) \u2264\n      \u222b\u207b y, \u2016f y\u2016\u208a + \u2016(A.set n).indicator (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc := by\n    apply lintegral_mono\n    intro x\n    dsimp\n    rw [\u2190 ENNReal.coe_add]\n    exact ENNReal.coe_le_coe.2 (nnnorm_sub_le _ _)\n  _ = (\u222b\u207b y, \u2016f y\u2016\u208a \u2202\u03bc) + \u222b\u207b y, \u2016(A.set n).indicator (fun _ : \u03b1 => c) y\u2016\u208a \u2202\u03bc :=\n    (lintegral_add_left h'f.ennnorm _)\n  _ < \u221e + \u221e :=\n    haveI I : Integrable ((A.set n).indicator fun _ : \u03b1 => c) \u03bc := by\n      simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),\n        integrableOn_const, A.finite n, or_true_iff]\n    ENNReal.add_lt_add hf.2 I.2 ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t \u22a2 \u222b\u207b (y : \u03b1), \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc \u2264     \u222b\u207b (y : \u03b1), \u2191\u2016f y\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc ** apply lintegral_mono ** case hfg \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t \u22a2 (fun a => \u2191\u2016f a - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a) \u2264 fun a =>     \u2191\u2016f a\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a ** intro x ** case hfg \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t x : \u03b1 \u22a2 (fun a => \u2191\u2016f a - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a) x \u2264     (fun a => \u2191\u2016f a\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set A n) (fun x => c) a\u2016\u208a) x ** dsimp ** case hfg \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t x : \u03b1 \u22a2 \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a \u2264     \u2191\u2016f x\u2016\u208a + \u2191\u2016indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a ** rw [\u2190 ENNReal.coe_add] ** case hfg \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t x : \u03b1 \u22a2 \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a \u2264     \u2191(\u2016f x\u2016\u208a + \u2016indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a) ** exact ENNReal.coe_le_coe.2 (nnnorm_sub_le _ _) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t n : \u2115 c : E hi\u271d : c \u2208 t \u22a2 Integrable (indicator (FiniteSpanningSetsIn.set A n) fun x => c) ** simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),\n  integrableOn_const, A.finite n, or_true_iff] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a \u22a2 \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) ** intro c hc ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t \u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) ** obtain \u27e8n, xn\u27e9 : \u2203 n, x \u2208 A.set n := by simpa [\u2190 A.spanning] using mem_univ x ** case intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t n : \u2115 xn : x \u2208 FiniteSpanningSetsIn.set A n \u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) ** specialize hx n c hc ** case intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t n : \u2115 xn : x \u2208 FiniteSpanningSetsIn.set A n hx :   Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)     (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) \u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) ** simp only [xn, indicator_of_mem] at hx ** case intro \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t n : \u2115 xn : x \u2208 FiniteSpanningSetsIn.set A n hx :   Tendsto     (fun a =>       (\u222b\u207b (y : \u03b1) in a,           \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /         \u2191\u2191\u03bc a)     (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u22a2 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) ** apply hx.congr' _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t n : \u2115 xn : x \u2208 FiniteSpanningSetsIn.set A n hx :   Tendsto     (fun a =>       (\u222b\u207b (y : \u03b1) in a,           \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /         \u2191\u2191\u03bc a)     (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u22a2 (fun a =>       (\u222b\u207b (y : \u03b1) in a,           \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /         \u2191\u2191\u03bc a) =\u1da0[filterAt v x]     fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a ** filter_upwards [v.eventually_filterAt_subset_of_nhds (IsOpen.mem_nhds (A.set_mem n) xn),\n  v.eventually_filterAt_measurableSet x] with a ha h'a ** case h \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t n : \u2115 xn : x \u2208 FiniteSpanningSetsIn.set A n hx :   Tendsto     (fun a =>       (\u222b\u207b (y : \u03b1) in a,           \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /         \u2191\u2191\u03bc a)     (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) a : Set \u03b1 ha : a \u2286 FiniteSpanningSetsIn.set A n h'a : MeasurableSet a \u22a2 (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /       \u2191\u2191\u03bc a =     (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a ** congr 1 ** case h.e_a \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t n : \u2115 xn : x \u2208 FiniteSpanningSetsIn.set A n hx :   Tendsto     (fun a =>       (\u222b\u207b (y : \u03b1) in a,           \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /         \u2191\u2191\u03bc a)     (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) a : Set \u03b1 ha : a \u2286 FiniteSpanningSetsIn.set A n h'a : MeasurableSet a \u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc =     \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc ** apply set_lintegral_congr_fun h'a ** case h.e_a \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t n : \u2115 xn : x \u2208 FiniteSpanningSetsIn.set A n hx :   Tendsto     (fun a =>       (\u222b\u207b (y : \u03b1) in a,           \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /         \u2191\u2191\u03bc a)     (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) a : Set \u03b1 ha : a \u2286 FiniteSpanningSetsIn.set A n h'a : MeasurableSet a \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     x \u2208 a \u2192 \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) x\u2016\u208a = \u2191\u2016f x - c\u2016\u208a ** apply eventually_of_forall fun y => ?_ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t n : \u2115 xn : x \u2208 FiniteSpanningSetsIn.set A n hx :   Tendsto     (fun a =>       (\u222b\u207b (y : \u03b1) in a,           \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /         \u2191\u2191\u03bc a)     (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) a : Set \u03b1 ha : a \u2286 FiniteSpanningSetsIn.set A n h'a : MeasurableSet a y : \u03b1 \u22a2 y \u2208 a \u2192 \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a = \u2191\u2016f y - c\u2016\u208a ** intro hy ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t n : \u2115 xn : x \u2208 FiniteSpanningSetsIn.set A n hx :   Tendsto     (fun a =>       (\u222b\u207b (y : \u03b1) in a,           \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a \u2202\u03bc) /         \u2191\u2191\u03bc a)     (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) a : Set \u03b1 ha : a \u2286 FiniteSpanningSetsIn.set A n h'a : MeasurableSet a y : \u03b1 hy : y \u2208 a \u22a2 \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set (finiteSpanningSetsInOpen' \u03bc) n) (fun x => c) y\u2016\u208a = \u2191\u2016f y - c\u2016\u208a ** simp only [ha hy, indicator_of_mem] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a c : E hc : c \u2208 t \u22a2 \u2203 n, x \u2208 FiniteSpanningSetsIn.set A n ** simpa [\u2190 A.spanning] using mem_univ x ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 \u22a2 \u2203 c, c \u2208 t \u2227 \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 ** simp_rw [\u2190 edist_eq_coe_nnnorm_sub] ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 \u22a2 \u2203 c, c \u2208 t \u2227 edist (f x) c < \u03b5 / 2 ** have : f x \u2208 closure t := ht (mem_range_self _) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 this : f x \u2208 closure t \u22a2 \u2203 c, c \u2208 t \u2227 edist (f x) c < \u03b5 / 2 ** exact EMetric.mem_closure_iff.1 this (\u03b5 / 2) (ENNReal.half_pos (ne_of_gt \u03b5pos)) ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 a : Set \u03b1 ha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2 h'a : 0 < \u2191\u2191\u03bc a h''a : \u2191\u2191\u03bc a < \u22a4 \u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - f x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a + \u2191\u2016f x - c\u2016\u208a \u2202\u03bc ** apply lintegral_mono fun x => ?_ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x\u271d : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x\u271d) (\ud835\udcdd \u2191\u2016f x\u271d - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u271d\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x\u271d, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x\u271d) (\ud835\udcdd \u2191\u2016f x\u271d - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x\u271d - c\u2016\u208a < \u03b5 / 2 a : Set \u03b1 ha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2 h'a : 0 < \u2191\u2191\u03bc a h''a : \u2191\u2191\u03bc a < \u22a4 x : \u03b1 \u22a2 \u2191\u2016f x - f x\u271d\u2016\u208a \u2264 \u2191\u2016f x - c\u2016\u208a + \u2191\u2016f x\u271d - c\u2016\u208a ** simpa only [\u2190 edist_eq_coe_nnnorm_sub] using edist_triangle_right _ _ _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 a : Set \u03b1 ha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2 h'a : 0 < \u2191\u2191\u03bc a h''a : \u2191\u2191\u03bc a < \u22a4 \u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc + \u222b\u207b (x_1 : \u03b1) in a, \u2191\u2016f x - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a + \u03b5 / 2 * \u2191\u2191\u03bc a ** refine' add_le_add _ _ ** case refine'_1 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 a : Set \u03b1 ha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2 h'a : 0 < \u2191\u2191\u03bc a h''a : \u2191\u2191\u03bc a < \u22a4 \u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a ** rw [ENNReal.div_lt_iff (Or.inl h'a.ne') (Or.inl h''a.ne)] at ha ** case refine'_1 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 a : Set \u03b1 ha : \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc < \u03b5 / 2 * \u2191\u2191\u03bc a h'a : 0 < \u2191\u2191\u03bc a h''a : \u2191\u2191\u03bc a < \u22a4 \u22a2 \u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a ** exact ha.le ** case refine'_2 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 a : Set \u03b1 ha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2 h'a : 0 < \u2191\u2191\u03bc a h''a : \u2191\u2191\u03bc a < \u22a4 \u22a2 \u222b\u207b (x_1 : \u03b1) in a, \u2191\u2016f x - c\u2016\u208a \u2202\u03bc \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a ** simp only [lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter] ** case refine'_2 \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 a : Set \u03b1 ha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2 h'a : 0 < \u2191\u2191\u03bc a h''a : \u2191\u2191\u03bc a < \u22a4 \u22a2 \u2191\u2016f x - c\u2016\u208a * \u2191\u2191\u03bc a \u2264 \u03b5 / 2 * \u2191\u2191\u03bc a ** exact mul_le_mul_right' xc.le _ ** \u03b1 : Type u_1 inst\u271d\u2075 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : SecondCountableTopology \u03b1 inst\u271d\u00b2 : BorelSpace \u03b1 inst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc \u03c1 : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03c1 f : \u03b1 \u2192 E hf : Integrable f h'f : StronglyMeasurable f A : FiniteSpanningSetsIn \u03bc {K | IsOpen K} := finiteSpanningSetsInOpen' \u03bc t : Set E t_count : Set.Countable t ht : range f \u2286 closure t main :   \u2200\u1d50 (x : \u03b1) \u2202\u03bc,     \u2200 (n : \u2115) (c : E),       c \u2208 t \u2192         Tendsto           (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)           (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) x : \u03b1 hx :   \u2200 (n : \u2115) (c : E),     c \u2208 t \u2192       Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) y\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a)         (filterAt v x) (\ud835\udcdd \u2191\u2016f x - indicator (FiniteSpanningSetsIn.set A n) (fun x => c) x\u2016\u208a) h'x : \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a M : \u2200 (c : E), c \u2208 t \u2192 Tendsto (fun a => (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd \u2191\u2016f x - c\u2016\u208a) \u03b5 : \u211d\u22650\u221e \u03b5pos : \u03b5 > 0 c : E ct : c \u2208 t xc : \u2191\u2016f x - c\u2016\u208a < \u03b5 / 2 a : Set \u03b1 ha : (\u222b\u207b (y : \u03b1) in a, \u2191\u2016f y - c\u2016\u208a \u2202\u03bc) / \u2191\u2191\u03bc a < \u03b5 / 2 h'a : 0 < \u2191\u2191\u03bc a h''a : \u2191\u2191\u03bc a < \u22a4 \u22a2 \u03b5 / 2 * \u2191\u2191\u03bc a + \u03b5 / 2 * \u2191\u2191\u03bc a = \u03b5 * \u2191\u2191\u03bc a ** rw [\u2190 add_mul, ENNReal.add_halves] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.pairwise_disjoint_Ico_zpow ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 a b : \u03b1 \u22a2 Pairwise (Disjoint on fun n => Ico (b ^ n) (b ^ (n + 1))) ** simpa only [one_mul] using pairwise_disjoint_Ico_mul_zpow 1 b ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.empty_pow ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d : Monoid \u03b1 s t : Set \u03b1 a : \u03b1 m n\u271d n : \u2115 hn : n \u2260 0 \u22a2 \u2205 ^ n = \u2205 ** rw [\u2190 tsub_add_cancel_of_le (Nat.succ_le_of_lt <| Nat.pos_of_ne_zero hn), pow_succ, empty_mul] ** Qed",
        "informal": ""
    },
    {
        "formal": "Primrec\u2082.unpaired ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 f : \u2115 \u2192 \u2115 \u2192 \u03b1 h : Primrec (Nat.unpaired f) \u22a2 Primrec\u2082 f ** simpa using h.comp natPair ** Qed",
        "informal": ""
    },
    {
        "formal": "Num.gcd_to_nat_aux ** n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : natSize (pos a * b) \u2264 Nat.succ n \u22a2 \u2191(gcdAux (Nat.succ n) (pos a) b) = Nat.gcd \u2191(pos a) \u2191b ** simp only [gcdAux, cast_pos] ** n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : natSize (pos a * b) \u2264 Nat.succ n \u22a2 \u2191(gcdAux n (b % pos a) (pos a)) = Nat.gcd \u2191a \u2191b ** rw [Nat.gcd_rec, gcd_to_nat_aux, mod_to_nat] ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : natSize (pos a * b) \u2264 Nat.succ n \u22a2 natSize (b % pos a * pos a) \u2264 n ** rw [natSize_to_nat, mul_to_nat, Nat.size_le] at h \u22a2 ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : \u2191(pos a) * \u2191b < 2 ^ Nat.succ n \u22a2 \u2191(b % pos a) * \u2191(pos a) < 2 ^ n ** rw [mod_to_nat, mul_comm] ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : \u2191(pos a) * \u2191b < 2 ^ Nat.succ n \u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) < 2 ^ n ** rw [pow_succ', \u2190 Nat.mod_add_div b (pos a)] at h ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2 \u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) < 2 ^ n ** refine' lt_of_mul_lt_mul_right (lt_of_le_of_lt _ h) (Nat.zero_le 2) ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2 \u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) * 2 \u2264 \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) ** rw [mul_two, mul_add] ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2 \u22a2 \u2191(pos a) * (\u2191b % \u2191(pos a)) + \u2191(pos a) * (\u2191b % \u2191(pos a)) \u2264     \u2191(pos a) * (\u2191b % \u2191(pos a)) + \u2191(pos a) * (\u2191(pos a) * (\u2191b / \u2191(pos a))) ** refine'\n  add_le_add_left\n    (Nat.mul_le_mul_left _ (le_trans (le_of_lt (Nat.mod_lt _ (PosNum.cast_pos _))) _)) _ ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2 \u22a2 \u2191a \u2264 \u2191(pos a) * (\u2191b / \u2191(pos a)) ** suffices 1 \u2264 _ by simpa using Nat.mul_le_mul_left (pos a) this ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2 \u22a2 1 \u2264 \u2191b / \u2191a ** rw [Nat.le_div_iff_mul_le a.cast_pos, one_mul] ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2 \u22a2 \u2191a \u2264 \u2191b ** exact le_to_nat.2 ab ** n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : natSize (pos a * b) \u2264 Nat.succ n \u22a2 Nat.gcd (\u2191b % \u2191(pos a)) \u2191(pos a) = Nat.gcd (\u2191b % \u2191a) \u2191a ** rfl ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : natSize (pos a * b) \u2264 Nat.succ n \u22a2 b % pos a \u2264 pos a ** rw [\u2190 le_to_nat, mod_to_nat] ** case a n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : natSize (pos a * b) \u2264 Nat.succ n \u22a2 \u2191b % \u2191(pos a) \u2264 \u2191(pos a) ** exact le_of_lt (Nat.mod_lt _ (PosNum.cast_pos _)) ** n : \u2115 a : PosNum b : Num ab : pos a \u2264 b h : \u2191(pos a) * (\u2191b % \u2191(pos a) + \u2191(pos a) * (\u2191b / \u2191(pos a))) < 2 ^ n * 2 this : 1 \u2264 ?m.1181137 \u22a2 \u2191a \u2264 \u2191(pos a) * (\u2191b / \u2191(pos a)) ** simpa using Nat.mul_le_mul_left (pos a) this ** Qed",
        "informal": ""
    },
    {
        "formal": "Rel.image_univ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : Rel \u03b1 \u03b2 \u22a2 image r Set.univ = codom r ** ext y ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : Rel \u03b1 \u03b2 y : \u03b2 \u22a2 y \u2208 image r Set.univ \u2194 y \u2208 codom r ** simp [mem_image, codom] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ae_restrict_biUnion_finset_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b9 : MeasurableSpace \u03b2 inst\u271d : MeasurableSpace \u03b3 \u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s\u271d s' t\u271d : Set \u03b1 s : \u03b9 \u2192 Set \u03b1 t : Finset \u03b9 p : \u03b1 \u2192 Prop \u22a2 (\u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (\u22c3 i \u2208 t, s i), p x) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s i), p x ** simp_rw [Filter.Eventually, ae_restrict_biUnion_finset_eq s, mem_iSup] ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.disjoint_union_left ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 x y z : Finmap \u03b2 \u22a2 Disjoint (x \u222a y) z \u2194 Disjoint x z \u2227 Disjoint y z ** simp [Disjoint, Finmap.mem_union, or_imp, forall_and] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.condexpIndL1Fin_add ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x y : G \u22a2 condexpIndL1Fin hm hs h\u03bcs (x + y) = condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y ** ext1 ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x y : G \u22a2 \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y) ** refine' (Mem\u2112p.coeFn_toLp q).trans _ ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x y : G \u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x + condexpIndL1Fin hm hs h\u03bcs y) ** refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x y : G \u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs x) + \u2191\u2191(condexpIndL1Fin hm hs h\u03bcs y) ** refine' EventuallyEq.trans _\n  (EventuallyEq.add (Mem\u2112p.coeFn_toLp q).symm (Mem\u2112p.coeFn_toLp q).symm) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x y : G \u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs (x + y)) =\u1d50[\u03bc] fun x_1 =>     \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y) x_1 ** rw [condexpIndSMul_add] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x y : G \u22a2 \u2191\u2191(condexpIndSMul hm hs h\u03bcs x + condexpIndSMul hm hs h\u03bcs y) =\u1d50[\u03bc] fun x_1 =>     \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y) x_1 ** refine' (Lp.coeFn_add _ _).trans (eventually_of_forall fun a => _) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F inst\u271d\u2079 : NormedAddCommGroup F' inst\u271d\u2078 : NormedSpace \ud835\udd5c F' inst\u271d\u2077 : NormedSpace \u211d F' inst\u271d\u2076 : CompleteSpace F' inst\u271d\u2075 : NormedAddCommGroup G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \u211d G' inst\u271d\u00b2 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 inst\u271d\u00b9 : NormedSpace \u211d G hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 x y : G a : \u03b1 \u22a2 (\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y)) a =     (fun x_1 => \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1 + \u2191\u2191(condexpIndSMul hm hs h\u03bcs y) x_1) a ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MvQPF.liftR_map_last ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F lawful : LawfulMvFunctor F \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u R : \u03b9' \u2192 \u03b9' \u2192 Prop x : F (\u03b1 ::: \u03b9) f g : \u03b9 \u2192 \u03b9' hh : \u2200 (x : \u03b9), R (f x) (g x) h : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) } b : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h c : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9   (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=   ofSubtype (repeatEq \u03b1) ::: _root_.id \u22a2 subtypeVal (RelLast' \u03b1 R) \u229a toSubtype (RelLast' \u03b1 R) \u229a fromAppend1DropLast \u229a c \u229a b =     ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag ** dsimp ** n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F lawful : LawfulMvFunctor F \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u R : \u03b9' \u2192 \u03b9' \u2192 Prop x : F (\u03b1 ::: \u03b9) f g : \u03b9 \u2192 \u03b9' hh : \u2200 (x : \u03b9), R (f x) (g x) h : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) } b : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h c : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9   (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=   ofSubtype (repeatEq \u03b1) ::: _root_.id \u22a2 subtypeVal (RelLast' \u03b1 R) \u229a       toSubtype (RelLast' \u03b1 R) \u229a         fromAppend1DropLast \u229a           (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a             (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) =     ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag ** apply eq_of_drop_last_eq ** case h\u2081 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F lawful : LawfulMvFunctor F \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u R : \u03b9' \u2192 \u03b9' \u2192 Prop x : F (\u03b1 ::: \u03b9) f g : \u03b9 \u2192 \u03b9' hh : \u2200 (x : \u03b9), R (f x) (g x) h : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) } b : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h c : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9   (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=   ofSubtype (repeatEq \u03b1) ::: _root_.id \u22a2 lastFun       (subtypeVal (RelLast' \u03b1 R) \u229a         toSubtype (RelLast' \u03b1 R) \u229a           fromAppend1DropLast \u229a             (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a               (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) })) =     lastFun (((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag) ** simp only [lastFun_from_append1_drop_last, lastFun_toSubtype, lastFun_appendFun,\n  lastFun_subtypeVal, comp.left_id, lastFun_comp, lastFun_prod] ** case h\u2081 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F lawful : LawfulMvFunctor F \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u R : \u03b9' \u2192 \u03b9' \u2192 Prop x : F (\u03b1 ::: \u03b9) f g : \u03b9 \u2192 \u03b9' hh : \u2200 (x : \u03b9), R (f x) (g x) h : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) } b : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h c : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9   (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=   ofSubtype (repeatEq \u03b1) ::: _root_.id \u22a2 (Subtype.val \u2218 fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) = Prod.map f g \u2218 lastFun prod.diag ** ext1 ** case h\u2081.h n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F lawful : LawfulMvFunctor F \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u R : \u03b9' \u2192 \u03b9' \u2192 Prop x : F (\u03b1 ::: \u03b9) f g : \u03b9 \u2192 \u03b9' hh : \u2200 (x : \u03b9), R (f x) (g x) h : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) } b : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h c : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9   (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=   ofSubtype (repeatEq \u03b1) ::: _root_.id x\u271d : TypeVec.last (\u03b1 ::: \u03b9) \u22a2 (Subtype.val \u2218 fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) }) x\u271d =     (Prod.map f g \u2218 lastFun prod.diag) x\u271d ** rfl ** case h\u2080 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F lawful : LawfulMvFunctor F \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u R : \u03b9' \u2192 \u03b9' \u2192 Prop x : F (\u03b1 ::: \u03b9) f g : \u03b9 \u2192 \u03b9' hh : \u2200 (x : \u03b9), R (f x) (g x) h : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) } b : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h c : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9   (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=   ofSubtype (repeatEq \u03b1) ::: _root_.id \u22a2 dropFun       (subtypeVal (RelLast' \u03b1 R) \u229a         toSubtype (RelLast' \u03b1 R) \u229a           fromAppend1DropLast \u229a             (ofSubtype (repeatEq \u03b1) ::: _root_.id) \u229a               (diagSub ::: fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) })) =     dropFun (((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a prod.diag) ** dsimp ** case h\u2080 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F lawful : LawfulMvFunctor F \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u R : \u03b9' \u2192 \u03b9' \u2192 Prop x : F (\u03b1 ::: \u03b9) f g : \u03b9 \u2192 \u03b9' hh : \u2200 (x : \u03b9), R (f x) (g x) h : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) } b : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h c : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9   (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=   ofSubtype (repeatEq \u03b1) ::: _root_.id \u22a2 subtypeVal (repeatEq \u03b1) \u229a dropFun (toSubtype (RelLast' \u03b1 R)) \u229a ofSubtype (repeatEq \u03b1) \u229a diagSub =     dropFun ((TypeVec.id ::: f) \u2297' (TypeVec.id ::: g)) \u229a dropFun prod.diag ** simp only [prod_map_id, dropFun_prod, dropFun_appendFun, dropFun_diag, id_comp,\n  dropFun_toSubtype] ** case h\u2080 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F lawful : LawfulMvFunctor F \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u R : \u03b9' \u2192 \u03b9' \u2192 Prop x : F (\u03b1 ::: \u03b9) f g : \u03b9 \u2192 \u03b9' hh : \u2200 (x : \u03b9), R (f x) (g x) h : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) } b : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h c : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9   (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=   ofSubtype (repeatEq \u03b1) ::: _root_.id \u22a2 subtypeVal (repeatEq \u03b1) \u229a (toSubtype fun i x => RelLast' \u03b1 R (Fin2.fs i) x) \u229a ofSubtype (repeatEq \u03b1) \u229a diagSub =     prod.diag ** erw [toSubtype_of_subtype_assoc, id_comp] ** case h\u2080 n : \u2115 F : TypeVec.{u} (n + 1) \u2192 Type u mvf : MvFunctor F q : MvQPF F lawful : LawfulMvFunctor F \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u R : \u03b9' \u2192 \u03b9' \u2192 Prop x : F (\u03b1 ::: \u03b9) f g : \u03b9 \u2192 \u03b9' hh : \u2200 (x : \u03b9), R (f x) (g x) h : \u03b9 \u2192 { x // uncurry R x } := fun x => { val := (f x, g x), property := (_ : R (f x) (g x)) } b : \u03b1 ::: \u03b9 \u27f9 Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } := diagSub ::: h c : Subtype_ (repeatEq \u03b1) ::: { x // uncurry R x } \u27f9   (fun i => { x // ofRepeat (RelLast' \u03b1 R (Fin2.fs i) x) }) ::: Subtype (uncurry R) :=   ofSubtype (repeatEq \u03b1) ::: _root_.id \u22a2 (fun i x => subtypeVal (repeatEq \u03b1) i (diagSub i x)) = prod.diag ** clear liftR_map_last q mvf lawful F x R f g hh h b c ** case h\u2080 n : \u2115 \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u \u22a2 (fun i x => subtypeVal (repeatEq \u03b1) i (diagSub i x)) = prod.diag ** ext (i x) : 2 ** case h\u2080.a.h n : \u2115 \u03b1 : TypeVec.{u} n \u03b9 \u03b9' : Type u i : Fin2 n x : TypeVec.drop (\u03b1 ::: \u03b9) i \u22a2 subtypeVal (repeatEq \u03b1) i (diagSub i x) = prod.diag i x ** induction i with\n| fz => rfl\n| fs _ ih =>\n  apply ih ** case h\u2080.a.h.fz n : \u2115 \u03b9 \u03b9' : Type u n\u271d : \u2115 \u03b1 : TypeVec.{u} (Nat.succ n\u271d) x : TypeVec.drop (\u03b1 ::: \u03b9) Fin2.fz \u22a2 subtypeVal (repeatEq \u03b1) Fin2.fz (diagSub Fin2.fz x) = prod.diag Fin2.fz x ** rfl ** case h\u2080.a.h.fs n : \u2115 \u03b9 \u03b9' : Type u n\u271d : \u2115 a\u271d : Fin2 n\u271d ih : \u2200 {\u03b1 : TypeVec.{u} n\u271d} (x : TypeVec.drop (\u03b1 ::: \u03b9) a\u271d), subtypeVal (repeatEq \u03b1) a\u271d (diagSub a\u271d x) = prod.diag a\u271d x \u03b1 : TypeVec.{u} (Nat.succ n\u271d) x : TypeVec.drop (\u03b1 ::: \u03b9) (Fin2.fs a\u271d) \u22a2 subtypeVal (repeatEq \u03b1) (Fin2.fs a\u271d) (diagSub (Fin2.fs a\u271d) x) = prod.diag (Fin2.fs a\u271d) x ** apply ih ** Qed",
        "informal": ""
    },
    {
        "formal": "WithBot.image_coe_Ico ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some '' Ico a b = Ico \u2191a \u2191b ** rw [\u2190 preimage_coe_Ico, image_preimage_eq_inter_range, range_coe,\n  inter_eq_self_of_subset_left\n    (Subset.trans Ico_subset_Ici_self <| Ici_subset_Ioi.2 <| bot_lt_coe a)] ** Qed",
        "informal": ""
    },
    {
        "formal": "order_eq_card_zpowers' ** n : \u2115 A : Type u_1 R : Type u_2 inst\u271d\u00b2 : AddGroup A inst\u271d\u00b9 : Ring R \u03b1 : Type u_3 inst\u271d : Group \u03b1 a : \u03b1 \u22a2 orderOf a = Nat.card { x // x \u2208 zpowers a } ** have := Nat.card_congr (MulAction.orbitZpowersEquiv a (1 : \u03b1)) ** n : \u2115 A : Type u_1 R : Type u_2 inst\u271d\u00b2 : AddGroup A inst\u271d\u00b9 : Ring R \u03b1 : Type u_3 inst\u271d : Group \u03b1 a : \u03b1 this :   Nat.card \u2191(MulAction.orbit { x // x \u2208 zpowers a } 1) =     Nat.card (ZMod (Function.minimalPeriod ((fun x x_1 => x \u2022 x_1) a) 1)) \u22a2 orderOf a = Nat.card { x // x \u2208 zpowers a } ** rwa [Nat.card_zmod, orbit_subgroup_one_eq_self, eq_comm] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.pred_neg_pred ** a : \u2124 \u22a2 pred (-pred a) = -a ** rw [neg_pred, pred_succ] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.ncard_inter_add_ncard_union ** \u03b1 : Type u_1 s\u271d t\u271d s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d ht : autoParam (Set.Finite t) _auto\u271d \u22a2 ncard (s \u2229 t) + ncard (s \u222a t) = ncard s + ncard t ** rw [add_comm, ncard_union_add_ncard_inter _ _ hs ht] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.indep_iSup_limsup ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 \u03b1 : Type u_3 p : Set \u03b9 \u2192 Prop f : Filter \u03b9 ns : \u03b1 \u2192 Set \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s hf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f hns : Directed (fun x x_1 => x \u2264 x_1) ns hnsp : \u2200 (a : \u03b1), p (ns a) hns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a \u22a2 Indep (\u2a06 n, s n) (limsup s f) ** suffices (\u2a06 a, \u2a06 n \u2208 ns a, s n) = \u2a06 n, s n by\n  rw [\u2190 this]\n  exact indep_iSup_directed_limsup h_le h_indep hf hns hnsp ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 \u03b1 : Type u_3 p : Set \u03b9 \u2192 Prop f : Filter \u03b9 ns : \u03b1 \u2192 Set \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s hf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f hns : Directed (fun x x_1 => x \u2264 x_1) ns hnsp : \u2200 (a : \u03b1), p (ns a) hns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a \u22a2 \u2a06 a, \u2a06 n \u2208 ns a, s n = \u2a06 n, s n ** rw [iSup_comm] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 \u03b1 : Type u_3 p : Set \u03b9 \u2192 Prop f : Filter \u03b9 ns : \u03b1 \u2192 Set \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s hf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f hns : Directed (fun x x_1 => x \u2264 x_1) ns hnsp : \u2200 (a : \u03b1), p (ns a) hns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a \u22a2 \u2a06 j, \u2a06 i, \u2a06 (_ : j \u2208 ns i), s j = \u2a06 n, s n ** refine' iSup_congr fun n => _ ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 \u03b1 : Type u_3 p : Set \u03b9 \u2192 Prop f : Filter \u03b9 ns : \u03b1 \u2192 Set \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s hf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f hns : Directed (fun x x_1 => x \u2264 x_1) ns hnsp : \u2200 (a : \u03b1), p (ns a) hns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a n : \u03b9 \u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = s n ** have h : \u2a06 (i : \u03b1) (_ : n \u2208 ns i), s n = \u2a06 _ : \u2203 i, n \u2208 ns i, s n := by rw [iSup_exists] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 \u03b1 : Type u_3 p : Set \u03b9 \u2192 Prop f : Filter \u03b9 ns : \u03b1 \u2192 Set \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s hf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f hns : Directed (fun x x_1 => x \u2264 x_1) ns hnsp : \u2200 (a : \u03b1), p (ns a) hns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a n : \u03b9 h : \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = \u2a06 (_ : \u2203 i, n \u2208 ns i), s n \u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = s n ** haveI : Nonempty (\u2203 i : \u03b1, n \u2208 ns i) := \u27e8hns_univ n\u27e9 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 \u03b1 : Type u_3 p : Set \u03b9 \u2192 Prop f : Filter \u03b9 ns : \u03b1 \u2192 Set \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s hf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f hns : Directed (fun x x_1 => x \u2264 x_1) ns hnsp : \u2200 (a : \u03b1), p (ns a) hns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a n : \u03b9 h : \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = \u2a06 (_ : \u2203 i, n \u2208 ns i), s n this : Nonempty (\u2203 i, n \u2208 ns i) \u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = s n ** rw [h, iSup_const] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 \u03b1 : Type u_3 p : Set \u03b9 \u2192 Prop f : Filter \u03b9 ns : \u03b1 \u2192 Set \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s hf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f hns : Directed (fun x x_1 => x \u2264 x_1) ns hnsp : \u2200 (a : \u03b1), p (ns a) hns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a this : \u2a06 a, \u2a06 n \u2208 ns a, s n = \u2a06 n, s n \u22a2 Indep (\u2a06 n, s n) (limsup s f) ** rw [\u2190 this] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 \u03b1 : Type u_3 p : Set \u03b9 \u2192 Prop f : Filter \u03b9 ns : \u03b1 \u2192 Set \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s hf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f hns : Directed (fun x x_1 => x \u2264 x_1) ns hnsp : \u2200 (a : \u03b1), p (ns a) hns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a this : \u2a06 a, \u2a06 n \u2208 ns a, s n = \u2a06 n, s n \u22a2 Indep (\u2a06 a, \u2a06 n \u2208 ns a, s n) (limsup s f) ** exact indep_iSup_directed_limsup h_le h_indep hf hns hnsp ** \u03a9 : Type u_1 \u03b9 : Type u_2 m m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d : IsProbabilityMeasure \u03bc s : \u03b9 \u2192 MeasurableSpace \u03a9 \u03b1 : Type u_3 p : Set \u03b9 \u2192 Prop f : Filter \u03b9 ns : \u03b1 \u2192 Set \u03b9 h_le : \u2200 (n : \u03b9), s n \u2264 m0 h_indep : iIndep s hf : \u2200 (t : Set \u03b9), p t \u2192 t\u1d9c \u2208 f hns : Directed (fun x x_1 => x \u2264 x_1) ns hnsp : \u2200 (a : \u03b1), p (ns a) hns_univ : \u2200 (n : \u03b9), \u2203 a, n \u2208 ns a n : \u03b9 \u22a2 \u2a06 i, \u2a06 (_ : n \u2208 ns i), s n = \u2a06 (_ : \u2203 i, n \u2208 ns i), s n ** rw [iSup_exists] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.exists_superset_subset_encard_eq ** \u03b1 : Type u_1 s t : Set \u03b1 k : (fun x => \u2115\u221e) (PartENat.card \u2191s) hst : s \u2286 t hsk : encard s \u2264 k hkt : k \u2264 encard t \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k ** obtain (hs | hs) := eq_or_ne s.encard \u22a4 ** case inr \u03b1 : Type u_1 s t : Set \u03b1 k : (fun x => \u2115\u221e) (PartENat.card \u2191s) hst : s \u2286 t hsk : encard s \u2264 k hkt : k \u2264 encard t hs : encard s \u2260 \u22a4 \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k ** obtain \u27e8k, rfl\u27e9 := exists_add_of_le hsk ** case inr.intro \u03b1 : Type u_1 s t : Set \u03b1 hst : s \u2286 t hs : encard s \u2260 \u22a4 k : \u2115\u221e hsk : encard s \u2264 encard s + k hkt : encard s + k \u2264 encard t \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + k ** obtain \u27e8k', hk'\u27e9 := exists_add_of_le hkt ** case inr.intro.intro \u03b1 : Type u_1 s t : Set \u03b1 hst : s \u2286 t hs : encard s \u2260 \u22a4 k : \u2115\u221e hsk : encard s \u2264 encard s + k hkt : encard s + k \u2264 encard t k' : \u2115\u221e hk' : encard t = encard s + k + k' \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + k ** have hk : k \u2264 encard (t \\ s) ** case inr.intro.intro \u03b1 : Type u_1 s t : Set \u03b1 hst : s \u2286 t hs : encard s \u2260 \u22a4 k : \u2115\u221e hsk : encard s \u2264 encard s + k hkt : encard s + k \u2264 encard t k' : \u2115\u221e hk' : encard t = encard s + k + k' hk : k \u2264 encard (t \\ s) \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + k ** obtain \u27e8r', hr', rfl\u27e9 := exists_subset_encard_eq hk ** case inr.intro.intro.intro.intro \u03b1 : Type u_1 s t : Set \u03b1 hst : s \u2286 t hs : encard s \u2260 \u22a4 k' : \u2115\u221e r' : Set \u03b1 hr' : r' \u2286 t \\ s hsk : encard s \u2264 encard s + encard r' hkt : encard s + encard r' \u2264 encard t hk' : encard t = encard s + encard r' + k' hk : encard r' \u2264 encard (t \\ s) \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = encard s + encard r' ** refine' \u27e8s \u222a r', subset_union_left _ _, union_subset hst (hr'.trans (diff_subset _ _)), _\u27e9 ** case inr.intro.intro.intro.intro \u03b1 : Type u_1 s t : Set \u03b1 hst : s \u2286 t hs : encard s \u2260 \u22a4 k' : \u2115\u221e r' : Set \u03b1 hr' : r' \u2286 t \\ s hsk : encard s \u2264 encard s + encard r' hkt : encard s + encard r' \u2264 encard t hk' : encard t = encard s + encard r' + k' hk : encard r' \u2264 encard (t \\ s) \u22a2 encard (s \u222a r') = encard s + encard r' ** rw [encard_union_eq (disjoint_of_subset_right hr' disjoint_sdiff_right)] ** case inl \u03b1 : Type u_1 s t : Set \u03b1 k : (fun x => \u2115\u221e) (PartENat.card \u2191s) hst : s \u2286 t hsk : encard s \u2264 k hkt : k \u2264 encard t hs : encard s = \u22a4 \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k ** rw [hs, top_le_iff] at hsk ** case inl \u03b1 : Type u_1 s t : Set \u03b1 k : (fun x => \u2115\u221e) (PartENat.card \u2191s) hst : s \u2286 t hsk : k = \u22a4 hkt : k \u2264 encard t hs : encard s = \u22a4 \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = k ** subst hsk ** case inl \u03b1 : Type u_1 s t : Set \u03b1 hst : s \u2286 t hs : encard s = \u22a4 hkt : \u22a4 \u2264 encard t \u22a2 \u2203 r, s \u2286 r \u2227 r \u2286 t \u2227 encard r = \u22a4 ** exact \u27e8s, Subset.rfl, hst, hs\u27e9 ** case hk \u03b1 : Type u_1 s t : Set \u03b1 hst : s \u2286 t hs : encard s \u2260 \u22a4 k : \u2115\u221e hsk : encard s \u2264 encard s + k hkt : encard s + k \u2264 encard t k' : \u2115\u221e hk' : encard t = encard s + k + k' \u22a2 k \u2264 encard (t \\ s) ** rw [\u2190encard_diff_add_encard_of_subset hst, add_comm] at hkt ** case hk \u03b1 : Type u_1 s t : Set \u03b1 hst : s \u2286 t hs : encard s \u2260 \u22a4 k : \u2115\u221e hsk : encard s \u2264 encard s + k hkt : k + encard s \u2264 encard (t \\ s) + encard s k' : \u2115\u221e hk' : encard t = encard s + k + k' \u22a2 k \u2264 encard (t \\ s) ** exact WithTop.le_of_add_le_add_right hs hkt ** Qed",
        "informal": ""
    },
    {
        "formal": "Finmap.insert_insert ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v inst\u271d : DecidableEq \u03b1 a : \u03b1 b b' : \u03b2 a s\u271d : Finmap \u03b2 s : AList \u03b2 \u22a2 insert a b' (insert a b \u27e6s\u27e7) = insert a b' \u27e6s\u27e7 ** simp only [insert_toFinmap, AList.insert_insert] ** Qed",
        "informal": ""
    },
    {
        "formal": "Turing.TM2to1.tr_respects_aux ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3 v : \u03c3 T : ListBlank ((i : K) \u2192 Option (\u0393 i)) k : K S : (k : K) \u2192 List (\u0393 k) hT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},     (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b \u22a2 \u2203 b,     TrCfg (TM2.stepAux (stRun o q) v S) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun o q)) v (Tape.mk' \u2205 (addBottom T))) b ** simp only [trNormal_run, step_run] ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3 v : \u03c3 T : ListBlank ((i : K) \u2192 Option (\u0393 i)) k : K S : (k : K) \u2192 List (\u0393 k) hT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},     (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b \u22a2 \u2203 b,     TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b ** have hgo := tr_respects_aux\u2081 M o q v (hT k) _ le_rfl ** K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3 v : \u03c3 T : ListBlank ((i : K) \u2192 Option (\u0393 i)) k : K S : (k : K) \u2192 List (\u0393 k) hT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},     (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b hgo :   Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) } \u22a2 \u2203 b,     TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b ** obtain \u27e8T', hT', hrun\u27e9 := tr_respects_aux\u2082 hT o ** case intro.intro K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3 v : \u03c3 T : ListBlank ((i : K) \u2192 Option (\u0393 i)) k : K S : (k : K) \u2192 List (\u0393 k) hT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},     (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b hgo :   Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) } T' : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT' :   \u2200 (k_1 : K),     ListBlank.map (proj k_1) T' =       ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k_1))) hrun :   TM1.stepAux (trStAct ?m.723095 o) ?m.723096 ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =     TM1.stepAux ?m.723095 (stVar ?m.723096 (S k) o)       ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k)]         (Tape.mk' \u2205 (addBottom T'))) \u22a2 \u2203 b,     TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b ** have := hgo.tail' rfl ** case intro.intro K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3 v : \u03c3 T : ListBlank ((i : K) \u2192 Option (\u0393 i)) k : K S : (k : K) \u2192 List (\u0393 k) hT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},     (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b hgo :   Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) } T' : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT' :   \u2200 (k_1 : K),     ListBlank.map (proj k_1) T' =       ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k_1))) hrun :   TM1.stepAux (trStAct ?m.723095 o) ?m.723096 ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =     TM1.stepAux ?m.723095 (stVar ?m.723096 (S k) o)       ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite ?m.723096 (S k) o) k)]         (Tape.mk' \u2205 (addBottom T'))) this :   Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     (TM1.stepAux (tr M (go k o q)) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))) \u22a2 \u2203 b,     TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b ** rw [tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_snd,\n  stk_nth_val _ (hT k), List.get?_len_le (le_of_eq (List.length_reverse _)), Option.isNone, cond,\n  hrun, TM1.stepAux] at this ** case intro.intro K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3 v : \u03c3 T : ListBlank ((i : K) \u2192 Option (\u0393 i)) k : K S : (k : K) \u2192 List (\u0393 k) hT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},     (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b hgo :   Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) } T' : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT' :   \u2200 (k_1 : K),     ListBlank.map (proj k_1) T' =       ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1))) hrun :   TM1.stepAux (trStAct (goto fun x x => ret q) o) v       ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =     TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)       ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T'))) this :   Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     (match       match none with       | some val => false       | none => true with     | true =>       TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)         ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]           (Tape.mk' \u2205 (addBottom T')))     | false =>       TM1.stepAux (goto fun x x => go k o q) v         (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))))) \u22a2 \u2203 b,     TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b ** obtain \u27e8c, gc, rc\u27e9 := IH hT' ** case intro.intro.intro.intro K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3 v : \u03c3 T : ListBlank ((i : K) \u2192 Option (\u0393 i)) k : K S : (k : K) \u2192 List (\u0393 k) hT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},     (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b hgo :   Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) } T' : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT' :   \u2200 (k_1 : K),     ListBlank.map (proj k_1) T' =       ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1))) hrun :   TM1.stepAux (trStAct (goto fun x x => ret q) o) v       ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =     TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)       ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T'))) this :   Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     (match       match none with       | some val => false       | none => true with     | true =>       TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)         ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]           (Tape.mk' \u2205 (addBottom T')))     | false =>       TM1.stepAux (goto fun x x => go k o q) v         (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))))) c : TM1.Cfg \u0393' \u039b' \u03c3 gc : TrCfg (TM2.stepAux q ?m.726996 fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c rc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) ?m.726996 (Tape.mk' \u2205 (addBottom T'))) c \u22a2 \u2203 b,     TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227       Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b ** refine' \u27e8c, gc, (this.to\u2080.trans (tr_respects_aux\u2083 M _) c (TransGen.head' rfl _)).to_reflTransGen\u27e9 ** case intro.intro.intro.intro K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3 v : \u03c3 T : ListBlank ((i : K) \u2192 Option (\u0393 i)) k : K S : (k : K) \u2192 List (\u0393 k) hT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},     (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b hgo :   Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) } T' : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT' :   \u2200 (k_1 : K),     ListBlank.map (proj k_1) T' =       ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1))) hrun :   TM1.stepAux (trStAct (goto fun x x => ret q) o) v       ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =     TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)       ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T'))) this :   Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     (match       match none with       | some val => false       | none => true with     | true =>       TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)         ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]           (Tape.mk' \u2205 (addBottom T')))     | false =>       TM1.stepAux (goto fun x x => go k o q) v         (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))))) c : TM1.Cfg \u0393' \u039b' \u03c3 gc : TrCfg (TM2.stepAux q (stVar v (S k) o) fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c rc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c \u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)     (TM1.stepAux (tr M (ret q)) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c ** rw [tr, TM1.stepAux, Tape.mk'_head, addBottom_head_fst] ** case intro.intro.intro.intro K : Type u_1 inst\u271d\u00b2 : DecidableEq K \u0393 : K \u2192 Type u_2 \u039b : Type u_3 inst\u271d\u00b9 : Inhabited \u039b \u03c3 : Type u_4 inst\u271d : Inhabited \u03c3 M : \u039b \u2192 Stmt\u2082 q : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3 v : \u03c3 T : ListBlank ((i : K) \u2192 Option (\u0393 i)) k : K S : (k : K) \u2192 List (\u0393 k) hT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k IH :   \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},     (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192       \u2203 b,         TrCfg (TM2.stepAux q v S) b \u2227           Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b hgo :   Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) } T' : ListBlank ((k : K) \u2192 Option (\u0393 k)) hT' :   \u2200 (k_1 : K),     ListBlank.map (proj k_1) T' =       ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1))) hrun :   TM1.stepAux (trStAct (goto fun x x => ret q) o) v       ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =     TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)       ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T'))) this :   Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }     (match       match none with       | some val => false       | none => true with     | true =>       TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)         ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]           (Tape.mk' \u2205 (addBottom T')))     | false =>       TM1.stepAux (goto fun x x => go k o q) v         (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))))) c : TM1.Cfg \u0393' \u039b' \u03c3 gc : TrCfg (TM2.stepAux q (stVar v (S k) o) fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c rc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c \u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)     (bif true then TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))     else TM1.stepAux (move Dir.left (goto fun x x => ret q)) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T')))     c ** exact rc ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.IndepSets.indep ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba \u22a2 Indep m1 m2 \u03ba ** intros t1 t2 ht1 ht2 ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t1 \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t1 * \u2191\u2191(\u2191\u03ba a) t2 ** refine @induction_on_inter _ (fun t \u21a6 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u03ba a (t \u2229 t2) = \u03ba a t * \u03ba a t2) _ m1 hpm1 hp1\n  ?_ ?_ ?_ ?_ _ ht1 ** case refine_1 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} \u22a2 (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) \u2205 ** simp only [Set.empty_inter, measure_empty, zero_mul, eq_self_iff_true,\n  Filter.eventually_true] ** case refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} \u22a2 \u2200 (t : Set \u03a9), t \u2208 p1 \u2192 (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) t ** intros t ht_mem_p1 ** case refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} t : Set \u03a9 ht_mem_p1 : t \u2208 p1 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2 ** have ht1 : MeasurableSet[m] t := by\n  refine h1 _ ?_\n  rw [hpm1]\n  exact measurableSet_generateFrom ht_mem_p1 ** case refine_2 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1\u271d : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} t : Set \u03a9 ht_mem_p1 : t \u2208 p1 ht1 : MeasurableSet t \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2 ** exact IndepSets.indep_aux h2 hp2 hpm2 hyp ht_mem_p1 ht1 ht2 ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} t : Set \u03a9 ht_mem_p1 : t \u2208 p1 \u22a2 MeasurableSet t ** exact measurableSet_generateFrom ht_mem_p1 ** case refine_3 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} \u22a2 \u2200 (t : Set \u03a9),     MeasurableSet t \u2192       (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) t \u2192         (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) t\u1d9c ** intros t ht h ** case refine_3 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} t : Set \u03a9 ht : MeasurableSet t h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t\u1d9c \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t\u1d9c * \u2191\u2191(\u2191\u03ba a) t2 ** filter_upwards [h] with a ha ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} t : Set \u03a9 ht : MeasurableSet t h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 \u2191\u2191(\u2191\u03ba a) (t\u1d9c \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t\u1d9c * \u2191\u2191(\u2191\u03ba a) t2 ** have : t\u1d9c \u2229 t2 = t2 \\ (t \u2229 t2) := by\n  rw [Set.inter_comm t, Set.diff_self_inter, Set.diff_eq_compl_inter] ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} t : Set \u03a9 ht : MeasurableSet t h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2 this : t\u1d9c \u2229 t2 = t2 \\ (t \u2229 t2) \u22a2 \u2191\u2191(\u2191\u03ba a) (t\u1d9c \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t\u1d9c * \u2191\u2191(\u2191\u03ba a) t2 ** rw [this, Set.inter_comm t t2,\n  measure_diff (Set.inter_subset_left _ _) ((h2 _ ht2).inter (h1 _ ht))\n    (measure_ne_top (\u03ba a) _),\n  Set.inter_comm, ha, measure_compl (h1 _ ht) (measure_ne_top (\u03ba a) t), measure_univ,\n  mul_comm (1 - \u03ba a t), ENNReal.mul_sub (fun _ _ \u21a6 measure_ne_top (\u03ba a) _), mul_one, mul_comm] ** \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} t : Set \u03a9 ht : MeasurableSet t h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 t\u1d9c \u2229 t2 = t2 \\ (t \u2229 t2) ** rw [Set.inter_comm t, Set.diff_self_inter, Set.diff_eq_compl_inter] ** case refine_4 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} \u22a2 \u2200 (f : \u2115 \u2192 Set \u03a9),     Pairwise (Disjoint on f) \u2192       (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192         (\u2200 (i : \u2115), (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) (f i)) \u2192           (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) (\u22c3 i, f i) ** intros f hf_disj hf_meas h ** case refine_4 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200 (i : \u2115), (fun t => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) (t \u2229 t2) = \u2191\u2191(\u2191\u03ba a) t * \u2191\u2191(\u2191\u03ba a) t2) (f i) \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) ((\u22c3 i, f i) \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2 ** rw [\u2190 ae_all_iff] at h ** case refine_4 \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(\u2191\u03ba a) ((\u22c3 i, f i) \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2 ** filter_upwards [h] with a ha ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 \u2191\u2191(\u2191\u03ba a) ((\u22c3 i, f i) \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2 ** rw [Set.inter_comm, Set.inter_iUnion, measure_iUnion] ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = \u2191\u2191(\u2191\u03ba a) (\u22c3 i, f i) * \u2191\u2191(\u2191\u03ba a) t2 ** rw [measure_iUnion hf_disj (fun i \u21a6 h1 _ (hf_meas i))] ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = (\u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i)) * \u2191\u2191(\u2191\u03ba a) t2 ** rw [\u2190 ENNReal.tsum_mul_right] ** case h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = \u2211' (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 ** congr 1 with i ** case h.e_f.h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 i : \u2115 \u22a2 \u2191\u2191(\u2191\u03ba a) (t2 \u2229 f i) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 ** rw [Set.inter_comm t2, ha i] ** case h.hn \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 Pairwise (Disjoint on fun i => t2 \u2229 f i) ** intros i j hij ** case h.hn \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 i j : \u2115 hij : i \u2260 j \u22a2 (Disjoint on fun i => t2 \u2229 f i) i j ** rw [Function.onFun, Set.inter_comm t2, Set.inter_comm t2] ** case h.hn \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 i j : \u2115 hij : i \u2260 j \u22a2 Disjoint (f i \u2229 t2) (f j \u2229 t2) ** exact Disjoint.inter_left _ (Disjoint.inter_right _ (hf_disj hij)) ** case h.h \u03b1 : Type u_1 \u03a9 : Type u_2 \u03b9 : Type u_3 _m\u03b1 : MeasurableSpace \u03b1 m1 m2 m : MeasurableSpace \u03a9 \u03ba : { x // x \u2208 kernel \u03b1 \u03a9 } \u03bc : Measure \u03b1 inst\u271d : IsMarkovKernel \u03ba p1 p2 : Set (Set \u03a9) h1 : m1 \u2264 m h2 : m2 \u2264 m hp1 : IsPiSystem p1 hp2 : IsPiSystem p2 hpm1 : m1 = generateFrom p1 hpm2 : m2 = generateFrom p2 hyp : IndepSets p1 p2 \u03ba t1 t2 : Set \u03a9 ht1 : t1 \u2208 {s | MeasurableSet s} ht2 : t2 \u2208 {s | MeasurableSet s} f : \u2115 \u2192 Set \u03a9 hf_disj : Pairwise (Disjoint on f) hf_meas : \u2200 (i : \u2115), MeasurableSet (f i) h : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 a : \u03b1 ha : \u2200 (i : \u2115), \u2191\u2191(\u2191\u03ba a) (f i \u2229 t2) = \u2191\u2191(\u2191\u03ba a) (f i) * \u2191\u2191(\u2191\u03ba a) t2 \u22a2 \u2200 (i : \u2115), MeasurableSet (t2 \u2229 f i) ** exact fun i \u21a6 (h2 _ ht2).inter (h1 _ (hf_meas i)) ** Qed",
        "informal": ""
    },
    {
        "formal": "Partrec.sum_casesOn_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f : \u03b1 \u2192 \u03b2 \u2295 \u03b3 g : \u03b1 \u2192 \u03b2 \u2192 \u03c3 h : \u03b1 \u2192 \u03b3 \u2192. \u03c3 hf : Computable f hg : Computable\u2082 g hh : Partrec\u2082 h this :   Partrec fun a =>     Option.casesOn (Sum.casesOn (f a) (fun x => Option.none) Option.some)       (Part.some (Sum.casesOn (f a) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>       Part.map Option.some (h a c) a : \u03b1 \u22a2 (Option.casesOn (Sum.casesOn (f a) (fun x => Option.none) Option.some)       (Part.some (Sum.casesOn (f a) (fun b => Option.some (g a b)) fun x => Option.none)) fun c =>       Part.map Option.some (h a c)) =     Part.map Option.some (Sum.casesOn (f a) (fun b => Part.some (g a b)) (h a)) ** cases f a <;> simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.lintegral_iSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f \u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** set c : \u211d\u22650 \u2192 \u211d\u22650\u221e := (\u2191) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some \u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** set F := fun a : \u03b1 => \u2a06 n, f n a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a \u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** have _ : Measurable F := measurable_iSup hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F \u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** refine' le_antisymm _ (iSup_lintegral_le _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F \u22a2 lintegral \u03bc F \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** rw [lintegral_eq_nnreal] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F \u22a2 \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2a06 n, f n x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2264     \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** refine' iSup_le fun s => iSup_le fun hsf => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x \u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** refine' ENNReal.le_of_forall_lt_one_mul_le fun a ha => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x a : \u211d\u22650\u221e ha : a < 1 \u22a2 a * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** rcases ENNReal.lt_iff_exists_coe.1 ha with \u27e8r, rfl, _\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha : \u2191r < 1 \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** have ha : r < 1 := ENNReal.coe_lt_coe.1 ha ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** let rs := s.map fun a => r * a ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** have eq_rs : (const \u03b1 r : \u03b1 \u2192\u209b \u211d\u22650\u221e) * map c s = rs.map c := by\n  ext1 a\n  exact ENNReal.coe_mul.symm ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** have mono : \u2200 r : \u211d\u22650\u221e, Monotone fun n => rs.map c \u207b\u00b9' {r} \u2229 { a | r \u2264 f n a } := by\n  intro r i j h\n  refine' inter_subset_inter (Subset.refl _) _\n  intro x (hx : r \u2264 f i x)\n  exact le_trans hx (h_mono h x) ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** have h_meas : \u2200 n, MeasurableSet { a : \u03b1 | (\u21d1(map c rs)) a \u2264 f n a } := fun n =>\n  measurableSet_le (SimpleFunc.measurable _) (hf n) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s \u22a2 const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs ** ext1 a ** case H \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s a : \u03b1 \u22a2 \u2191(const \u03b1 \u2191r * SimpleFunc.map c s) a = \u2191(SimpleFunc.map c rs) a ** exact ENNReal.coe_mul.symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs \u22a2 \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} ** intro p ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e \u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} ** rw [\u2190 inter_iUnion] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e \u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a} ** nth_rw 1 [\u2190 inter_univ (map c rs \u207b\u00b9' {p})] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e \u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 univ = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a} ** refine' Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e x : \u03b1 hx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a} ** by_cases p_eq : p = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e x : \u03b1 hx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} p_eq : \u00acp = 0 \u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a} ** simp only [coe_map, mem_preimage, Function.comp_apply, mem_singleton_iff] at hx ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e x : \u03b1 p_eq : \u00acp = 0 hx : \u2191(r * \u2191s x) = p \u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a} ** subst hx ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** have : r * s x \u2260 0 := by rwa [Ne, \u2190 ENNReal.coe_eq_zero] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this : r * \u2191s x \u2260 0 \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** have : s x \u2260 0 := by\n  refine' mt _ this\n  intro h\n  rw [h, mul_zero] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d : r * \u2191s x \u2260 0 this : \u2191s x \u2260 0 \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** have : (rs.map c) x < \u2a06 n : \u2115, f n x := by\n  refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x)\n  suffices r * s x < 1 * s x by simpa\n  exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d\u00b9 : r * \u2191s x \u2260 0 this\u271d : \u2191s x \u2260 0 this : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** rcases lt_iSup_iff.1 this with \u27e8i, hi\u27e9 ** case neg.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d\u00b9 : r * \u2191s x \u2260 0 this\u271d : \u2191s x \u2260 0 this : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x i : \u2115 hi : \u2191(SimpleFunc.map c rs) x < f i x \u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a} ** exact mem_iUnion.2 \u27e8i, le_of_lt hi\u27e9 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs p : \u211d\u22650\u221e x : \u03b1 hx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} p_eq : p = 0 \u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a} ** simp [p_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 \u22a2 r * \u2191s x \u2260 0 ** rwa [Ne, \u2190 ENNReal.coe_eq_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this : r * \u2191s x \u2260 0 \u22a2 \u2191s x \u2260 0 ** refine' mt _ this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this : r * \u2191s x \u2260 0 \u22a2 \u2191s x = 0 \u2192 r * \u2191s x = 0 ** intro h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this : r * \u2191s x \u2260 0 h : \u2191s x = 0 \u22a2 r * \u2191s x = 0 ** rw [h, mul_zero] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d : r * \u2191s x \u2260 0 this : \u2191s x \u2260 0 \u22a2 \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x ** refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d : r * \u2191s x \u2260 0 this : \u2191s x \u2260 0 \u22a2 \u2191rs x < \u2191s x ** suffices r * s x < 1 * s x by simpa ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d : r * \u2191s x \u2260 0 this : \u2191s x \u2260 0 \u22a2 r * \u2191s x < 1 * \u2191s x ** exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs x : \u03b1 p_eq : \u00ac\u2191(r * \u2191s x) = 0 this\u271d\u00b9 : r * \u2191s x \u2260 0 this\u271d : \u2191s x \u2260 0 this : r * \u2191s x < 1 * \u2191s x \u22a2 \u2191rs x < \u2191s x ** simpa ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} \u22a2 \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} ** intro r i j h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} r : \u211d\u22650\u221e i j : \u2115 h : i \u2264 j \u22a2 (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) i \u2264     (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) j ** refine' inter_subset_inter (Subset.refl _) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} r : \u211d\u22650\u221e i j : \u2115 h : i \u2264 j \u22a2 {a | r \u2264 f i a} \u2286 {a | r \u2264 f j a} ** intro x (hx : r \u2264 f i x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} r : \u211d\u22650\u221e i j : \u2115 h : i \u2264 j x : \u03b1 hx : r \u2264 f i x \u22a2 x \u2208 {a | r \u2264 f j a} ** exact le_trans hx (h_mono h x) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map c s) \u03bc =     \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r}) ** rw [\u2190 const_mul_lintegral, eq_rs, SimpleFunc.lintegral] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r}) =     \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) ** simp only [(eq _).symm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d\u00b9 : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} x : \u211d\u22650\u221e x\u271d : x \u2208 SimpleFunc.range (SimpleFunc.map c rs) \u22a2 x * \u2191\u2191\u03bc (\u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a}) =     \u2a06 n, x * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a}) ** rw [measure_iUnion_eq_iSup (directed_of_sup <| mono x), ENNReal.mul_iSup] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), \u2a06 n, r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =     \u2a06 n, \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) ** rw [ENNReal.finset_sum_iSup_nat] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \u22a2 \u2200 (a : \u211d\u22650\u221e), Monotone fun n => a * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {a} \u2229 {a_1 | a \u2264 f n a_1}) ** intro p i j h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} p : \u211d\u22650\u221e i j : \u2115 h : i \u2264 j \u22a2 (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) i \u2264     (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) j ** exact mul_le_mul_left' (measure_mono <| mono p h) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \u22a2 \u2a06 n, \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264     \u2a06 n, SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc ** refine' iSup_mono fun n => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 \u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264     SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc ** rw [restrict_lintegral _ (h_meas n)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 \u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264     \u2211 r in SimpleFunc.range (SimpleFunc.map c rs),       r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) ** refine' le_of_eq (Finset.sum_congr rfl fun r _ => _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d\u00b9 : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 r : \u211d\u22650\u221e x\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs) \u22a2 r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =     r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) ** congr 2 with a ** case e_a.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d\u00b9 : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 r : \u211d\u22650\u221e x\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs) a : \u03b1 \u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} \u2194     a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} ** refine' and_congr_right _ ** case e_a.e_a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d\u00b9 : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r\u271d : \u211d\u22650 right\u271d ha\u271d : \u2191r\u271d < 1 ha : r\u271d < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s eq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 r : \u211d\u22650\u221e x\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs) a : \u03b1 \u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2192 (a \u2208 {a | r \u2264 f n a} \u2194 a \u2208 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) ** simp (config := { contextual := true }) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} \u22a2 \u2a06 n, SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264     \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** refine' iSup_mono fun n => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 \u22a2 SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** rw [\u2190 SimpleFunc.lintegral_eq_lintegral] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 \u22a2 \u222b\u207b (a : \u03b1), \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** refine' lintegral_mono fun a => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a} n : \u2115 a : \u03b1 \u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a ** simp only [map_apply] at h_meas ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), Measurable (f n) h_mono : Monotone f c : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some F : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a x\u271d : Measurable F s : \u03b1 \u2192\u209b \u211d\u22650 hsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x r : \u211d\u22650 right\u271d ha\u271d : \u2191r < 1 ha : r < 1 rs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s eq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs eq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a} mono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} h_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(r * \u2191s a) \u2264 f n a} n : \u2115 a : \u03b1 \u22a2 indicator {a | \u2191(r * \u2191s a) \u2264 f n a} (fun x => \u2191(r * \u2191s x)) a \u2264 f n a ** exact indicator_apply_le id ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_mul_left_Ioo ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a : \u03b1 h : 0 < a b c : \u03b1 \u22a2 (fun x x_1 => x * x_1) a '' Ioo b c = Ioo (a * b) (a * c) ** convert image_mul_right_Ioo b c h using 1 <;> simp only [mul_comm _ a] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.condCount_inter_self ** \u03a9 : Type u_1 inst\u271d\u00b9 : MeasurableSpace \u03a9 inst\u271d : MeasurableSingletonClass \u03a9 s t u : Set \u03a9 hs : Set.Finite s \u22a2 \u2191\u2191(condCount s) (s \u2229 t) = \u2191\u2191(condCount s) t ** rw [condCount, cond_inter_self _ hs.measurableSet] ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.dual_ordConnectedComponent ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x : \u03b1 \u22a2 \u2191toDual x \u2208 ordConnectedComponent (\u2191ofDual \u207b\u00b9' s) (\u2191toDual x\u271d) \u2194 \u2191toDual x \u2208 \u2191ofDual \u207b\u00b9' ordConnectedComponent s x\u271d ** rw [mem_ordConnectedComponent, dual_uIcc] ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 s t : Set \u03b1 x\u271d y z x : \u03b1 \u22a2 \u2191ofDual \u207b\u00b9' [[x\u271d, x]] \u2286 \u2191ofDual \u207b\u00b9' s \u2194 \u2191toDual x \u2208 \u2191ofDual \u207b\u00b9' ordConnectedComponent s x\u271d ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto ** \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) E : Set \u03a9 E_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0 \u22a2 Tendsto (fun i => (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(\u03bcs i) s)) E) L (\ud835\udcdd ((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) E)) ** have E_nullbdry' : (\u03bc : Measure \u03a9) (frontier E) = 0 := by\n  rw [\u2190 ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure, E_nullbdry, ENNReal.coe_zero] ** \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) E : Set \u03a9 E_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0 E_nullbdry' : \u2191\u2191\u2191\u03bc (frontier E) = 0 \u22a2 Tendsto (fun i => (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(\u03bcs i) s)) E) L (\ud835\udcdd ((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) E)) ** have key := ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto' \u03bcs_lim E_nullbdry' ** \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) E : Set \u03a9 E_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0 E_nullbdry' : \u2191\u2191\u2191\u03bc (frontier E) = 0 key : Tendsto (fun i => \u2191\u2191\u2191(\u03bcs i) E) L (\ud835\udcdd (\u2191\u2191\u2191\u03bc E)) \u22a2 Tendsto (fun i => (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(\u03bcs i) s)) E) L (\ud835\udcdd ((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) E)) ** exact (ENNReal.tendsto_toNNReal (measure_ne_top (\u2191\u03bc) E)).comp key ** \u03a9\u271d : Type u_1 inst\u271d\u00b3 : MeasurableSpace \u03a9\u271d \u03a9 : Type u_2 \u03b9 : Type u_3 L : Filter \u03b9 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : PseudoEMetricSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9 \u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc) E : Set \u03a9 E_nullbdry : (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (frontier E) = 0 \u22a2 \u2191\u2191\u2191\u03bc (frontier E) = 0 ** rw [\u2190 ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure, E_nullbdry, ENNReal.coe_zero] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.map_bind\u2081 ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S inst\u271d : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R f : R \u2192+* S g : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 \u2191(map f) (\u2191(bind\u2081 g) \u03c6) = \u2191(bind\u2081 fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6) ** rw [hom_bind\u2081, map_comp_C, \u2190 eval\u2082Hom_map_hom] ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u_3 S : Type u_4 T : Type u_5 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S inst\u271d : CommSemiring T f\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R f : R \u2192+* S g : \u03c3 \u2192 MvPolynomial \u03c4 R \u03c6 : MvPolynomial \u03c3 R \u22a2 \u2191(eval\u2082Hom C fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6) ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "List.isSublist_iff_sublist ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u2081 l\u2082 : List \u03b1 \u22a2 isSublist l\u2081 l\u2082 = true \u2194 l\u2081 <+ l\u2082 ** cases l\u2081 <;> cases l\u2082 <;> simp [isSublist] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : hd\u2081 = hd\u2082 \u22a2 (if hd\u2081 = hd\u2082 then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2194 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082 ** simp [h_eq, cons_sublist_cons, isSublist_iff_sublist] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2082 \u22a2 (if hd\u2081 = hd\u2082 then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2194 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082 ** simp only [h_eq] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2082 \u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2194 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082 ** constructor ** case mp \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2082 \u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u2192 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082 ** intro h_sub ** case mp \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2082 h_sub : (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u22a2 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082 ** apply Sublist.cons ** case mp.a \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2082 h_sub : (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true \u22a2 hd\u2081 :: tl\u2081 <+ tl\u2082 ** exact isSublist_iff_sublist.mp h_sub ** case mpr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2082 \u22a2 hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082 \u2192 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true ** intro h_sub ** case mpr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2082 h_sub : hd\u2081 :: tl\u2081 <+ hd\u2082 :: tl\u2082 \u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true ** cases h_sub ** case mpr.cons \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2082 a\u271d : hd\u2081 :: tl\u2081 <+ tl\u2082 \u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true  case mpr.cons\u2082 \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2081 a\u271d : tl\u2081 <+ tl\u2082 \u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true ** case cons h_sub =>\n  exact isSublist_iff_sublist.mpr h_sub ** case mpr.cons\u2082 \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2081 a\u271d : tl\u2081 <+ tl\u2082 \u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true ** case cons\u2082 =>\n  contradiction ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 : List \u03b1 hd\u2082 : \u03b1 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2082 h_sub : hd\u2081 :: tl\u2081 <+ tl\u2082 \u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true ** exact isSublist_iff_sublist.mpr h_sub ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 hd\u2081 : \u03b1 tl\u2081 tl\u2082 : List \u03b1 h_eq : \u00achd\u2081 = hd\u2081 a\u271d : tl\u2081 <+ tl\u2082 \u22a2 (if False then isSublist tl\u2081 tl\u2082 else isSublist (hd\u2081 :: tl\u2081) tl\u2082) = true ** contradiction ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.continuous_testAgainstNN_eval ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2075 : SMul R \u211d\u22650 inst\u271d\u2074 : SMul R \u211d\u22650\u221e inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 Continuous fun \u03bc => testAgainstNN \u03bc f ** show Continuous ((fun \u03c6 : WeakDual \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650) => \u03c6 f) \u2218 toWeakDualBCNN) ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2075 : SMul R \u211d\u22650 inst\u271d\u2074 : SMul R \u211d\u22650\u221e inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 Continuous ((fun \u03c6 => \u2191\u03c6 f) \u2218 toWeakDualBCNN) ** refine Continuous.comp ?_ (toWeakDualBCNN_continuous (\u03a9 := \u03a9)) ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u2075 : SMul R \u211d\u22650 inst\u271d\u2074 : SMul R \u211d\u22650\u221e inst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e inst\u271d\u00b9 : TopologicalSpace \u03a9 inst\u271d : OpensMeasurableSpace \u03a9 f : \u03a9 \u2192\u1d47 \u211d\u22650 \u22a2 Continuous fun \u03c6 => \u2191\u03c6 f ** exact @WeakBilin.eval_continuous _ _ _ _ _ _ ContinuousLinearMap.module _ _ _ _ ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.set_lintegral_deterministic' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 a : \u03b1 hg : Measurable g hf : Measurable f s : Set \u03b2 hs : MeasurableSet s inst\u271d : Decidable (g a \u2208 s) \u22a2 \u222b\u207b (x : \u03b2) in s, f x \u2202\u2191(deterministic g hg) a = if g a \u2208 s then f (g a) else 0 ** rw [kernel.deterministic_apply, set_lintegral_dirac' hf hs] ** Qed",
        "informal": ""
    },
    {
        "formal": "MonoidHom.pi_ext ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u2074 : Monoid \u03b2 inst\u271d\u00b3 : Monoid \u03b3 M : \u03b9 \u2192 Type u_6 inst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i) inst\u271d\u00b9 : Finite \u03b9 inst\u271d : DecidableEq \u03b9 f g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3 h : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x) \u22a2 f = g ** cases nonempty_fintype \u03b9 ** case intro F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u2074 : Monoid \u03b2 inst\u271d\u00b3 : Monoid \u03b3 M : \u03b9 \u2192 Type u_6 inst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i) inst\u271d\u00b9 : Finite \u03b9 inst\u271d : DecidableEq \u03b9 f g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3 h : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x) val\u271d : Fintype \u03b9 \u22a2 f = g ** ext x ** case intro.h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u2074 : Monoid \u03b2 inst\u271d\u00b3 : Monoid \u03b3 M : \u03b9 \u2192 Type u_6 inst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i) inst\u271d\u00b9 : Finite \u03b9 inst\u271d : DecidableEq \u03b9 f g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3 h : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x) val\u271d : Fintype \u03b9 x : (i : \u03b9) \u2192 M i \u22a2 \u2191f x = \u2191g x ** rw [\u2190 noncommProd_mul_single x, univ.noncommProd_map, univ.noncommProd_map] ** case intro.h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u2074 : Monoid \u03b2 inst\u271d\u00b3 : Monoid \u03b3 M : \u03b9 \u2192 Type u_6 inst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i) inst\u271d\u00b9 : Finite \u03b9 inst\u271d : DecidableEq \u03b9 f g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3 h : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x) val\u271d : Fintype \u03b9 x : (i : \u03b9) \u2192 M i \u22a2 noncommProd univ (fun i => \u2191f (Pi.mulSingle i (x i)))       (_ :         \u2200 (x_1 : \u03b9),           x_1 \u2208 \u2191univ \u2192             \u2200 (y : \u03b9), y \u2208 \u2191univ \u2192 x_1 \u2260 y \u2192 Commute (\u2191f (Pi.mulSingle x_1 (x x_1))) (\u2191f (Pi.mulSingle y (x y)))) =     noncommProd univ (fun i => \u2191g (Pi.mulSingle i (x i)))       (_ :         \u2200 (x_1 : \u03b9),           x_1 \u2208 \u2191univ \u2192             \u2200 (y : \u03b9), y \u2208 \u2191univ \u2192 x_1 \u2260 y \u2192 Commute (\u2191g (Pi.mulSingle x_1 (x x_1))) (\u2191g (Pi.mulSingle y (x y)))) ** congr 1 with i ** case intro.h.e_f.h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b2 op : \u03b1 \u2192 \u03b1 \u2192 \u03b1 inst\u271d\u2074 : Monoid \u03b2 inst\u271d\u00b3 : Monoid \u03b3 M : \u03b9 \u2192 Type u_6 inst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i) inst\u271d\u00b9 : Finite \u03b9 inst\u271d : DecidableEq \u03b9 f g : ((i : \u03b9) \u2192 M i) \u2192* \u03b3 h : \u2200 (i : \u03b9) (x : M i), \u2191f (Pi.mulSingle i x) = \u2191g (Pi.mulSingle i x) val\u271d : Fintype \u03b9 x : (i : \u03b9) \u2192 M i i : \u03b9 \u22a2 \u2191f (Pi.mulSingle i (x i)) = \u2191g (Pi.mulSingle i (x i)) ** exact h i (x i) ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.setToFun_finset_sum ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C \u03b9 : Type u_7 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 E hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) \u22a2 (setToFun \u03bc T hT fun a => \u2211 i in s, f i a) = \u2211 i in s, setToFun \u03bc T hT (f i) ** convert setToFun_finset_sum' hT s hf with a ** case h.e'_2.h.e'_14.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : CompleteSpace F T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F C C' C'' : \u211d f\u271d g : \u03b1 \u2192 E hT : DominatedFinMeasAdditive \u03bc T C \u03b9 : Type u_7 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 E hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) a : \u03b1 \u22a2 \u2211 i in s, f i a = Finset.sum s (fun i => f i) a ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Measure.isFiniteMeasure_map ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 \u22a2 IsFiniteMeasure (map f \u03bc) ** by_cases hf : AEMeasurable f \u03bc ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 hf : AEMeasurable f \u22a2 IsFiniteMeasure (map f \u03bc) ** constructor ** case pos.measure_univ_lt_top \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 hf : AEMeasurable f \u22a2 \u2191\u2191(map f \u03bc) univ < \u22a4 ** rw [map_apply_of_aemeasurable hf MeasurableSet.univ] ** case pos.measure_univ_lt_top \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 hf : AEMeasurable f \u22a2 \u2191\u2191\u03bc (f \u207b\u00b9' univ) < \u22a4 ** exact measure_lt_top \u03bc _ ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 hf : \u00acAEMeasurable f \u22a2 IsFiniteMeasure (map f \u03bc) ** rw [map_of_not_aemeasurable hf] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 hf : \u00acAEMeasurable f \u22a2 IsFiniteMeasure 0 ** exact MeasureTheory.isFiniteMeasureZero ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.BinomialHeap.Imp.Heap.WF.deleteMin ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n : Nat a : \u03b1 s' s : Heap \u03b1 h : WF le n s eq : Heap.deleteMin le s = some (a, s') \u22a2 WF le 0 s' ** cases s with cases eq | cons r a c s => ?_ ** case cons.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 h : WF le n (cons r a c s) \u22a2 WF le 0     (Heap.merge le       (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)       (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })         (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) ** have : (s.findMin le (cons r a c) \u27e8id, a, c, s\u27e9).WF le :=\n  let \u27e8_, h\u2082, h\u2083\u27e9 := h\n  h\u2083.findMin \u27e8_, fun h => h.of_le (Nat.zero_le _), h\u2082, h\u2083\u27e9\n    fun h => \u27e8Nat.zero_le _, h\u2082, h\u27e9 ** case cons.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 h : WF le n (cons r a c s) this : FindMin.WF le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) \u22a2 WF le 0     (Heap.merge le       (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)       (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })         (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) ** revert this ** case cons.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 h : WF le n (cons r a c s) \u22a2 FindMin.WF le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) \u2192     WF le 0       (Heap.merge le         (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)         (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })           (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)) ** let { before, val, node, next } := s.findMin le (cons r a c) \u27e8id, a, c, s\u27e9 ** case cons.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 h : WF le n (cons r a c s) before : Heap \u03b1 \u2192 Heap \u03b1 val : \u03b1 node : HeapNode \u03b1 next : Heap \u03b1 \u22a2 FindMin.WF le { before := before, val := val, node := node, next := next } \u2192     WF le 0       (Heap.merge le (HeapNode.toHeap { before := before, val := val, node := node, next := next }.node)         (FindMin.before { before := before, val := val, node := node, next := next }           { before := before, val := val, node := node, next := next }.next)) ** intro \u27e8_, hk, ih\u2081, ih\u2082\u27e9 ** case cons.refl \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool n r : Nat a : \u03b1 c : HeapNode \u03b1 s : Heap \u03b1 h : WF le n (cons r a c s) before : Heap \u03b1 \u2192 Heap \u03b1 val : \u03b1 node : HeapNode \u03b1 next : Heap \u03b1 rank\u271d : Nat hk :   \u2200 {s : Heap \u03b1},     WF le rank\u271d s \u2192 WF le 0 (FindMin.before { before := before, val := val, node := node, next := next } s) ih\u2081 :   HeapNode.WF le { before := before, val := val, node := node, next := next }.val     { before := before, val := val, node := node, next := next }.node rank\u271d ih\u2082 : WF le (rank\u271d + 1) { before := before, val := val, node := node, next := next }.next \u22a2 WF le 0     (Heap.merge le (HeapNode.toHeap { before := before, val := val, node := node, next := next }.node)       (FindMin.before { before := before, val := val, node := node, next := next }         { before := before, val := val, node := node, next := next }.next)) ** exact ih\u2081.toHeap.merge <| hk (ih\u2082.of_le (Nat.le_succ _)) ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.kernel.withDensity_tsum ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) \u22a2 withDensity \u03ba (\u2211' (n : \u03b9), f n) = kernel.sum fun n => withDensity \u03ba (f n) ** have h_sum_a : \u2200 a, Summable fun n => f n a := fun a => Pi.summable.mpr fun b => ENNReal.summable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a \u22a2 withDensity \u03ba (\u2211' (n : \u03b9), f n) = kernel.sum fun n => withDensity \u03ba (f n) ** have h_sum : Summable fun n => f n := Pi.summable.mpr h_sum_a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n \u22a2 withDensity \u03ba (\u2211' (n : \u03b9), f n) = kernel.sum fun n => withDensity \u03ba (f n) ** ext a s hs ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(withDensity \u03ba (\u2211' (n : \u03b9), f n)) a) s = \u2191\u2191(\u2191(kernel.sum fun n => withDensity \u03ba (f n)) a) s ** rw [sum_apply' _ a hs, withDensity_apply' \u03ba _ a hs] ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s  \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 Measurable (Function.uncurry (\u2211' (n : \u03b9), f n)) ** swap ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s ** have : \u222b\u207b b in s, (\u2211' n, f n) a b \u2202\u03ba a = \u222b\u207b b in s, \u2211' n, (fun b => f n a b) b \u2202\u03ba a := by\n  congr with b\n  rw [tsum_apply h_sum, tsum_apply (h_sum_a a)] ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s this : \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a \u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s ** rw [this, lintegral_tsum fun n => (Measurable.of_uncurry_left (hf n)).aemeasurable] ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s this : \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a \u22a2 \u2211' (i : \u03b9), \u222b\u207b (a_1 : \u03b2) in s, f i a a_1 \u2202\u2191\u03ba a = \u2211' (n : \u03b9), \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s ** congr with n ** case h.h.e_f.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s this : \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a n : \u03b9 \u22a2 \u222b\u207b (a_1 : \u03b2) in s, f n a a_1 \u2202\u2191\u03ba a = \u2191\u2191(\u2191(withDensity \u03ba (f n)) a) s ** rw [withDensity_apply' _ (hf n) a hs] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 Measurable (Function.uncurry (\u2211' (n : \u03b9), f n)) ** have : Function.uncurry (\u2211' n, f n) = \u2211' n, Function.uncurry (f n) := by\n  ext1 p\n  simp only [Function.uncurry_def]\n  rw [tsum_apply h_sum, tsum_apply (h_sum_a _), tsum_apply]\n  exact Pi.summable.mpr fun p => ENNReal.summable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s this : Function.uncurry (\u2211' (n : \u03b9), f n) = \u2211' (n : \u03b9), Function.uncurry (f n) \u22a2 Measurable (Function.uncurry (\u2211' (n : \u03b9), f n)) ** rw [this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s this : Function.uncurry (\u2211' (n : \u03b9), f n) = \u2211' (n : \u03b9), Function.uncurry (f n) \u22a2 Measurable (\u2211' (n : \u03b9), Function.uncurry (f n)) ** exact Measurable.ennreal_tsum' hf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 Function.uncurry (\u2211' (n : \u03b9), f n) = \u2211' (n : \u03b9), Function.uncurry (f n) ** ext1 p ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s p : \u03b1 \u00d7 \u03b2 \u22a2 Function.uncurry (\u2211' (n : \u03b9), f n) p = tsum (fun n => Function.uncurry (f n)) p ** simp only [Function.uncurry_def] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s p : \u03b1 \u00d7 \u03b2 \u22a2 tsum (fun n => f n) p.1 p.2 = tsum (fun n p => f n p.1 p.2) p ** rw [tsum_apply h_sum, tsum_apply (h_sum_a _), tsum_apply] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s p : \u03b1 \u00d7 \u03b2 \u22a2 Summable fun n p => f n p.1 p.2 ** exact Pi.summable.mpr fun p => ENNReal.summable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s \u22a2 \u222b\u207b (b : \u03b2) in s, tsum (fun n => f n) a b \u2202\u2191\u03ba a = \u222b\u207b (b : \u03b2) in s, \u2211' (n : \u03b9), (fun b => f n a b) b \u2202\u2191\u03ba a ** congr with b ** case e_f.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba\u271d : { x // x \u2208 kernel \u03b1 \u03b2 } f\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e inst\u271d\u00b9 : Countable \u03b9 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d : IsSFiniteKernel \u03ba f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e hf : \u2200 (i : \u03b9), Measurable (Function.uncurry (f i)) h_sum_a : \u2200 (a : \u03b1), Summable fun n => f n a h_sum : Summable fun n => f n a : \u03b1 s : Set \u03b2 hs : MeasurableSet s b : \u03b2 \u22a2 tsum (fun n => f n) a b = \u2211' (n : \u03b9), (fun b => f n a b) b ** rw [tsum_apply h_sum, tsum_apply (h_sum_a a)] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.FiniteMeasure.restrict_mass ** \u03a9 : Type u_1 inst\u271d\u2074 : MeasurableSpace \u03a9 R : Type u_2 inst\u271d\u00b3 : SMul R \u211d\u22650 inst\u271d\u00b2 : SMul R \u211d\u22650\u221e inst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e inst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e \u03bc : FiniteMeasure \u03a9 A : Set \u03a9 \u22a2 mass (restrict \u03bc A) = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) A ** simp only [mass, restrict_apply \u03bc A MeasurableSet.univ, univ_inter] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_lintegral_div_measure ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 f : G \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 \u2191\u2191\u03bc s * \u222b\u207b (y : G), f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u2202\u03bd = \u222b\u207b (x : G), f x \u2202\u03bc ** set g := fun y => f y\u207b\u00b9 / \u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 f : G \u2192 \u211d\u22650\u221e hf : Measurable f g : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc ** have hg : Measurable g :=\n  (hf.comp measurable_inv).div ((measurable_measure_mul_right \u03bd sm).comp measurable_inv) ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 f : G \u2192 \u211d\u22650\u221e hf : Measurable f g : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) hg : Measurable g \u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc ** simp_rw [measure_mul_lintegral_eq \u03bc \u03bd sm g hg, inv_inv] ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 f : G \u2192 \u211d\u22650\u221e hf : Measurable f g : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) hg : Measurable g \u22a2 \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s)) \u2202\u03bc = \u222b\u207b (x : G), f x \u2202\u03bc ** refine' lintegral_congr_ae _ ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 f : G \u2192 \u211d\u22650\u221e hf : Measurable f g : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) hg : Measurable g \u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) =\u1da0[ae \u03bc] fun x => f x ** refine' (ae_measure_preimage_mul_right_lt_top_of_ne_zero \u03bc \u03bd sm h2s h3s).mono fun x hx => _ ** G : Type u_1 inst\u271d\u2077 : MeasurableSpace G inst\u271d\u2076 : Group G inst\u271d\u2075 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u2074 : SigmaFinite \u03bd inst\u271d\u00b3 : SigmaFinite \u03bc s : Set G inst\u271d\u00b2 : MeasurableInv G inst\u271d\u00b9 : IsMulLeftInvariant \u03bc inst\u271d : IsMulLeftInvariant \u03bd sm : MeasurableSet s h2s : \u2191\u2191\u03bd s \u2260 0 h3s : \u2191\u2191\u03bd s \u2260 \u22a4 f : G \u2192 \u211d\u22650\u221e hf : Measurable f g : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) hg : Measurable g x : G hx : \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4 \u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) x = (fun x => f x) x ** simp_rw [ENNReal.mul_div_cancel' (measure_mul_right_ne_zero \u03bd h2s _) hx.ne] ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.iIndepFun.mgf_sum ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) s : Finset \u03b9 \u22a2 mgf (\u2211 i in s, X i) \u03bc t = \u220f i in s, mgf (X i) \u03bc t ** induction' s using Finset.induction_on with i s hi_notin_s h_rec h_int ** case empty \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) \u22a2 mgf (\u2211 i in \u2205, X i) \u03bc t = \u220f i in \u2205, mgf (X i) \u03bc t ** simp only [sum_empty, mgf_zero_fun, measure_univ, ENNReal.one_toReal, prod_empty] ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_rec : mgf (\u2211 i in s, X i) \u03bc t = \u220f i in s, mgf (X i) \u03bc t \u22a2 mgf (\u2211 i in insert i s, X i) \u03bc t = \u220f i in insert i s, mgf (X i) \u03bc t ** have h_int' : \u2200 i : \u03b9, AEStronglyMeasurable (fun \u03c9 : \u03a9 => exp (t * X i \u03c9)) \u03bc := fun i =>\n  ((h_meas i).const_mul t).exp.aestronglyMeasurable ** case insert \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X\u271d : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d inst\u271d : IsProbabilityMeasure \u03bc X : \u03b9 \u2192 \u03a9 \u2192 \u211d h_indep : iIndepFun (fun i => inferInstance) X h_meas : \u2200 (i : \u03b9), Measurable (X i) i : \u03b9 s : Finset \u03b9 hi_notin_s : \u00aci \u2208 s h_rec : mgf (\u2211 i in s, X i) \u03bc t = \u220f i in s, mgf (X i) \u03bc t h_int' : \u2200 (i : \u03b9), AEStronglyMeasurable (fun \u03c9 => rexp (t * X i \u03c9)) \u03bc \u22a2 mgf (\u2211 i in insert i s, X i) \u03bc t = \u220f i in insert i s, mgf (X i) \u03bc t ** rw [sum_insert hi_notin_s,\n  IndepFun.mgf_add (h_indep.indepFun_finset_sum_of_not_mem h_meas hi_notin_s).symm (h_int' i)\n    (aestronglyMeasurable_exp_mul_sum fun i _ => h_int' i),\n  h_rec, prod_insert hi_notin_s] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.OuterMeasure.map_ofFunction ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Injective f \u22a2 \u2191(map f) (OuterMeasure.ofFunction m m_empty) = OuterMeasure.ofFunction (fun s => m (f \u207b\u00b9' s)) m_empty ** refine' (map_ofFunction_le _).antisymm fun s => _ ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Injective f s : Set \u03b2 \u22a2 \u2191(OuterMeasure.ofFunction (fun s => m (f \u207b\u00b9' s)) m_empty) s \u2264 \u2191(\u2191(map f) (OuterMeasure.ofFunction m m_empty)) s ** simp only [ofFunction_apply, map_apply, le_iInf_iff] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Injective f s : Set \u03b2 \u22a2 \u2200 (i : \u2115 \u2192 Set \u03b1), f \u207b\u00b9' s \u2286 iUnion i \u2192 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f \u207b\u00b9' t n) \u2264 \u2211' (n : \u2115), m (i n) ** intro t ht ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Injective f s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t \u22a2 \u2a05 t, \u2a05 (_ : s \u2286 iUnion t), \u2211' (n : \u2115), m (f \u207b\u00b9' t n) \u2264 \u2211' (n : \u2115), m (t n) ** refine' iInf_le_of_le (fun n => (range f)\u1d9c \u222a f '' t n) (iInf_le_of_le _ _) ** case refine'_1 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Injective f s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t \u22a2 s \u2286 \u22c3 n, (range f)\u1d9c \u222a f '' t n ** rw [\u2190 union_iUnion, \u2190 inter_subset, \u2190 image_preimage_eq_inter_range, \u2190 image_iUnion] ** case refine'_1 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Injective f s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t \u22a2 f '' (f \u207b\u00b9' s) \u2286 f '' \u22c3 i, t i ** exact image_subset _ ht ** case refine'_2 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Injective f s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t \u22a2 \u2211' (n : \u2115), m (f \u207b\u00b9' (fun n => (range f)\u1d9c \u222a f '' t n) n) \u2264 \u2211' (n : \u2115), m (t n) ** refine' ENNReal.tsum_le_tsum fun n => le_of_eq _ ** case refine'_2 \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Injective f s : Set \u03b2 t : \u2115 \u2192 Set \u03b1 ht : f \u207b\u00b9' s \u2286 iUnion t n : \u2115 \u22a2 m (f \u207b\u00b9' (fun n => (range f)\u1d9c \u222a f '' t n) n) = m (t n) ** simp [hf.preimage_image] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.Integrable.bdd_mul ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc hfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C \u22a2 Integrable fun x => f x * g x ** cases' isEmpty_or_nonempty \u03b1 with h\u03b1 h\u03b1 ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc hfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C h\u03b1 : IsEmpty \u03b1 \u22a2 Integrable fun x => f x * g x ** exact integrable_zero_measure ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc hfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C h\u03b1 : Nonempty \u03b1 \u22a2 Integrable fun x => f x * g x ** refine' \u27e8hm.mul hint.1, _\u27e9 ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc hfbdd : \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C h\u03b1 : Nonempty \u03b1 \u22a2 HasFiniteIntegral fun x => f x * g x ** obtain \u27e8C, hC\u27e9 := hfbdd ** case inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc h\u03b1 : Nonempty \u03b1 C : \u211d hC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C \u22a2 HasFiniteIntegral fun x => f x * g x ** have hCnonneg : 0 \u2264 C := le_trans (norm_nonneg _) (hC h\u03b1.some) ** case inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc h\u03b1 : Nonempty \u03b1 C : \u211d hC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C hCnonneg : 0 \u2264 C \u22a2 HasFiniteIntegral fun x => f x * g x ** have : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => \u27e8C, hCnonneg\u27e9 * \u2016g x\u2016\u208a := by\n  intro x\n  simp only [nnnorm_mul]\n  exact mul_le_mul_of_nonneg_right (hC x) (zero_le _) ** case inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc h\u03b1 : Nonempty \u03b1 C : \u211d hC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C hCnonneg : 0 \u2264 C this : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a \u22a2 HasFiniteIntegral fun x => f x * g x ** refine' lt_of_le_of_lt (lintegral_mono_nnreal this) _ ** case inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc h\u03b1 : Nonempty \u03b1 C : \u211d hC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C hCnonneg : 0 \u2264 C this : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a \u22a2 \u222b\u207b (a : \u03b1), \u2191({ val := C, property := hCnonneg } * \u2016g a\u2016\u208a) \u2202\u03bc < \u22a4 ** simp only [ENNReal.coe_mul] ** case inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc h\u03b1 : Nonempty \u03b1 C : \u211d hC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C hCnonneg : 0 \u2264 C this : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a \u22a2 \u222b\u207b (a : \u03b1), \u2191{ val := C, property := hCnonneg } * \u2191\u2016g a\u2016\u208a \u2202\u03bc < \u22a4 ** rw [lintegral_const_mul' _ _ ENNReal.coe_ne_top] ** case inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc h\u03b1 : Nonempty \u03b1 C : \u211d hC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C hCnonneg : 0 \u2264 C this : (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a \u22a2 \u2191{ val := C, property := hCnonneg } * \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a \u2202\u03bc < \u22a4 ** exact ENNReal.mul_lt_top ENNReal.coe_ne_top (ne_of_lt hint.2) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc h\u03b1 : Nonempty \u03b1 C : \u211d hC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C hCnonneg : 0 \u2264 C \u22a2 (fun x => \u2016f x * g x\u2016\u208a) \u2264 fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a ** intro x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc h\u03b1 : Nonempty \u03b1 C : \u211d hC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C hCnonneg : 0 \u2264 C x : \u03b1 \u22a2 (fun x => \u2016f x * g x\u2016\u208a) x \u2264 (fun x => { val := C, property := hCnonneg } * \u2016g x\u2016\u208a) x ** simp only [nnnorm_mul] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 F : Type u_5 inst\u271d : NormedDivisionRing F f g : \u03b1 \u2192 F hint : Integrable g hm : AEStronglyMeasurable f \u03bc h\u03b1 : Nonempty \u03b1 C : \u211d hC : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C hCnonneg : 0 \u2264 C x : \u03b1 \u22a2 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a \u2264 { val := C, property := hCnonneg } * \u2016g x\u2016\u208a ** exact mul_le_mul_of_nonneg_right (hC x) (zero_le _) ** Qed",
        "informal": ""
    },
    {
        "formal": "Array.reverse_data ** \u03b1 : Type u_1 a : Array \u03b1 \u22a2 (reverse a).data = List.reverse a.data ** simp only [reverse] ** \u03b1 : Type u_1 a : Array \u03b1 \u22a2 (if h : size a \u2264 1 then a       else         reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =     List.reverse a.data ** split ** \u03b1 : Type u_1 a as : Array \u03b1 i j : Nat hj : j < size as h : i + j + 1 = size a h\u2082 : size as = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k k : Nat \u22a2 List.get? (reverse.loop as i { val := j, isLt := hj }).data k = List.get? (List.reverse a.data) k ** rw [reverse.loop] ** \u03b1 : Type u_1 a as : Array \u03b1 i j : Nat hj : j < size as h : i + j + 1 = size a h\u2082 : size as = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k k : Nat \u22a2 List.get?       (if h : i < { val := j, isLt := hj }.val then           let_fun this := (_ : { val := j, isLt := hj }.val - 1 - (i + 1) < { val := j, isLt := hj }.val - i);           let as_1 := swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj };           let_fun this := (_ : { val := j, isLt := hj }.val - 1 < size as_1);           reverse.loop as_1 (i + 1) { val := { val := j, isLt := hj }.val - 1, isLt := this }         else as).data       k =     List.get? (List.reverse a.data) k ** dsimp ** \u03b1 : Type u_1 a as : Array \u03b1 i j : Nat hj : j < size as h : i + j + 1 = size a h\u2082 : size as = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k k : Nat \u22a2 List.get?       (if h : i < j then           reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)             { val := j - 1,               isLt := (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }         else as).data       k =     List.get? (List.reverse a.data) k ** split <;> rename_i h\u2081 ** case inl \u03b1 : Type u_1 a as : Array \u03b1 i j : Nat hj : j < size as h : i + j + 1 = size a h\u2082 : size as = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k k : Nat h\u2081 : i < j \u22a2 List.get?       (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)           { val := j - 1,             isLt :=               (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data       k =     List.get? (List.reverse a.data) k ** have := reverse.termination h\u2081 ** case inl \u03b1 : Type u_1 a as : Array \u03b1 i j : Nat hj : j < size as h : i + j + 1 = size a h\u2082 : size as = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k k : Nat h\u2081 : i < j this : j - 1 - (i + 1) < j - i \u22a2 List.get?       (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj }) (i + 1)           { val := j - 1,             isLt :=               (_ : j - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j, isLt := hj })) }).data       k =     List.get? (List.reverse a.data) k ** match j with | j+1 => ?_ ** case inl \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082 : size as = size a k j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j + 1 - 1 - (i + 1) < j + 1 - i \u22a2 List.get?       (reverse.loop (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) (i + 1)           { val := j + 1 - 1,             isLt :=               (_ :                 j + 1 - 1 < size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj })) }).data       k =     List.get? (List.reverse a.data) k ** simp at * ** case inl.h \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082 : size as = size a k j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i \u22a2 i + 1 + j + 1 = size a ** rwa [Nat.add_right_comm i] ** case inl.h\u2082 \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082 : size as = size a k j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i \u22a2 size (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }) = size a ** simp [size_swap, h\u2082] ** case inl.H \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082 : size as = size a k j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i \u22a2 \u2200 (k : Nat),     List.get? (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }).data k =       if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k ** intro k ** case inl.H \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082 : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat \u22a2 List.get? (swap as { val := i, isLt := (_ : i < size as) } { val := j + 1, isLt := hj }).data k =     if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k ** rw [\u2190 getElem?_eq_data_get?, get?_swap] ** case inl.H \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082 : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat \u22a2 (if { val := j + 1, isLt := hj }.val = k then some as[{ val := i, isLt := (_ : i < size as) }.val]     else       if { val := i, isLt := (_ : i < size as) }.val = k then some as[{ val := j + 1, isLt := hj }.val] else as[k]?) =     if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k ** simp [getElem?_eq_data_get?, getElem_eq_data_get, \u2190 List.get?_eq_get, H, Nat.le_of_lt h\u2081] ** case inl.H \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082 : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat \u22a2 (if j + 1 = k then List.get? a.data i     else       if i = k then List.get? a.data (j + 1)       else if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =     if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k ** split <;> rename_i h\u2082 ** case inl.H.inr \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082\u271d : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat h\u2082 : \u00acj + 1 = k \u22a2 (if i = k then List.get? a.data (j + 1)     else if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =     if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k ** split <;> rename_i h\u2083 ** case inl.H.inr.inr \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082\u271d : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat h\u2082 : \u00acj + 1 = k h\u2083 : \u00aci = k \u22a2 (if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k) =     if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k ** simp only [Nat.succ_le, Nat.lt_iff_le_and_ne.trans (and_iff_left h\u2083),\n  Nat.lt_succ.symm.trans (Nat.lt_iff_le_and_ne.trans (and_iff_left (Ne.symm h\u2082)))] ** case inl.H.inl \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082\u271d : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat h\u2082 : j + 1 = k \u22a2 List.get? a.data i = if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k ** simp [\u2190 h\u2082, Nat.not_le.2 (Nat.lt_succ_self _)] ** case inl.H.inl \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082\u271d : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat h\u2082 : j + 1 = k \u22a2 List.get? a.data i = List.get? (List.reverse a.data) (j + 1) ** exact (List.get?_reverse' _ _ (Eq.trans (by simp_arith) h)).symm ** \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082\u271d : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat h\u2082 : j + 1 = k \u22a2 j + 1 + i + 1 = i + (j + 1) + 1 ** simp_arith ** case inl.H.inr.inl \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082\u271d : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat h\u2082 : \u00acj + 1 = k h\u2083 : i = k \u22a2 List.get? a.data (j + 1) = if i + 1 \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k ** simp [\u2190 h\u2083, Nat.not_le.2 (Nat.lt_succ_self _)] ** case inl.H.inr.inl \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082\u271d : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat h\u2082 : \u00acj + 1 = k h\u2083 : i = k \u22a2 List.get? a.data (j + 1) = List.get? (List.reverse a.data) i ** exact (List.get?_reverse' _ _ (Eq.trans (by simp_arith) h)).symm ** \u03b1 : Type u_1 a as : Array \u03b1 i j\u271d : Nat h\u2082\u271d : size as = size a k\u271d j : Nat hj : j + 1 < size as h : i + (j + 1) + 1 = size a H :   \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j + 1 then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : i < j + 1 this : j - (i + 1) < j + 1 - i k : Nat h\u2082 : \u00acj + 1 = k h\u2083 : i = k \u22a2 i + (j + 1) + 1 = i + (j + 1) + 1 ** simp_arith ** case inr \u03b1 : Type u_1 a as : Array \u03b1 i j : Nat hj : j < size as h : i + j + 1 = size a h\u2082 : size as = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k k : Nat h\u2081 : \u00aci < j \u22a2 List.get? as.data k = List.get? (List.reverse a.data) k ** rw [H] ** case inr \u03b1 : Type u_1 a as : Array \u03b1 i j : Nat hj : j < size as h : i + j + 1 = size a h\u2082 : size as = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k k : Nat h\u2081 : \u00aci < j \u22a2 (if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k) = List.get? (List.reverse a.data) k ** split <;> rename_i h\u2082 ** case inr.inl \u03b1 : Type u_1 a as : Array \u03b1 i j : Nat hj : j < size as h : i + j + 1 = size a h\u2082\u271d : size as = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k k : Nat h\u2081 : \u00aci < j h\u2082 : i \u2264 k \u2227 k \u2264 j \u22a2 List.get? a.data k = List.get? (List.reverse a.data) k ** cases Nat.le_antisymm (Nat.not_lt.1 h\u2081) (Nat.le_trans h\u2082.1 h\u2082.2) ** case inr.inl.refl \u03b1 : Type u_1 a as : Array \u03b1 i : Nat h\u2082\u271d : size as = size a k : Nat hj : i < size as h : i + i + 1 = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 i then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : \u00aci < i h\u2082 : i \u2264 k \u2227 k \u2264 i \u22a2 List.get? a.data k = List.get? (List.reverse a.data) k ** cases Nat.le_antisymm h\u2082.1 h\u2082.2 ** case inr.inl.refl.refl \u03b1 : Type u_1 a as : Array \u03b1 i : Nat h\u2082\u271d : size as = size a hj : i < size as h : i + i + 1 = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 i then List.get? a.data k else List.get? (List.reverse a.data) k h\u2081 : \u00aci < i h\u2082 : i \u2264 i \u2227 i \u2264 i \u22a2 List.get? a.data i = List.get? (List.reverse a.data) i ** exact (List.get?_reverse' _ _ h).symm ** case inr.inr \u03b1 : Type u_1 a as : Array \u03b1 i j : Nat hj : j < size as h : i + j + 1 = size a h\u2082\u271d : size as = size a H : \u2200 (k : Nat), List.get? as.data k = if i \u2264 k \u2227 k \u2264 j then List.get? a.data k else List.get? (List.reverse a.data) k k : Nat h\u2081 : \u00aci < j h\u2082 : \u00ac(i \u2264 k \u2227 k \u2264 j) \u22a2 List.get? (List.reverse a.data) k = List.get? (List.reverse a.data) k ** rfl ** case inl \u03b1 : Type u_1 a : Array \u03b1 h\u271d : size a \u2264 1 \u22a2 a.data = List.reverse a.data ** match a with | \u27e8[]\u27e9 | \u27e8[_]\u27e9 => rfl ** \u03b1 : Type u_1 a : Array \u03b1 head\u271d : \u03b1 h\u271d : size { data := [head\u271d] } \u2264 1 \u22a2 { data := [head\u271d] }.data = List.reverse { data := [head\u271d] }.data ** rfl ** case inr \u03b1 : Type u_1 a : Array \u03b1 h\u271d : \u00acsize a \u2264 1 \u22a2 (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =     List.reverse a.data ** have := Nat.sub_add_cancel (Nat.le_of_not_le \u2039_\u203a) ** case inr \u03b1 : Type u_1 a : Array \u03b1 h\u271d : \u00acsize a \u2264 1 this : size a - 1 + 1 = size a \u22a2 (reverse.loop a 0 { val := size a - 1, isLt := (_ : Nat.pred (Nat.sub (size a) 0) < Nat.sub (size a) 0) }).data =     List.reverse a.data ** refine List.ext <| go _ _ _ _ (by simp [this]) rfl fun k => ?_ ** case inr \u03b1 : Type u_1 a : Array \u03b1 h\u271d : \u00acsize a \u2264 1 this : size a - 1 + 1 = size a k : Nat \u22a2 List.get? a.data k = if 0 \u2264 k \u2227 k \u2264 size a - 1 then List.get? a.data k else List.get? (List.reverse a.data) k ** split ** case inr.inl \u03b1 : Type u_1 a : Array \u03b1 h\u271d\u00b9 : \u00acsize a \u2264 1 this : size a - 1 + 1 = size a k : Nat h\u271d : 0 \u2264 k \u2227 k \u2264 size a - 1 \u22a2 List.get? a.data k = List.get? a.data k  case inr.inr \u03b1 : Type u_1 a : Array \u03b1 h\u271d\u00b9 : \u00acsize a \u2264 1 this : size a - 1 + 1 = size a k : Nat h\u271d : \u00ac(0 \u2264 k \u2227 k \u2264 size a - 1) \u22a2 List.get? a.data k = List.get? (List.reverse a.data) k ** {rfl} ** case inr.inr \u03b1 : Type u_1 a : Array \u03b1 h\u271d\u00b9 : \u00acsize a \u2264 1 this : size a - 1 + 1 = size a k : Nat h\u271d : \u00ac(0 \u2264 k \u2227 k \u2264 size a - 1) \u22a2 List.get? a.data k = List.get? (List.reverse a.data) k ** rename_i h ** case inr.inr \u03b1 : Type u_1 a : Array \u03b1 h\u271d : \u00acsize a \u2264 1 this : size a - 1 + 1 = size a k : Nat h : \u00ac(0 \u2264 k \u2227 k \u2264 size a - 1) \u22a2 List.get? a.data k = List.get? (List.reverse a.data) k ** simp [\u2190 show k < _ + 1 \u2194 _ from Nat.lt_succ (n := a.size - 1), this] at h ** case inr.inr \u03b1 : Type u_1 a : Array \u03b1 h\u271d : \u00acsize a \u2264 1 this : size a - 1 + 1 = size a k : Nat h : size a \u2264 k \u22a2 List.get? a.data k = List.get? (List.reverse a.data) k ** rw [List.get?_eq_none.2 \u2039_\u203a, List.get?_eq_none.2 (a.data.length_reverse \u25b8 \u2039_\u203a)] ** \u03b1 : Type u_1 a : Array \u03b1 h\u271d : \u00acsize a \u2264 1 this : size a - 1 + 1 = size a \u22a2 0 + (size a - 1) + 1 = size a ** simp [this] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.integral_eq_norm_posPart_sub ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } \u22a2 integral f = \u2016posPart f\u2016 - \u2016negPart f\u2016 ** have ae_eq\u2081 : (toSimpleFunc f).posPart =\u1d50[\u03bc] (toSimpleFunc (posPart f)).map norm := by\n  filter_upwards [posPart_toSimpleFunc f] with _ h\n  rw [SimpleFunc.map_apply, h]\n  conv_lhs => rw [\u2190 SimpleFunc.posPart_map_norm, SimpleFunc.map_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) \u22a2 integral f = \u2016posPart f\u2016 - \u2016negPart f\u2016 ** have ae_eq\u2082 : (toSimpleFunc f).negPart =\u1d50[\u03bc] (toSimpleFunc (negPart f)).map norm := by\n  filter_upwards [negPart_toSimpleFunc f] with _ h\n  rw [SimpleFunc.map_apply, h]\n  conv_lhs => rw [\u2190 SimpleFunc.negPart_map_norm, SimpleFunc.map_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) \u22a2 integral f = \u2016posPart f\u2016 - \u2016negPart f\u2016 ** rw [integral, norm_eq_integral, norm_eq_integral, \u2190 SimpleFunc.integral_sub] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } \u22a2 \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ** filter_upwards [posPart_toSimpleFunc f] with _ h ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } a\u271d : \u03b1 h : \u2191(toSimpleFunc (posPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d \u22a2 \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d ** rw [SimpleFunc.map_apply, h] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } a\u271d : \u03b1 h : \u2191(toSimpleFunc (posPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d \u22a2 \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2016\u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d\u2016 ** conv_lhs => rw [\u2190 SimpleFunc.posPart_map_norm, SimpleFunc.map_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) \u22a2 \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) ** filter_upwards [negPart_toSimpleFunc f] with _ h ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d : \u03b1 h : \u2191(toSimpleFunc (negPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d \u22a2 \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d ** rw [SimpleFunc.map_apply, h] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d : \u03b1 h : \u2191(toSimpleFunc (negPart f)) a\u271d = \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d \u22a2 \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2016\u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d\u2016 ** conv_lhs => rw [\u2190 SimpleFunc.negPart_map_norm, SimpleFunc.map_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) \u22a2 MeasureTheory.SimpleFunc.integral \u03bc (toSimpleFunc f) =     MeasureTheory.SimpleFunc.integral \u03bc       (SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f))) ** apply MeasureTheory.SimpleFunc.integral_congr (SimpleFunc.integrable f) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) \u22a2 \u2191(toSimpleFunc f) =\u1d50[\u03bc]     \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f))) ** filter_upwards [ae_eq\u2081, ae_eq\u2082] with _ h\u2081 h\u2082 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d : \u03b1 h\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d h\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d \u22a2 \u2191(toSimpleFunc f) a\u271d =     \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f)) - SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d ** show _ = _ - _ ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d : \u03b1 h\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d h\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d \u22a2 \u2191(toSimpleFunc f) a\u271d =     \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d - \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d ** rw [\u2190 h\u2081, \u2190 h\u2082] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d : \u03b1 h\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d h\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d \u22a2 \u2191(toSimpleFunc f) a\u271d =     \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d - \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d ** have := (toSimpleFunc f).posPart_sub_negPart ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d : \u03b1 h\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) a\u271d h\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d = \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) a\u271d this :   MeasureTheory.SimpleFunc.posPart (toSimpleFunc f) - MeasureTheory.SimpleFunc.negPart (toSimpleFunc f) = toSimpleFunc f \u22a2 \u2191(toSimpleFunc f) a\u271d =     \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) a\u271d - \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) a\u271d ** conv_lhs => rw [\u2190 this] ** case hf \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) \u22a2 Integrable \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ** exact (SimpleFunc.integrable f).pos_part.congr ae_eq\u2081 ** case hg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2078 : NormedField \ud835\udd5c inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u2074 : NormedAddCommGroup F' inst\u271d\u00b3 : NormedSpace \u211d F' E' : Type u_6 inst\u271d\u00b2 : NormedAddCommGroup E' inst\u271d\u00b9 : NormedSpace \u211d E' inst\u271d : NormedSpace \ud835\udd5c E' f : { x // x \u2208 simpleFunc \u211d 1 \u03bc } ae_eq\u2081 : \u2191(MeasureTheory.SimpleFunc.posPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (posPart f))) ae_eq\u2082 : \u2191(MeasureTheory.SimpleFunc.negPart (toSimpleFunc f)) =\u1d50[\u03bc] \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) \u22a2 Integrable \u2191(SimpleFunc.map norm (toSimpleFunc (negPart f))) ** exact (SimpleFunc.integrable f).neg_part.congr ae_eq\u2082 ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.iSup\u2082_lintegral_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 \u03b9 : Sort u_5 \u03b9' : \u03b9 \u2192 Sort u_6 f : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u2a06 i, \u2a06 j, \u222b\u207b (a : \u03b1), f i j a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2a06 i, \u2a06 j, f i j a \u2202\u03bc ** convert (monotone_lintegral \u03bc).le_map_iSup\u2082 f with a ** case h.e'_4.h.e'_4.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 \u03b9 : Sort u_5 \u03b9' : \u03b9 \u2192 Sort u_6 f : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 \u211d\u22650\u221e a : \u03b1 \u22a2 \u2a06 i, \u2a06 j, f i j a = iSup (fun i => \u2a06 j, f i j) a ** simp only [iSup_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_condexpL2_eq ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc ** rw [\u2190 sub_eq_zero, lpMeas_coe, \u2190\n  integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs)\n    (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs)] ** \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u222b (a : \u03b1) in s, (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) a \u2202\u03bc = 0 ** refine' integral_eq_zero_of_forall_integral_inner_eq_zero \ud835\udd5c _ _ _ ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2200 (c : E'), \u222b (x : \u03b1) in s, inner c ((\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) x) \u2202\u03bc = 0 ** intro c ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E' \u22a2 \u222b (x : \u03b1) in s, inner c ((\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) x) \u2202\u03bc = 0 ** simp_rw [Pi.sub_apply, inner_sub_right] ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E' \u22a2 \u222b (x : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) x) - inner c (\u2191\u2191f x) \u2202\u03bc = 0 ** rw [integral_sub\n    ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs).const_inner c)\n    ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs).const_inner c)] ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E' \u22a2 \u222b (a : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a) \u2202\u03bc - \u222b (a : \u03b1) in s, inner c (\u2191\u2191f a) \u2202\u03bc = 0 ** have h_ae_eq_f := Mem\u2112p.coeFn_toLp (E := \ud835\udd5c) ((Lp.mem\u2112p f).const_inner c) ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E' h_ae_eq_f :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) =\u1d50[\u03bc] fun a => inner c (\u2191\u2191f a) \u22a2 \u222b (a : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a) \u2202\u03bc - \u222b (a : \u03b1) in s, inner c (\u2191\u2191f a) \u2202\u03bc = 0 ** rw [\u2190 lpMeas_coe, sub_eq_zero, \u2190\n  set_integral_congr_ae (hm s hs) ((condexpL2_const_inner hm f c).mono fun x hx _ => hx), \u2190\n  set_integral_congr_ae (hm s hs) (h_ae_eq_f.mono fun x hx _ => hx)] ** case refine'_2 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 c : E' h_ae_eq_f :   \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) =\u1d50[\u03bc] fun a => inner c (\u2191\u2191f a) \u22a2 \u222b (x : \u03b1) in s,       \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) x \u2202\u03bc =     \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) x \u2202\u03bc ** exact integral_condexpL2_eq_of_fin_meas_real _ hs h\u03bcs ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 Integrable fun a => (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) a ** rw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (\u2191(condexpL2 E' \ud835\udd5c hm f)) f).symm)] ** case refine'_1 \u03b1 : Type u_1 E : Type u_2 E' : Type u_3 F : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9\u2070 : CompleteSpace E inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E' inst\u271d\u2077 : CompleteSpace E' inst\u271d\u2076 : NormedSpace \u211d E' inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedAddCommGroup G' inst\u271d\u00b9 : NormedSpace \u211d G' inst\u271d : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s t : Set \u03b1 hm : m \u2264 m0 f : { x // x \u2208 Lp E' 2 } hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 Integrable fun x => \u2191\u2191(\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - f) x ** exact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.measure_iUnion_eq_iSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) s \u22a2 \u2191\u2191\u03bc (\u22c3 i, s i) = \u2a06 i, \u2191\u2191\u03bc (s i) ** cases nonempty_encodable \u03b9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) s val\u271d : Encodable \u03b9 \u22a2 \u2191\u2191\u03bc (\u22c3 i, s i) = \u2a06 i, \u2191\u2191\u03bc (s i) ** generalize ht : Function.extend Encodable.encode s \u22a5 = t ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t\u271d : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 val\u271d : Encodable \u03b9 t : \u2115 \u2192 Set \u03b1 ht : Function.extend Encodable.encode s \u22a5 = t hd : Directed (fun x x_1 => x \u2286 x_1) t \u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) = \u2a06 n, \u2191\u2191\u03bc (t n) ** clear! \u03b9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 t : \u2115 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) t \u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) = \u2a06 n, \u2191\u2191\u03bc (t n) ** refine' le_antisymm _ (iSup_le fun i => measure_mono <| subset_iUnion _ _) ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 t : \u2115 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) t \u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n) ** set T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n) ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 t : \u2115 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) t T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n) \u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n) ** set Td : \u2115 \u2192 Set \u03b1 := disjointed T ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 t : \u2115 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) t T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n) Td : \u2115 \u2192 Set \u03b1 := disjointed T \u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n) ** have hm : \u2200 n, MeasurableSet (Td n) :=\n  MeasurableSet.disjointed fun n => measurableSet_toMeasurable _ _ ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 t : \u2115 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) t T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n) Td : \u2115 \u2192 Set \u03b1 := disjointed T hm : \u2200 (n : \u2115), MeasurableSet (Td n) \u22a2 \u2191\u2191\u03bc (\u22c3 n, t n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n) ** calc\n  \u03bc (\u22c3 n, t n) \u2264 \u03bc (\u22c3 n, T n) := measure_mono (iUnion_mono fun i => subset_toMeasurable _ _)\n  _ = \u03bc (\u22c3 n, Td n) := by rw [iUnion_disjointed]\n  _ \u2264 \u2211' n, \u03bc (Td n) := (measure_iUnion_le _)\n  _ = \u2a06 I : Finset \u2115, \u2211 n in I, \u03bc (Td n) := ENNReal.tsum_eq_iSup_sum\n  _ \u2264 \u2a06 n, \u03bc (t n) := iSup_le fun I => by\n    rcases hd.finset_le I with \u27e8N, hN\u27e9\n    calc\n      (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n        (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm\n      _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (measure_mono (iUnion\u2082_mono fun n _hn => disjointed_subset _ _))\n      _ = \u03bc (\u22c3 n \u2208 I, t n) := (measure_biUnion_toMeasurable I.countable_toSet _)\n      _ \u2264 \u03bc (t N) := (measure_mono (iUnion\u2082_subset hN))\n      _ \u2264 \u2a06 n, \u03bc (t n) := le_iSup (\u03bc \u2218 t) N ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s\u271d s\u2081 s\u2082 t\u271d : Set \u03b1 inst\u271d : Countable \u03b9 s : \u03b9 \u2192 Set \u03b1 val\u271d : Encodable \u03b9 t : \u2115 \u2192 Set \u03b1 ht : Function.extend Encodable.encode s \u22a5 = t hd : Directed (fun x x_1 => x \u2286 x_1) t this : \u2191\u2191\u03bc (\u2a06 i, s i) = \u2a06 n, Function.extend Encodable.encode (fun x => \u2191\u2191\u03bc (s x)) (fun x => 0) n \u22a2 \u2191\u2191\u03bc (\u22c3 i, s i) = \u2a06 i, \u2191\u2191\u03bc (s i) ** exact this.trans (iSup_extend_bot Encodable.encode_injective _) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 t : \u2115 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) t T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n) Td : \u2115 \u2192 Set \u03b1 := disjointed T hm : \u2200 (n : \u2115), MeasurableSet (Td n) \u22a2 \u2191\u2191\u03bc (\u22c3 n, T n) = \u2191\u2191\u03bc (\u22c3 n, Td n) ** rw [iUnion_disjointed] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 t : \u2115 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) t T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n) Td : \u2115 \u2192 Set \u03b1 := disjointed T hm : \u2200 (n : \u2115), MeasurableSet (Td n) I : Finset \u2115 \u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n) ** rcases hd.finset_le I with \u27e8N, hN\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t\u271d : Set \u03b1 t : \u2115 \u2192 Set \u03b1 hd : Directed (fun x x_1 => x \u2286 x_1) t T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n) Td : \u2115 \u2192 Set \u03b1 := disjointed T hm : \u2200 (n : \u2115), MeasurableSet (Td n) I : Finset \u2115 N : \u2115 hN : \u2200 (i : \u2115), i \u2208 I \u2192 t i \u2286 t N \u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 n, \u2191\u2191\u03bc (t n) ** calc\n  (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n    (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm\n  _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (measure_mono (iUnion\u2082_mono fun n _hn => disjointed_subset _ _))\n  _ = \u03bc (\u22c3 n \u2208 I, t n) := (measure_biUnion_toMeasurable I.countable_toSet _)\n  _ \u2264 \u03bc (t N) := (measure_mono (iUnion\u2082_subset hN))\n  _ \u2264 \u2a06 n, \u03bc (t n) := le_iSup (\u03bc \u2218 t) N ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.norm_integral_le_of_norm_le ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b : \u211d f g\u271d : \u211d \u2192 E \u03bc : Measure \u211d g : \u211d \u2192 \u211d h : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict \u03bc (\u0399 a b), \u2016f t\u2016 \u2264 g t hbound : IntervalIntegrable g \u03bc a b \u22a2 \u2016\u222b (t : \u211d) in a..b, f t \u2202\u03bc\u2016 \u2264 |\u222b (t : \u211d) in a..b, g t \u2202\u03bc| ** simp_rw [norm_intervalIntegral_eq, abs_intervalIntegral_eq,\n  abs_eq_self.mpr (integral_nonneg_of_ae <| h.mono fun _t ht => (norm_nonneg _).trans ht),\n  norm_integral_le_of_norm_le hbound.def h] ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.eval\u2082_mul ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 \u22a2 \u2200 {p : MvPolynomial \u03c3 R}, eval\u2082 f g (p * q) = eval\u2082 f g p * eval\u2082 f g q ** apply MvPolynomial.induction_on q ** case h_C R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 \u22a2 \u2200 (a : R) {p : MvPolynomial \u03c3 R}, eval\u2082 f g (p * \u2191C a) = eval\u2082 f g p * eval\u2082 f g (\u2191C a) ** simp [eval\u2082_C, eval\u2082_mul_C] ** case h_add R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 \u22a2 \u2200 (p q : MvPolynomial \u03c3 R),     (\u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * p) = eval\u2082 f g p_1 * eval\u2082 f g p) \u2192       (\u2200 {p : MvPolynomial \u03c3 R}, eval\u2082 f g (p * q) = eval\u2082 f g p * eval\u2082 f g q) \u2192         \u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * (p + q)) = eval\u2082 f g p_1 * eval\u2082 f g (p + q) ** simp (config := { contextual := true }) [mul_add, eval\u2082_add] ** case h_X R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f : R \u2192+* S\u2081 g : \u03c3 \u2192 S\u2081 \u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),     (\u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * p) = eval\u2082 f g p_1 * eval\u2082 f g p) \u2192       \u2200 {p_1 : MvPolynomial \u03c3 R}, eval\u2082 f g (p_1 * (p * X n)) = eval\u2082 f g p_1 * eval\u2082 f g (p * X n) ** simp (config := { contextual := true }) [X, eval\u2082_monomial, eval\u2082_mul_monomial, \u2190 mul_assoc] ** Qed",
        "informal": ""
    },
    {
        "formal": "List.cons_eq_append ** \u03b1\u271d : Type u_1 x : \u03b1\u271d c a b : List \u03b1\u271d \u22a2 x :: c = a ++ b \u2194 a = [] \u2227 b = x :: c \u2228 \u2203 a', a = x :: a' \u2227 c = a' ++ b ** rw [eq_comm, append_eq_cons] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.snorm'_eq_zero_of_ae_zero' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G hq0_ne : q \u2260 0 h\u03bc : \u03bc \u2260 0 f : \u03b1 \u2192 F hf_zero : f =\u1d50[\u03bc] 0 \u22a2 snorm' f q \u03bc = 0 ** rw [snorm'_congr_ae hf_zero, snorm'_zero' hq0_ne h\u03bc] ** Qed",
        "informal": ""
    },
    {
        "formal": "Prod.map_bijective ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : Nonempty \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b4 \u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g ** haveI := Nonempty.map f \u2039_\u203a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : Nonempty \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b4 this : Nonempty \u03b3 \u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g ** haveI := Nonempty.map g \u2039_\u203a ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : Nonempty \u03b1 inst\u271d : Nonempty \u03b2 f : \u03b1 \u2192 \u03b3 g : \u03b2 \u2192 \u03b4 this\u271d : Nonempty \u03b3 this : Nonempty \u03b4 \u22a2 Bijective (map f g) \u2194 Bijective f \u2227 Bijective g ** exact (map_injective.and map_surjective).trans (and_and_and_comm) ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.finSuccEquiv_eq ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 \u22a2 \u2191(finSuccEquiv R n) =     eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i ** ext i : 2 ** case hC.a R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 i : R \u22a2 \u2191(RingHom.comp (\u2191(finSuccEquiv R n)) C) i =     \u2191(RingHom.comp           (eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i) C)       i ** simp only [finSuccEquiv, optionEquivLeft_apply, aeval_C, AlgEquiv.coe_trans, RingHom.coe_coe,\n  coe_eval\u2082Hom, comp_apply, renameEquiv_apply, eval\u2082_C, RingHom.coe_comp, rename_C] ** case hC.a R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 i : R \u22a2 \u2191(algebraMap R (MvPolynomial (Fin n) R)[X]) i = \u2191Polynomial.C (\u2191C i) ** rfl ** case hX.a R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R n : \u2115 i : Fin (n + 1) n\u271d : \u2115 \u22a2 Polynomial.coeff (\u2191\u2191(finSuccEquiv R n) (X i)) n\u271d =     Polynomial.coeff       (\u2191(eval\u2082Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i) (X i))       n\u271d ** refine' Fin.cases _ _ i <;> simp [finSuccEquiv] ** Qed",
        "informal": ""
    },
    {
        "formal": "Nat.gcd_div ** m n k : Nat H1 : k \u2223 m H2 : k \u2223 n H0 : k = 0 \u22a2 gcd (m / k) (n / k) = gcd m n / k ** simp [H0] ** m n k : Nat H1 : k \u2223 m H2 : k \u2223 n H3 : k > 0 \u22a2 gcd (m / k) (n / k) = gcd m n / k ** apply Nat.eq_of_mul_eq_mul_right H3 ** m n k : Nat H1 : k \u2223 m H2 : k \u2223 n H3 : k > 0 \u22a2 gcd (m / k) (n / k) * k = gcd m n / k * k ** rw [Nat.div_mul_cancel (dvd_gcd H1 H2), \u2190 gcd_mul_right,\n    Nat.div_mul_cancel H1, Nat.div_mul_cancel H2] ** Qed",
        "informal": ""
    },
    {
        "formal": "measurable_iSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) \u22a2 Measurable fun b => \u2a06 i, f i b ** rcases isEmpty_or_nonempty \u03b9 with h\u03b9|h\u03b9 ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 \u22a2 Measurable fun b => \u2a06 i, f i b ** have A : MeasurableSet {b | BddAbove (range (fun i \u21a6 f i b))} :=\n  measurableSet_bddAbove_range hf ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} \u22a2 Measurable fun b => \u2a06 i, f i b ** have : Measurable (fun (_b : \u03b4) \u21a6 sSup (\u2205 : Set \u03b1)) := measurable_const ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} this : Measurable fun _b => sSup \u2205 \u22a2 Measurable fun b => \u2a06 i, f i b ** apply Measurable.isLUB_of_mem hf A _ _ this ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : IsEmpty \u03b9 \u22a2 Measurable fun b => \u2a06 i, f i b ** simp [iSup_of_empty'] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} this : Measurable fun _b => sSup \u2205 \u22a2 \u2200 (b : \u03b4), b \u2208 {b | BddAbove (range fun i => f i b)} \u2192 IsLUB {a | \u2203 i, f i b = a} (\u2a06 i, f i b) ** rintro b \u27e8c, hc\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} this : Measurable fun _b => sSup \u2205 b : \u03b4 c : \u03b1 hc : c \u2208 upperBounds (range fun i => f i b) \u22a2 IsLUB {a | \u2203 i, f i b = a} (\u2a06 i, f i b) ** apply isLUB_ciSup ** case intro.H \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} this : Measurable fun _b => sSup \u2205 b : \u03b4 c : \u03b1 hc : c \u2208 upperBounds (range fun i => f i b) \u22a2 BddAbove (range fun y => f y b) ** refine \u27e8c, ?_\u27e9 ** case intro.H \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} this : Measurable fun _b => sSup \u2205 b : \u03b4 c : \u03b1 hc : c \u2208 upperBounds (range fun i => f i b) \u22a2 c \u2208 upperBounds (range fun y => f y b) ** rintro d \u27e8i, rfl\u27e9 ** case intro.H.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} this : Measurable fun _b => sSup \u2205 b : \u03b4 c : \u03b1 hc : c \u2208 upperBounds (range fun i => f i b) i : \u03b9 \u22a2 (fun y => f y b) i \u2264 c ** exact hc (mem_range_self i) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} this : Measurable fun _b => sSup \u2205 \u22a2 EqOn (fun b => \u2a06 i, f i b) (fun _b => sSup \u2205) {b | BddAbove (range fun i => f i b)}\u1d9c ** intro b hb ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} this : Measurable fun _b => sSup \u2205 b : \u03b4 hb : b \u2208 {b | BddAbove (range fun i => f i b)}\u1d9c \u22a2 (fun b => \u2a06 i, f i b) b = (fun _b => sSup \u2205) b ** apply csSup_of_not_bddAbove ** case hs \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9\u00b9 : BorelSpace \u03b1 inst\u271d\u00b9\u2070 : TopologicalSpace \u03b2 inst\u271d\u2079 : MeasurableSpace \u03b2 inst\u271d\u2078 : BorelSpace \u03b2 inst\u271d\u2077 : TopologicalSpace \u03b3 inst\u271d\u2076 : MeasurableSpace \u03b3 inst\u271d\u2075 : BorelSpace \u03b3 inst\u271d\u2074 : MeasurableSpace \u03b4 inst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : SecondCountableTopology \u03b1 \u03b9 : Sort u_6 inst\u271d : Countable \u03b9 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 hf : \u2200 (i : \u03b9), Measurable (f i) h\u03b9 : Nonempty \u03b9 A : MeasurableSet {b | BddAbove (range fun i => f i b)} this : Measurable fun _b => sSup \u2205 b : \u03b4 hb : b \u2208 {b | BddAbove (range fun i => f i b)}\u1d9c \u22a2 \u00acBddAbove (range fun i => f i b) ** exact hb ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_Iic_sub_Iic ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d ha : IntegrableOn f (Iic a) hb : IntegrableOn f (Iic b) \u22a2 \u222b (x : \u211d) in Iic b, f x \u2202\u03bc - \u222b (x : \u211d) in Iic a, f x \u2202\u03bc = \u222b (x : \u211d) in a..b, f x \u2202\u03bc ** wlog hab : a \u2264 b generalizing a b ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a\u271d b\u271d c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d a b : \u211d ha : IntegrableOn f (Iic a) hb : IntegrableOn f (Iic b) hab : a \u2264 b \u22a2 \u222b (x : \u211d) in Iic b, f x \u2202\u03bc - \u222b (x : \u211d) in Iic a, f x \u2202\u03bc = \u222b (x : \u211d) in a..b, f x \u2202\u03bc ** rw [sub_eq_iff_eq_add', integral_of_le hab, \u2190 integral_union (Iic_disjoint_Ioc le_rfl),\n  Iic_union_Ioc_eq_Iic hab] ** case ht \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a\u271d b\u271d c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d a b : \u211d ha : IntegrableOn f (Iic a) hb : IntegrableOn f (Iic b) hab : a \u2264 b \u22a2 MeasurableSet (Ioc a b)  case hfs \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a\u271d b\u271d c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d a b : \u211d ha : IntegrableOn f (Iic a) hb : IntegrableOn f (Iic b) hab : a \u2264 b \u22a2 IntegrableOn (fun x => f x) (Iic a)  case hft \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a\u271d b\u271d c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d a b : \u211d ha : IntegrableOn f (Iic a) hb : IntegrableOn f (Iic b) hab : a \u2264 b \u22a2 IntegrableOn (fun x => f x) (Ioc a b) ** exacts [measurableSet_Ioc, ha, hb.mono_set fun _ => And.right] ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d ha : IntegrableOn f (Iic a) hb : IntegrableOn f (Iic b) this :   \u2200 {a b : \u211d},     IntegrableOn f (Iic a) \u2192       IntegrableOn f (Iic b) \u2192         a \u2264 b \u2192 \u222b (x : \u211d) in Iic b, f x \u2202\u03bc - \u222b (x : \u211d) in Iic a, f x \u2202\u03bc = \u222b (x : \u211d) in a..b, f x \u2202\u03bc hab : \u00aca \u2264 b \u22a2 \u222b (x : \u211d) in Iic b, f x \u2202\u03bc - \u222b (x : \u211d) in Iic a, f x \u2202\u03bc = \u222b (x : \u211d) in a..b, f x \u2202\u03bc ** rw [integral_symm, \u2190 this hb ha (le_of_not_le hab), neg_sub] ** Qed",
        "informal": ""
    },
    {
        "formal": "Int.shiftLeft_coe_nat ** m n : \u2115 \u22a2 \u2191m <<< \u2191n = \u2191(m <<< n) ** simp [instShiftLeftInt, HShiftLeft.hShiftLeft] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.L1.SimpleFunc.norm_integral_le_norm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup F' inst\u271d : NormedSpace \u211d F' f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2016integral f\u2016 \u2264 \u2016f\u2016 ** rw [integral, norm_eq_integral] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedAddCommGroup F m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2075 : NormedField \ud835\udd5c inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E F' : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup F' inst\u271d : NormedSpace \u211d F' f : { x // x \u2208 simpleFunc E 1 \u03bc } \u22a2 \u2016MeasureTheory.SimpleFunc.integral \u03bc (toSimpleFunc f)\u2016 \u2264     MeasureTheory.SimpleFunc.integral \u03bc (SimpleFunc.map norm (toSimpleFunc f)) ** exact (toSimpleFunc f).norm_integral_le_integral_norm (SimpleFunc.integrable f) ** Qed",
        "informal": ""
    },
    {
        "formal": "Part.ofOption_eq_get ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b1 : Type u_4 o : Option \u03b1 h\u2081 : (\u2191o).Dom h\u2082 : { Dom := Option.isSome o = true, get := Option.get o }.Dom \u22a2 get (\u2191o) h\u2081 = get { Dom := Option.isSome o = true, get := Option.get o } h\u2082 ** cases o ** case none \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b1 : Type u_4 h\u2081 : (\u2191Option.none).Dom h\u2082 : { Dom := Option.isSome Option.none = true, get := Option.get Option.none }.Dom \u22a2 get (\u2191Option.none) h\u2081 = get { Dom := Option.isSome Option.none = true, get := Option.get Option.none } h\u2082 ** simp at h\u2082 ** case some \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b1 : Type u_4 val\u271d : \u03b1 h\u2081 : (\u2191(Option.some val\u271d)).Dom h\u2082 : { Dom := Option.isSome (Option.some val\u271d) = true, get := Option.get (Option.some val\u271d) }.Dom \u22a2 get (\u2191(Option.some val\u271d)) h\u2081 =     get { Dom := Option.isSome (Option.some val\u271d) = true, get := Option.get (Option.some val\u271d) } h\u2082 ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "Set.image_affine_Icc' ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a : \u03b1 h : 0 < a b c d : \u03b1 \u22a2 (fun x => a * x + b) '' Icc c d = Icc (a * c + b) (a * d + b) ** suffices (fun x => x + b) '' ((fun x => a * x) '' Icc c d) = Icc (a * c + b) (a * d + b) by\n  rwa [Set.image_image] at this ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a : \u03b1 h : 0 < a b c d : \u03b1 \u22a2 (fun x => x + b) '' ((fun x => a * x) '' Icc c d) = Icc (a * c + b) (a * d + b) ** rw [image_mul_left_Icc' h, image_add_const_Icc] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a : \u03b1 h : 0 < a b c d : \u03b1 this : (fun x => x + b) '' ((fun x => a * x) '' Icc c d) = Icc (a * c + b) (a * d + b) \u22a2 (fun x => a * x + b) '' Icc c d = Icc (a * c + b) (a * d + b) ** rwa [Set.image_image] at this ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.expand_size ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 \u22a2 Buckets.size (expand sz buckets).buckets = Buckets.size buckets ** rw [expand, go] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) buckets.val.data) +       Buckets.size (Buckets.mk (Array.size buckets.val * 2)) =     Buckets.size buckets ** rw [Buckets.mk_size] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) buckets.val.data) + 0 = Buckets.size buckets ** simp [Buckets.size] ** case hs \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 \u22a2 \u2200 (j : Nat), j < 0 \u2192 List.getD buckets.val.data j AssocList.nil = AssocList.nil ** intro. ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil \u22a2 Buckets.size (expand.go i source target) =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target ** unfold expand.go ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil \u22a2 Buckets.size       (if h : i < Array.size source then         let idx := { val := i, isLt := h };         let es := Array.get source idx;         let source := Array.set source idx AssocList.nil;         let target := AssocList.foldl reinsertAux target es;         expand.go (i + 1) source target       else target) =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target ** split ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source \u22a2 Buckets.size       (let idx := { val := i, isLt := H };       let es := Array.get source idx;       let source := Array.set source idx AssocList.nil;       let target := AssocList.foldl reinsertAux target es;       expand.go (i + 1) source target) =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target ** refine (go (i+1) _ _ fun j hj => ?a).trans ?b <;> simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source j : Nat hj : j < i + 1 \u22a2 List.getD (List.set source.data i AssocList.nil) j AssocList.nil = AssocList.nil ** simp [List.getD_eq_get?, List.get?_set] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source j : Nat hj : j < i + 1 \u22a2 Option.getD (if i = j then Option.map (fun x => AssocList.nil) (List.get? source.data j) else List.get? source.data j)       AssocList.nil =     AssocList.nil ** split ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source j : Nat hj : j < i + 1 h\u271d : i = j \u22a2 Option.getD (Option.map (fun x => AssocList.nil) (List.get? source.data j)) AssocList.nil = AssocList.nil ** cases List.get? .. <;> rfl ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source j : Nat hj : j < i + 1 h\u271d : \u00aci = j \u22a2 Option.getD (List.get? source.data j) AssocList.nil = AssocList.nil ** next H => exact hs _ (Nat.lt_of_le_of_ne (Nat.le_of_lt_succ hj) (Ne.symm H)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H\u271d : i < Array.size source j : Nat hj : j < i + 1 H : \u00aci = j \u22a2 Option.getD (List.get? source.data j) AssocList.nil = AssocList.nil ** exact hs _ (Nat.lt_of_le_of_ne (Nat.le_of_lt_succ hj) (Ne.symm H)) ** case b \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.set source.data i AssocList.nil)) +       Buckets.size (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target ** case b =>\nrefine have \u27e8l\u2081, l\u2082, h\u2081, _, eq\u27e9 := List.exists_of_set' H; eq \u25b8 ?_\nsimp [h\u2081, Buckets.size_eq]\nrw [Nat.add_assoc, Nat.add_assoc, Nat.add_assoc]; congr 1\n(conv => rhs; rw [Nat.add_left_comm]); congr 1\nrw [\u2190 Array.getElem_eq_data_get]\nhave := @reinsertAux_size \u03b1 \u03b2 _; simp [Buckets.size] at this\ninduction source[i].toList generalizing target <;> simp [*, Nat.succ_add]; rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.set source.data i AssocList.nil)) +       Buckets.size (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target ** refine have \u27e8l\u2081, l\u2082, h\u2081, _, eq\u27e9 := List.exists_of_set' H; eq \u25b8 ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (l\u2081 ++ AssocList.nil :: l\u2082)) +       Buckets.size (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])) =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target ** simp [h\u2081, Buckets.size_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +         Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +       Nat.sum         (List.map (fun x => List.length (AssocList.toList x))           (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +         (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +           Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082)) +       Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data) ** rw [Nat.add_assoc, Nat.add_assoc, Nat.add_assoc] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +       (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +         Nat.sum           (List.map (fun x => List.length (AssocList.toList x))             (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data)) =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2081) +       (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +         (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +           Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data))) ** congr 1 ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +       Nat.sum         (List.map (fun x => List.length (AssocList.toList x))           (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =     List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +       (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +         Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)) ** (conv => rhs; rw [Nat.add_left_comm]) ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +       Nat.sum         (List.map (fun x => List.length (AssocList.toList x))           (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +       (List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +         Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)) ** congr 1 ** case e_a.e_a \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 \u22a2 Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =     List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +       Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data) ** rw [\u2190 Array.getElem_eq_data_get] ** case e_a.e_a \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 \u22a2 Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =     List.length (AssocList.toList source[i]) +       Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data) ** have := @reinsertAux_size \u03b1 \u03b2 _ ** case e_a.e_a \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 this : \u2200 (data : Buckets \u03b1 \u03b2) (a : \u03b1) (b : \u03b2), Buckets.size (reinsertAux data a b) = Nat.succ (Buckets.size data) \u22a2 Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =     List.length (AssocList.toList source[i]) +       Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data) ** simp [Buckets.size] at this ** case e_a.e_a \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 this :   \u2200 (data : Buckets \u03b1 \u03b2) (a : \u03b1) (b : \u03b2),     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (reinsertAux data a b).val.data) =       Nat.succ (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) data.val.data)) \u22a2 Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =     List.length (AssocList.toList source[i]) +       Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data) ** induction source[i].toList generalizing target <;> simp [*, Nat.succ_add] ** case e_a.e_a.cons \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 this :   \u2200 (data : Buckets \u03b1 \u03b2) (a : \u03b1) (b : \u03b2),     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (reinsertAux data a b).val.data) =       Nat.succ (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) data.val.data)) head\u271d : \u03b1 \u00d7 \u03b2 tail\u271d : List (\u03b1 \u00d7 \u03b2) tail_ih\u271d :   \u2200 (target : Buckets \u03b1 \u03b2),     Nat.sum         (List.map (fun x => List.length (AssocList.toList x))           (List.foldl (fun d x => reinsertAux d x.fst x.snd) target tail\u271d).val.data) =       List.length tail\u271d + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data) target : Buckets \u03b1 \u03b2 \u22a2 List.length tail\u271d + Nat.succ (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)) =     Nat.succ (List.length tail\u271d + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)) ** rfl ** case e_a \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : i < Array.size source l\u2081 l\u2082 : List (AssocList \u03b1 \u03b2) h\u2081 : source.data = l\u2081 ++ List.get source.data { val := i, isLt := H } :: l\u2082 left\u271d : List.length l\u2081 = i eq : List.set source.data i AssocList.nil = l\u2081 ++ AssocList.nil :: l\u2082 \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +       Nat.sum         (List.map (fun x => List.length (AssocList.toList x))           (List.foldl (fun d x => reinsertAux d x.fst x.snd) target (AssocList.toList source[i])).val.data) =     List.length (AssocList.toList (List.get source.data { val := i, isLt := H })) +       (Nat.sum (List.map (fun x => List.length (AssocList.toList x)) l\u2082) +         Nat.sum (List.map (fun x => List.length (AssocList.toList x)) target.val.data)) ** conv => rhs; rw [Nat.add_left_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : \u00aci < Array.size source \u22a2 Buckets.size target = Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) + Buckets.size target ** rw [(_ : Nat.sum _ = 0), Nat.zero_add] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : \u00aci < Array.size source \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) source.data) = 0 ** rw [\u2190 (_ : source.data.map (fun _ => .nil) = source.data)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : \u00aci < Array.size source \u22a2 List.map (fun x => AssocList.nil) source.data = source.data ** refine List.ext_get (by simp) fun j h\u2081 h\u2082 => ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : \u00aci < Array.size source j : Nat h\u2081 : j < List.length (List.map (fun x => AssocList.nil) source.data) h\u2082 : j < List.length source.data \u22a2 List.get (List.map (fun x => AssocList.nil) source.data) { val := j, isLt := h\u2081 } =     List.get source.data { val := j, isLt := h\u2082 } ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : \u00aci < Array.size source j : Nat h\u2081 : j < List.length (List.map (fun x => AssocList.nil) source.data) h\u2082 : j < List.length source.data \u22a2 AssocList.nil = List.get source.data { val := j, isLt := h\u2082 } ** have := (hs j (Nat.lt_of_lt_of_le h\u2082 (Nat.not_lt.1 H))).symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : \u00aci < Array.size source j : Nat h\u2081 : j < List.length (List.map (fun x => AssocList.nil) source.data) h\u2082 : j < List.length source.data this : AssocList.nil = List.getD source.data j AssocList.nil \u22a2 AssocList.nil = List.get source.data { val := j, isLt := h\u2082 } ** rwa [List.getD_eq_get?, List.get?_eq_get, Option.getD_some] at this ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : \u00aci < Array.size source \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) (List.map (fun x => AssocList.nil) source.data)) = 0 ** simp ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : \u00aci < Array.size source \u22a2 Nat.sum (List.map ((fun x => List.length (AssocList.toList x)) \u2218 fun x => AssocList.nil) source.data) = 0 ** induction source.data <;> simp [*] ** \u03b1 : Type u_1 \u03b2 : Type u_2 sz : Nat inst\u271d : Hashable \u03b1 buckets : Buckets \u03b1 \u03b2 i : Nat source : Array (AssocList \u03b1 \u03b2) target : Buckets \u03b1 \u03b2 hs : \u2200 (j : Nat), j < i \u2192 List.getD source.data j AssocList.nil = AssocList.nil H : \u00aci < Array.size source \u22a2 List.length (List.map (fun x => AssocList.nil) source.data) = List.length source.data ** simp ** Qed",
        "informal": ""
    },
    {
        "formal": "ENNReal.lintegral_Lp_add_le_aux ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** have hp_not_nonpos : \u00acp \u2264 0 := by simp [hpq.pos] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** have htop_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 := by\n  by_contra h\n  exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** have h0_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 := by\n  simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** suffices h :\n  1 \u2264\n    (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a : \u03b1, g a ^ p \u2202\u03bc) ^ (1 / p)) ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 \u22a2 1 \u2264     (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *       ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) ** have h :\n  (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) \u2264\n    ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a : \u03b1, g a ^ p \u2202\u03bc) ^ (1 / p)) *\n      (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) ^ (1 / q) :=\n  lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add hpq hf hf_top hg hg_top ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 h :   \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264     ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q) \u22a2 1 \u2264     (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *       ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) ** have h_one_div_q : 1 / q = 1 - 1 / p := by\n  nth_rw 2 [\u2190 hpq.inv_add_inv_conj]\n  ring ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 h :   \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264     ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q) h_one_div_q : 1 / q = 1 - 1 / p \u22a2 1 \u2264     (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *       ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) ** simp_rw [h_one_div_q, sub_eq_add_neg 1 (1 / p), ENNReal.rpow_add _ _ h_add_zero h_add_top,\n  rpow_one] at h ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 h_one_div_q : 1 / q = 1 - 1 / p h :   \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264     ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *       ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p))) \u22a2 1 \u2264     (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *       ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) ** conv_rhs at h => enter [2]; rw [mul_comm] ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 h_one_div_q : 1 / q = 1 - 1 / p h :   \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264     ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *       ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u22a2 1 \u2264     (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *       ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) ** conv_lhs at h => rw [\u2190 one_mul (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc)] ** case h \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 h_one_div_q : 1 / q = 1 - 1 / p h :   1 * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264     ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *       ((\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u22a2 1 \u2264     (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *       ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) ** rwa [\u2190 mul_assoc, ENNReal.mul_le_mul_right h_add_zero h_add_top, mul_comm] at h ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 \u22a2 \u00acp \u2264 0 ** simp [hpq.pos] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 ** by_contra h ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 h : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4 \u22a2 False ** exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h) ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 h : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4 \u22a2 0 \u2264 1 / p ** simp [hpq.nonneg] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 ** simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 h :   1 \u2264     (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *       ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) \u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) ** rwa [\u2190 mul_le_mul_left h0_rpow htop_rpow, \u2190 mul_assoc, \u2190 rpow_add _ _ h_add_zero h_add_top, \u2190\n  sub_eq_add_neg, _root_.sub_self, rpow_zero, one_mul, mul_one] at h ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 h :   \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264     ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q) \u22a2 1 / q = 1 - 1 / p ** nth_rw 2 [\u2190 hpq.inv_add_inv_conj] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p q : \u211d hpq : Real.IsConjugateExponent p q f g : \u03b1 \u2192 \u211d\u22650\u221e hf : AEMeasurable f hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 hg : AEMeasurable g hg_top : \u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc \u2260 \u22a4 h_add_zero : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 0 h_add_top : \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2260 \u22a4 hp_not_nonpos : \u00acp \u2264 0 htop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 h0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 h :   \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc \u2264     ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q) \u22a2 1 / q = 1 / p + 1 / q - 1 / p ** ring ** Qed",
        "informal": ""
    },
    {
        "formal": "ZMod.natAbs_min_of_le_div_two ** n\u271d a n : \u2115 x y : \u2124 he : \u2191x = \u2191y hl : Int.natAbs x \u2264 n / 2 \u22a2 Int.natAbs x \u2264 Int.natAbs y ** rw [int_cast_eq_int_cast_iff_dvd_sub] at he ** n\u271d a n : \u2115 x y : \u2124 he : \u2191n \u2223 y - x hl : Int.natAbs x \u2264 n / 2 \u22a2 Int.natAbs x \u2264 Int.natAbs y ** obtain \u27e8m, he\u27e9 := he ** case intro n\u271d a n : \u2115 x y : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 he : y - x = \u2191n * m \u22a2 Int.natAbs x \u2264 Int.natAbs y ** rw [sub_eq_iff_eq_add] at he ** case intro n\u271d a n : \u2115 x y : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 he : y = \u2191n * m + x \u22a2 Int.natAbs x \u2264 Int.natAbs y ** subst he ** case intro n\u271d a n : \u2115 x : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 \u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x) ** obtain rfl | hm := eq_or_ne m 0 ** case intro.inr n\u271d a n : \u2115 x : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 hm : m \u2260 0 \u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x) ** apply hl.trans ** case intro.inr n\u271d a n : \u2115 x : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 hm : m \u2260 0 \u22a2 n / 2 \u2264 Int.natAbs (\u2191n * m + x) ** rw [\u2190 add_le_add_iff_right x.natAbs] ** case intro.inr n\u271d a n : \u2115 x : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 hm : m \u2260 0 \u22a2 n / 2 + Int.natAbs x \u2264 Int.natAbs (\u2191n * m + x) + Int.natAbs x ** refine' le_trans (le_trans ((add_le_add_iff_left _).2 hl) _) (Int.natAbs_sub_le _ _) ** case intro.inr n\u271d a n : \u2115 x : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 hm : m \u2260 0 \u22a2 n / 2 + n / 2 \u2264 Int.natAbs (\u2191n * m + x - x) ** rw [add_sub_cancel, Int.natAbs_mul, Int.natAbs_ofNat] ** case intro.inr n\u271d a n : \u2115 x : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 hm : m \u2260 0 \u22a2 n / 2 + n / 2 \u2264 n * Int.natAbs m ** refine' le_trans _ (Nat.le_mul_of_pos_right <| Int.natAbs_pos.2 hm) ** case intro.inr n\u271d a n : \u2115 x : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 hm : m \u2260 0 \u22a2 n / 2 + n / 2 \u2264 n ** rw [\u2190 mul_two] ** case intro.inr n\u271d a n : \u2115 x : \u2124 hl : Int.natAbs x \u2264 n / 2 m : \u2124 hm : m \u2260 0 \u22a2 n / 2 * 2 \u2264 n ** apply Nat.div_mul_le_self ** case intro.inl n\u271d a n : \u2115 x : \u2124 hl : Int.natAbs x \u2264 n / 2 \u22a2 Int.natAbs x \u2264 Int.natAbs (\u2191n * 0 + x) ** rw [mul_zero, zero_add] ** Qed",
        "informal": ""
    },
    {
        "formal": "intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedSpace \u211d E f : \u211d \u2192 E c ca cb : E l l' la la' lb lb' : Filter \u211d lt : Filter \u03b9 a b z : \u211d u v ua ub va vb : \u03b9 \u2192 \u211d inst\u271d\u00b9 : FTCFilter a la la' inst\u271d : FTCFilter b lb lb' hab : IntervalIntegrable f volume a b hmeas : StronglyMeasurableAtFilter f la' hf : Tendsto f (la' \u2293 Measure.ae volume) (\ud835\udcdd c) hu : Tendsto u lt la hv : Tendsto v lt la \u22a2 (fun t => ((\u222b (x : \u211d) in v t..b, f x) - \u222b (x : \u211d) in u t..b, f x) + (v t - u t) \u2022 c) =o[lt] (v - u) ** simpa only [integral_const, smul_eq_mul, mul_one] using\n  measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left hab hmeas hf hu hv ** Qed",
        "informal": ""
    },
    {
        "formal": "ProbabilityTheory.strong_law_Lp ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) \u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0) ** have hmeas : \u2200 i, AEStronglyMeasurable (X i) \u2119 := fun i =>\n  (hident i).aestronglyMeasurable_iff.2 h\u2112p.1 ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 \u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0) ** have hint : Integrable (X 0) \u2119 := h\u2112p.integrable hp ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) \u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0) ** have havg : \u2200 (n : \u2115),\n    AEStronglyMeasurable (fun \u03c9 => (n : \u211d) \u207b\u00b9 \u2022 (\u2211 i in range n, X i \u03c9)) \u2119 := by\n  intro n\n  exact AEStronglyMeasurable.const_smul (aestronglyMeasurable_sum _ fun i _ => hmeas i) _ ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) havg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119 \u22a2 Tendsto (fun n => snorm (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 - \u222b (a : \u03a9), X 0 a) p \u2119) atTop (\ud835\udcdd 0) ** refine' tendsto_Lp_of_tendstoInMeasure _ hp hp' havg (mem\u2112p_const _) _\n  (tendstoInMeasure_of_tendsto_ae havg (strong_law_ae _ hint hindep hident)) ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) havg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119 \u22a2 UnifIntegrable (fun n \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) p \u2119 ** rw [(_ : (fun (n : \u2115) \u03c9 => (n : \u211d)\u207b\u00b9 \u2022 (\u2211 i in range n, X i \u03c9))\n          = fun (n : \u2115) => (n : \u211d)\u207b\u00b9 \u2022 (\u2211 i in range n, X i))] ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) \u22a2 \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119 ** intro n ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) n : \u2115 \u22a2 AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119 ** exact AEStronglyMeasurable.const_smul (aestronglyMeasurable_sum _ fun i _ => hmeas i) _ ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) havg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119 \u22a2 UnifIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i) p \u2119 ** apply UniformIntegrable.unifIntegrable ** case hf \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) havg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119 \u22a2 UniformIntegrable (fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i) p \u2119 ** apply uniformIntegrable_average hp ** case hf \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) havg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119 \u22a2 UniformIntegrable (fun i => X i) p \u2119 ** exact Mem\u2112p.uniformIntegrable_of_identDistrib hp hp' h\u2112p hident ** \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) havg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119 \u22a2 (fun n \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) = fun n => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i ** ext n \u03c9 ** case h.h \u03a9 : Type u_1 inst\u271d\u2076 : MeasureSpace \u03a9 inst\u271d\u2075 : IsProbabilityMeasure \u2119 E : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : MeasurableSpace E inst\u271d : BorelSpace E p : \u211d\u22650\u221e hp : 1 \u2264 p hp' : p \u2260 \u22a4 X : \u2115 \u2192 \u03a9 \u2192 E h\u2112p : Mem\u2112p (X 0) p hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : \u2200 (i : \u2115), IdentDistrib (X i) (X 0) hmeas : \u2200 (i : \u2115), AEStronglyMeasurable (X i) \u2119 hint : Integrable (X 0) havg : \u2200 (n : \u2115), AEStronglyMeasurable (fun \u03c9 => (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9) \u2119 n : \u2115 \u03c9 : \u03a9 \u22a2 (\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i \u03c9 = ((\u2191n)\u207b\u00b9 \u2022 \u2211 i in range n, X i) \u03c9 ** simp only [Pi.smul_apply, sum_apply] ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integrable_prod_iff ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 E : Type u_6 inst\u271d\u2076 : MeasurableSpace \u03b1 inst\u271d\u2075 : MeasurableSpace \u03b1' inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b2' inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc \u03bc' : Measure \u03b1 \u03bd \u03bd' : Measure \u03b2 \u03c4 : Measure \u03b3 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : SigmaFinite \u03bd f : \u03b1 \u00d7 \u03b2 \u2192 E h1f : AEStronglyMeasurable f (Measure.prod \u03bc \u03bd) \u22a2 Integrable f \u2194 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, Integrable fun y => f (x, y)) \u2227 Integrable fun x => \u222b (y : \u03b2), \u2016f (x, y)\u2016 \u2202\u03bd ** simp [Integrable, h1f, hasFiniteIntegral_prod_iff', h1f.norm.integral_prod_right',\n  h1f.prod_mk_left] ** Qed",
        "informal": ""
    },
    {
        "formal": "Std.HashMap.Imp.erase_size ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets \u22a2 (erase m k).size = Buckets.size (erase m k).buckets ** dsimp [erase, cond] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets \u22a2 (match         AssocList.contains k           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with       | true =>         { size := m.size - 1,           buckets :=             Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.erase k                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val) }       | false => m).size =     Buckets.size       (match           AssocList.contains k             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] with         | true =>           { size := m.size - 1,             buckets :=               Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val                 (AssocList.erase k                   m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])                 (_ :                   USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                     Array.size m.buckets.val) }         | false => m).buckets ** split ** case h_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool heq\u271d :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true \u22a2 { size := m.size - 1,         buckets :=           Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.erase k               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val) }.size =     Buckets.size       { size := m.size - 1,           buckets :=             Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.erase k                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val) }.buckets ** next H =>\nsimp [h, Buckets.size]\nrefine have \u27e8_, _, h\u2081, _, eq\u27e9 := Buckets.exists_of_update ..; eq \u25b8 ?_\nsimp [h, h\u2081, Buckets.size_eq]\nrw [(_ : List.length _ = _ + 1), Nat.add_right_comm]; {rfl}\nclear h\u2081 eq\nsimp [AssocList.contains_eq] at H\nhave \u27e8a, h\u2081, h\u2082\u27e9 := H\nrefine have \u27e8_, _, _, _, _, h, eq\u27e9 := List.exists_of_eraseP h\u2081 h\u2082; eq \u25b8 ?_\nsimp [h]; rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool H :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true \u22a2 { size := m.size - 1,         buckets :=           Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val             (AssocList.erase k               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])             (_ :               USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                 Array.size m.buckets.val) }.size =     Buckets.size       { size := m.size - 1,           buckets :=             Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.erase k                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val) }.buckets ** simp [h, Buckets.size] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool H :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =     Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val               (AssocList.erase k                 m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])               (_ :                 USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <                   Array.size m.buckets.val)).val.data) ** refine have \u27e8_, _, h\u2081, _, eq\u27e9 := Buckets.exists_of_update ..; eq \u25b8 ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool H :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true w\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2) h\u2081 :   m.buckets.val.data =     w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d left\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val eq :   (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val           (AssocList.erase k             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])           (_ :             USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <               Array.size m.buckets.val)).val.data =     w\u271d\u00b9 ++       AssocList.erase k           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::         w\u271d \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) m.buckets.val.data) - 1 =     Nat.sum       (List.map (fun x => List.length (AssocList.toList x))         (w\u271d\u00b9 ++           AssocList.erase k               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::             w\u271d)) ** simp [h, h\u2081, Buckets.size_eq] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool H :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true w\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2) h\u2081 :   m.buckets.val.data =     w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d left\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val eq :   (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val           (AssocList.erase k             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])           (_ :             USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <               Array.size m.buckets.val)).val.data =     w\u271d\u00b9 ++       AssocList.erase k           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::         w\u271d \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +         (List.length             (AssocList.toList               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) +           Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d)) -       1 =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +       (List.length           (List.eraseP (fun x => x.fst == k)             (AssocList.toList               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +         Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d)) ** rw [(_ : List.length _ = _ + 1), Nat.add_right_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool H :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true w\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2) h\u2081 :   m.buckets.val.data =     w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d left\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val eq :   (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val           (AssocList.erase k             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])           (_ :             USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <               Array.size m.buckets.val)).val.data =     w\u271d\u00b9 ++       AssocList.erase k           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::         w\u271d \u22a2 Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +         (?n + Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d) + 1) -       1 =     Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d\u00b9) +       (List.length           (List.eraseP (fun x => x.fst == k)             (AssocList.toList               m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +         Nat.sum (List.map (fun x => List.length (AssocList.toList x)) w\u271d))  case n \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool H :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true w\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2) h\u2081 :   m.buckets.val.data =     w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d left\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val eq :   (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val           (AssocList.erase k             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])           (_ :             USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <               Array.size m.buckets.val)).val.data =     w\u271d\u00b9 ++       AssocList.erase k           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::         w\u271d \u22a2 Nat  \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool H :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true w\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2) h\u2081 :   m.buckets.val.data =     w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d left\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val eq :   (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val           (AssocList.erase k             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])           (_ :             USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <               Array.size m.buckets.val)).val.data =     w\u271d\u00b9 ++       AssocList.erase k           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::         w\u271d \u22a2 List.length       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =     ?n + 1 ** {rfl} ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool H :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true w\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2) h\u2081 :   m.buckets.val.data =     w\u271d\u00b9 ++ m.buckets.val[(mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] :: w\u271d left\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val eq :   (Buckets.update m.buckets (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val           (AssocList.erase k             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])           (_ :             USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val <               Array.size m.buckets.val)).val.data =     w\u271d\u00b9 ++       AssocList.erase k           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] ::         w\u271d \u22a2 List.length       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =     List.length         (List.eraseP (fun x => x.fst == k)           (AssocList.toList             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +       1 ** clear h\u2081 eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool H :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     true w\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2) left\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val \u22a2 List.length       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =     List.length         (List.eraseP (fun x => x.fst == k)           (AssocList.toList             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +       1 ** simp [AssocList.contains_eq] at H ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool w\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2) left\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val H :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true \u22a2 List.length       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =     List.length         (List.eraseP (fun x => x.fst == k)           (AssocList.toList             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +       1 ** have \u27e8a, h\u2081, h\u2082\u27e9 := H ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool w\u271d\u00b9 w\u271d : List (AssocList \u03b1 \u03b2) left\u271d : List.length w\u271d\u00b9 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val H :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true a : \u03b1 \u00d7 \u03b2 h\u2081 :   a \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] h\u2082 : (a.fst == k) = true \u22a2 List.length       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =     List.length         (List.eraseP (fun x => x.fst == k)           (AssocList.toList             m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val])) +       1 ** refine have \u27e8_, _, _, _, _, h, eq\u27e9 := List.exists_of_eraseP h\u2081 h\u2082; eq \u25b8 ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h\u271d : m.size = Buckets.size m.buckets c\u271d : Bool w\u271d\u2074 w\u271d\u00b3 : List (AssocList \u03b1 \u03b2) left\u271d\u00b2 : List.length w\u271d\u2074 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val H :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true a : \u03b1 \u00d7 \u03b2 h\u2081 :   a \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] h\u2082 : (a.fst == k) = true w\u271d\u00b2 : \u03b1 \u00d7 \u03b2 w\u271d\u00b9 w\u271d : List (\u03b1 \u00d7 \u03b2) left\u271d\u00b9 : \u2200 (b : \u03b1 \u00d7 \u03b2), b \u2208 w\u271d\u00b9 \u2192 \u00ac(b.fst == k) = true left\u271d : (w\u271d\u00b2.fst == k) = true h :   AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     w\u271d\u00b9 ++ w\u271d\u00b2 :: w\u271d eq :   List.eraseP (fun x => x.fst == k)       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =     w\u271d\u00b9 ++ w\u271d \u22a2 List.length       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =     List.length (w\u271d\u00b9 ++ w\u271d) + 1 ** simp [h] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h\u271d : m.size = Buckets.size m.buckets c\u271d : Bool w\u271d\u2074 w\u271d\u00b3 : List (AssocList \u03b1 \u03b2) left\u271d\u00b2 : List.length w\u271d\u2074 = USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val H :   \u2203 x,     x \u2208         AssocList.toList           m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] \u2227       (x.fst == k) = true a : \u03b1 \u00d7 \u03b2 h\u2081 :   a \u2208     AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] h\u2082 : (a.fst == k) = true w\u271d\u00b2 : \u03b1 \u00d7 \u03b2 w\u271d\u00b9 w\u271d : List (\u03b1 \u00d7 \u03b2) left\u271d\u00b9 : \u2200 (b : \u03b1 \u00d7 \u03b2), b \u2208 w\u271d\u00b9 \u2192 \u00ac(b.fst == k) = true left\u271d : (w\u271d\u00b2.fst == k) = true h :   AssocList.toList m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     w\u271d\u00b9 ++ w\u271d\u00b2 :: w\u271d eq :   List.eraseP (fun x => x.fst == k)       (AssocList.toList         m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val]) =     w\u271d\u00b9 ++ w\u271d \u22a2 List.length w\u271d\u00b9 + Nat.succ (List.length w\u271d) = List.length w\u271d\u00b9 + List.length w\u271d + 1 ** rfl ** case h_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : BEq \u03b1 inst\u271d : Hashable \u03b1 m : Imp \u03b1 \u03b2 k : \u03b1 h : m.size = Buckets.size m.buckets c\u271d : Bool heq\u271d :   AssocList.contains k       m.buckets.val[USize.toNat (mkIdx (_ : 0 < Array.size m.buckets.val) (UInt64.toUSize (hash k))).val] =     false \u22a2 m.size = Buckets.size m.buckets ** exact h ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable ** E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i) ** rcases em (\u2203 i, a i = b i) with (\u27e8i, hi\u27e9 | hne) ** case inl.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) i : Fin (n + 1) hi : a i = b i \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i) ** rw [volume_pi, \u2190 set_integral_congr_set_ae Measure.univ_pi_Ioc_ae_eq_Icc] ** case inl.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) i : Fin (n + 1) hi : a i = b i \u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),       \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i) ** have hi' : Ioc (a i) (b i) = \u2205 := Ioc_eq_empty hi.not_lt ** case inl.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) i : Fin (n + 1) hi : a i = b i hi' : Set.Ioc (a i) (b i) = \u2205 \u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),       \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i) ** have : (pi Set.univ fun j => Ioc (a j) (b j)) = \u2205 := univ_pi_eq_empty hi' ** case inl.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) i : Fin (n + 1) hi : a i = b i hi' : Set.Ioc (a i) (b i) = \u2205 this : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205 \u22a2 (\u222b (x : Fin (n + 1) \u2192 \u211d) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),       \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i \u2202Measure.pi fun x => volume) =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i) ** rw [this, integral_empty, sum_eq_zero] ** case inl.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) i : Fin (n + 1) hi : a i = b i hi' : Set.Ioc (a i) (b i) = \u2205 this : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205 \u22a2 \u2200 (x : Fin (n + 1)),     x \u2208 Finset.univ \u2192       (\u222b (x_1 : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove x) (b \u2218 Fin.succAbove x), f (Fin.insertNth x (b x) x_1) x) -           \u222b (x_1 : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove x) (b \u2218 Fin.succAbove x), f (Fin.insertNth x (a x) x_1) x =         0 ** rintro j - ** case inl.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) i : Fin (n + 1) hi : a i = b i hi' : Set.Ioc (a i) (b i) = \u2205 this : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205 j : Fin (n + 1) \u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -       \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =     0 ** rcases eq_or_ne i j with (rfl | hne) ** case inl.intro.inl E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) i : Fin (n + 1) hi : a i = b i hi' : Set.Ioc (a i) (b i) = \u2205 this : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205 \u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -       \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i =     0 ** simp [hi] ** case inl.intro.inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) i : Fin (n + 1) hi : a i = b i hi' : Set.Ioc (a i) (b i) = \u2205 this : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205 j : Fin (n + 1) hne : i \u2260 j \u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -       \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =     0 ** rcases Fin.exists_succAbove_eq hne with \u27e8i, rfl\u27e9 ** case inl.intro.inr.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) this : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205 j : Fin (n + 1) i : Fin n hi : a (Fin.succAbove j i) = b (Fin.succAbove j i) hi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205 hne : Fin.succAbove j i \u2260 j \u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -       \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =     0 ** have : Icc (a \u2218 j.succAbove) (b \u2218 j.succAbove) =\u1d50[volume] (\u2205 : Set \u211d\u207f) ** case inl.intro.inr.intro E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) this\u271d : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205 j : Fin (n + 1) i : Fin n hi : a (Fin.succAbove j i) = b (Fin.succAbove j i) hi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205 hne : Fin.succAbove j i \u2260 j this : Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j) =\u1d50[volume] \u2205 \u22a2 (\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (b j) x) j) -       \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j), f (Fin.insertNth j (a j) x) j =     0 ** rw [set_integral_congr_set_ae this, set_integral_congr_set_ae this, integral_empty,\n  integral_empty, sub_self] ** case this E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) this : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205 j : Fin (n + 1) i : Fin n hi : a (Fin.succAbove j i) = b (Fin.succAbove j i) hi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205 hne : Fin.succAbove j i \u2260 j \u22a2 Set.Icc (a \u2218 Fin.succAbove j) (b \u2218 Fin.succAbove j) =\u1d50[volume] \u2205 ** rw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)] ** case this E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) this : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = \u2205 j : Fin (n + 1) i : Fin n hi : a (Fin.succAbove j i) = b (Fin.succAbove j i) hi' : Set.Ioc (a (Fin.succAbove j i)) (b (Fin.succAbove j i)) = \u2205 hne : Fin.succAbove j i \u2260 j \u22a2 ENNReal.ofReal ((b \u2218 Fin.succAbove j) i - (a \u2218 Fin.succAbove j) i) = 0 ** simp [hi] ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) hne : \u00ac\u2203 i, a i = b i \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i) ** have hlt : \u2200 i, a i < b i := fun i => (hle i).lt_of_ne fun hi => hne \u27e8i, hi\u27e9 ** case inr E : Type u inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E n : \u2115 a b : Fin (n + 1) \u2192 \u211d hle : a \u2264 b f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] Fin (n + 1) \u2192 E s : Set (Fin (n + 1) \u2192 \u211d) hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : \u2200 (x : Fin (n + 1) \u2192 \u211d), x \u2208 (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s \u2192 HasFDerivAt f (f' x) x Hi : IntegrableOn (fun x => \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i) (Set.Icc a b) hne : \u00ac\u2203 i, a i = b i hlt : \u2200 (i : Fin (n + 1)), a i < b i \u22a2 \u222b (x : Fin (n + 1) \u2192 \u211d) in Set.Icc a b, \u2211 i : Fin (n + 1), \u2191(f' x) (e i) i =     \u2211 i : Fin (n + 1),       ((\u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (b i) x) i) -         \u222b (x : Fin n \u2192 \u211d) in Set.Icc (a \u2218 Fin.succAbove i) (b \u2218 Fin.succAbove i), f (Fin.insertNth i (a i) x) i) ** exact integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2082 \u27e8a, b, hlt\u27e9 f f' s hs Hc\n  Hd Hi ** Qed",
        "informal": ""
    },
    {
        "formal": "MvPolynomial.eval\u2082_eq' ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f\u271d : R \u2192+* S\u2081 g\u271d : \u03c3 \u2192 S\u2081 inst\u271d : Fintype \u03c3 g : R \u2192+* S\u2081 X : \u03c3 \u2192 S\u2081 f : MvPolynomial \u03c3 R \u22a2 eval\u2082 g X f = \u2211 d in support f, \u2191g (coeff d f) * \u220f i : \u03c3, X i ^ \u2191d i ** simp only [eval\u2082_eq, \u2190 Finsupp.prod_pow] ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R f\u271d : R \u2192+* S\u2081 g\u271d : \u03c3 \u2192 S\u2081 inst\u271d : Fintype \u03c3 g : R \u2192+* S\u2081 X : \u03c3 \u2192 S\u2081 f : MvPolynomial \u03c3 R \u22a2 \u2211 x in support f, \u2191g (coeff x f) * \u220f x_1 in x.support, X x_1 ^ \u2191x x_1 =     \u2211 x in support f, \u2191g (coeff x f) * Finsupp.prod x fun a b => X a ^ b ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "List.get_of_append ** \u03b1 : Type u_1 l\u2081 : List \u03b1 a : \u03b1 l\u2082 : List \u03b1 n : Nat l : List \u03b1 eq : l = l\u2081 ++ a :: l\u2082 h : length l\u2081 = n \u22a2 some (get l { val := n, isLt := (_ : n < length l) }) = some a ** rw [\u2190 get?_eq_get, eq, get?_append_right (h \u25b8 Nat.le_refl _), h, Nat.sub_self] ** \u03b1 : Type u_1 l\u2081 : List \u03b1 a : \u03b1 l\u2082 : List \u03b1 n : Nat l : List \u03b1 eq : l = l\u2081 ++ a :: l\u2082 h : length l\u2081 = n \u22a2 get? (a :: l\u2082) 0 = some a ** rfl ** Qed",
        "informal": ""
    },
    {
        "formal": "MeasureTheory.SimpleFunc.simpleFunc_bot ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1\u271d \u03b1 : Type u_5 f : \u03b1 \u2192\u209b \u03b2 inst\u271d : Nonempty \u03b2 \u22a2 \u2203 c, \u2200 (x : \u03b1), \u2191f x = c ** have hf_meas := @SimpleFunc.measurableSet_fiber \u03b1 _ \u22a5 f ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1\u271d \u03b1 : Type u_5 f : \u03b1 \u2192\u209b \u03b2 inst\u271d : Nonempty \u03b2 hf_meas : \u2200 (x : \u03b2), MeasurableSet (\u2191f \u207b\u00b9' {x}) \u22a2 \u2203 c, \u2200 (x : \u03b1), \u2191f x = c ** simp_rw [MeasurableSpace.measurableSet_bot_iff] at hf_meas ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1\u271d \u03b1 : Type u_5 f : \u03b1 \u2192\u209b \u03b2 inst\u271d : Nonempty \u03b2 hf_meas : \u2200 (x : \u03b2), \u2191f \u207b\u00b9' {x} = \u2205 \u2228 \u2191f \u207b\u00b9' {x} = univ \u22a2 \u2203 c, \u2200 (x : \u03b1), \u2191f x = c ** exact (exists_eq_const_of_preimage_singleton hf_meas).imp fun c hc \u21a6 congr_fun hc ** Qed",
        "informal": ""
    },
    {
        "formal": "Finset.inf_sdiff_right ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : BooleanAlgebra \u03b1 s : Finset \u03b9 hs : Finset.Nonempty s f : \u03b9 \u2192 \u03b1 a : \u03b1 \u22a2 (inf s fun b => f b \\ a) = inf s f \\ a ** induction' hs using Finset.Nonempty.cons_induction with b b t _ _ h ** case h\u2080 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : BooleanAlgebra \u03b1 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 a : \u03b1 b : \u03b9 \u22a2 (inf {b} fun b => f b \\ a) = inf {b} f \\ a ** rw [inf_singleton, inf_singleton] ** case h\u2081 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : BooleanAlgebra \u03b1 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 a : \u03b1 b : \u03b9 t : Finset \u03b9 h\u271d : \u00acb \u2208 t hs\u271d : Finset.Nonempty t h : (inf t fun b => f b \\ a) = inf t f \\ a \u22a2 (inf (cons b t h\u271d) fun b => f b \\ a) = inf (cons b t h\u271d) f \\ a ** rw [inf_cons, inf_cons, h, inf_sdiff] ** Qed",
        "informal": ""
    }
]